Research ArticleInformation Spreading on Weighted Multiplex Social Network

Xuzhen Zhu 1 Jinming Ma1 Xin Su 2 Hui Tian 1 Wei Wang 3 and Shimin Cai 4

1State Key Laboratory of Networking and Switching Technology Beijing University of Posts and TelecommunicationsBeijing 100876 China2School of Economics and Management Beijing University of Posts and Telecommunications Beijing 100876 China3Cybersecurity Research Institute Sichuan University Chengdu 610065 China4Web Sciences Center School of Computer Science and Engineering University of Electronic Science and Technology of ChinaChengdu 611731 China

Correspondence should be addressed to Wei Wang wwzqbxhotmailcom

Received 29 July 2019 Revised 15 October 2019 Accepted 28 October 2019 Published 11 November 2019

Academic Editor Eric Campos-Canton

Copyright copy 2019 Xuzhen Zhu et al shyis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Information spreading on multiplex networks has been investigated widely For multiplex networks the relations of each layerpossess dierent extents of intimacy which can be described as weighted multiplex networks Nevertheless the eect of weightedmultiplex network structures on information spreading has not been analyzed comprehensively We herein propose an in-formation spreading model on a weighted multiplex network shyen we develop an edge-weight-based compartmental theory todescribe the spreading dynamics We discover that under any adoption threshold of two subnetworks reducing weight dis-tribution heterogeneity does not alter the growth pattern of the nal adoption size versus information transmission probabilitywhile accelerating information spreading For xed weight distribution the growth pattern changes with the heterogeneous ofdegree distributionshyere is a critical initial seed size below which no global information outbreak can occur Extensive numericalsimulations arm that the theoretical predictions agree well with the numerical results

1 Introduction

shye Internet [1 2] web 20 [3] and intelligent mobileterminals [4] have completely transformed peoplersquos dailylives these infrastructures have assisted people in con-structing virtual social networks [5ndash8] which map peoplersquosconnections in real life into virtual cyberspaces and extendpeoplersquos social scope signicantly not only with school-mates colleagues and family members but also withstrangers in dierent countries Previously various types ofsocial networks such as Facebook [9] Twitter [10] Weibo[11] and Wechat [12] have been used Social networks areimportant in information transmission between users [13]such as commodity recommendation [14] news spreading[15 16] advertisement diusion [17 18] health behavioradoption [19] discovery of new friends [20] and socialinnovation propagation [21] In addition understanding themechanism of information spreading in social networks canfacilitate in the design of antivirus strategies [22ndash24] control

rumors [25] avoid economic risks [26 27] restrain socialunrest [28] etc Because of important values in economicactivities and engineering applications social informationspreading has attracted signicant attention [28ndash32]

Based on complex network theory [33 34] regarding auser as a node and a connective relation as an edge [35]researchers have mined various underlying factors to in-vestigate the information spreading mechanism and unveilthe fundamental laws Wide-ranging factors have been in-vestigated such as the initial seed size [36] clustering co-ecient [37] community structure [38] temporal network[39] role of synergy [40 41] heterogeneous adoptionthresholds [42] and limited imitation [43ndash45] Additionallyinformation spreading on a multiplex network [46ndash48] hasaroused the interest of researchers who perform numerousinvestigations about communication channel alternationson multiplex networks [49] heterogeneous behavioraladoption on multiplex networks [16] opinion competitionon multiplex networks [50] eect of multiple social

HindawiComplexityVolume 2019 Article ID 5920187 15 pageshttpsdoiorg10115520195920187

networks on user awareness [51] social support for sup-pressing epidemics onmultiplex networks [52] the interplayof social influence in multiplex networks [53] replicatordynamics on multiplex networks [54] etc

Moreover in social networks because of multipleconfirmations of the credibility and legitimacy of in-formation information spreading exhibits the effect of socialreinforcement [55 56] Repeated transmission of the sameinformation from one neighbor cannot enhance the credi-bility nor facilitate adoption-erefore social reinforcementderived from the memory of nonredundant information hasbeen investigated [57] which should be considered in in-formation spreading Apart from the abovementioned fac-tors edges represent the connective relations between usersand are always modeled in a binary state [58] In real socialnetworks different types of individuals such as colleaguesschoolmates family members and strangers may have in-timate or distant relations Even for the same type of in-dividuals they may exhibit different extents of intimacy-erefore similar to epidemic spreading on a weightednetwork [59] the relations in a social network should bereasonably modeled as weighted edges [60 61] Noticeablycompared with a one-layered complex network research oninformation spreading in multiplex networks is more com-plicated and challenging Apart from the intraheterogeneityof weights between edges in the same layer interheterogeneityof weight distributions between two layers exists as wellUnfortunately studies performed hitherto lack profoundanalysis regarding information spreading on weighted mul-tiplex networks with social reinforcement derived from thememory of nonredundant information

Herein in terms of social reinforcement effect generatedfrom the memory of nonredundant information and het-erogeneous weight distributions in two subnetworks wepropose an information spreading model on a weightedmultiplex network To further understand the intrinsicmechanism of information spreading and quantitativelypredict the effects of spreading in time evolution and finalsteady state we develop an edge-weight-based compart-mental theory Extensive simulations based on our proposednumerical method suggest that theoretical predictions agreewell with numerical simulations Combining the theoreticalpredictions and numerical simulations we discovered thatreducing weight distribution heterogeneity does not alter thegrowth pattern of the final adoption size but can accelerateinformation spreading and promote the outbreak of in-formation spreading Furthermore we discovered that acritical seed size determines the global information out-break above which reducing the weight distribution het-erogeneity materially affects the facilitation of informationspreading Additionally reducing the degree distributionheterogeneity can alter the growth pattern of the finaladoption size R(infin)

-e remainder of the paper is organized as follows InSection 2 we describe the information spreading model on aweighted multiplex network In Section 3 we propose anedge-weight-based compartmental theory to analyze in-formation spreading on a weighted multiplex network InSection 4 the numerical simulation method and parameter

settings are introduced In Section 5 we compare the the-oretical predictions with the numerical simulation resultsand unveil the effect mechanism of the weighted multiplexnetwork structure on information spreading In Section 6conclusions are presented

2 Model Descriptions

For investigating information spreading on a weightedmultiplex social network we constructed an informationspreading model based on an N-node two-layer weightedmultiplex network in which layers A and B represent twodifferent independent social networks (illustrated in Fig-ure 1) Correlated nodes in two layers indicate that they arethe same individual and edges between nodes in eachsubnetwork represent interindividual social relations Toeliminate the unnecessary intradegree-degree correlationthe uncorrelated configuration model [62] was manipulatedto construct the multiplex network following two in-dependent degree distributions PA(kA

i ) and PB(kBi ) in

which kAi and kB

i indicate the degrees of node i in layerA andB respectively -e degree vector of node i is set aski

rarr (kA

i kBi ) In this paper we assume that there is no

correlation between the degree and weight distributions andthe joint degree distribution is P(ki

rarr) PA(kA

i )PB(kBi ) Self-

loop and multiple edges are forbidden Regarding socialreinforcement we assume that each individual has differentadoption willingness in different social subnetworks Con-sequently node i representing an arbitrary individual hasadoption thresholds TA and TB in layers A and B re-spectively and a smaller adoption threshold suggests moreconsiderable information adoption willingness Moreoverthe intimate relations between individuals typically holddifferent strengths in the same social subnetwork and in twodistinctive subnetworks -erefore we further construct theweighted multiplex network with two dependent edge-weight distributions gA(w) and gB(w) Subsequently if anedge with weight wX

ij exists between nodes i and j in the layerX isin A B individual i in the subnetwork X transmits theinformation to j with probability

λX wXij1113872 1113873 1 minus 1 minus βX( 1113857

wXij (1)

where βX is the unit transmission probability in the sub-network X isin A B Given βX enlarging wX

ij can mono-tonically increase λX(wX

ij) which coincides with the positivecorrelation between relationship weight and transmissionpreference

In real social networks individuals always trustfullyadopt information after receiving a certain number of itfrom distinctive neighbors owing to social reinforcementoriginating from the memory of nonredundant informationtransmission [57] Accordingly we applied the generalizedsusceptible-adopted-recovered (SAR) model to describe thestate of an individual at each time step in the informationspreading process At the same time step an individual canonly remain in one state S-state (susceptible) A-state(adopted) or R-state (recovered) -e S-state implies that an

2 Complexity

individual has not adopted the information -e A-stateindicates that an individual has adopted the information andcan transmit the information to neighbors -e R-statemeans that an individual has lost interest in the informationand stops diffusion

Furthermore the detailed information spreading processis as follows Initially randomly selecting a fraction ρ0 ofnodes as initial seeds (in the A-state) At each time step eachA-state node i attempts to transmit the information to itsS-state neighbor j simultaneously through an edge of weightwA

ij with probability λA(wAij) in layer A and through an edge

of weight wBij with probability λB(wB

ij) in layer B according toequation (1) Because of nonredundant information trans-mission once an edge between nodes i and j in layer A or Bsuccessfully delivers the information from i to j such an edgeis not allowed for repeated information delivery An adopted

node however can perform many attempts until it turnsinto the R-state If the S-state node j in the layer X isin A B

successfully receives one piece of information from theA-state neighbor node j its accumulative number mX

j ofinformation pieces in the layer X isin A B will increase byone ie mX

j ⟶ mXj + 1 -en node j will compare the

accumulative number mXj with its adoption threshold TX

In reality for confirming the legitimacy and credibility of theinformation an individual can adopt the information only ifheshe corroborates with the information in both sub-networks -erefore the susceptible node j can enter theA-state when mA

j geTA and mBj geTB co-occur in both layers

A and B otherwise it remains in the S-state Regardingsocial reinforcement because the accumulative number ofinformation pieces determines the information adoptionthe information spreading process is characterized by the

1

2

3

4

5

1

2

3

4

5

Susceptible

Adopted

Recovered

A

B

(a)

1

2

3

4

5

1

2

3

4

5

Susceptible

Adopted

Recovered

A

B

(b)

1

1

2

2

3

3

4

4

5

5

Susceptible

Adopted

Recovered

A

B

(c)

1

2

3

4

5

1

2

3

4

5

Susceptible

Adopted

Recovered

A

B

(d)

Figure 1 Illustration of social information spreading with transformation of three states (susceptible adopted and recovered) on aweighted multiplex network Two different independent social networks A and B comprising degrees of intimate relations represent twolayers of the weighted multiplex network in which different degrees of edge-widths denote different edge-weights

Complexity 3

non-Markovian effect At the same time step after informationtransmission the A-state node j will lose interest in the in-formation and enter the R-state with probability γ ie it willstop diffusing information in both layers A and B Once thenode changes into the R-state it will retreat from furtherinformation spreading Finally the disappearance of all A-statenodes signals the termination of the spreading dynamics

3 Edge-Weight-Based Compartmental Theory

To include the social reinforcement effect the mean-filedtheory [63] and an advanced approach [64] can be usedInspired by References [57 65 66] an edge-weight-basedcompartmental theory is conceived to analyze the spreadingprocess with theoretically strong correlation among thethree states of nodes in a multiplex the network featured bystrong weight heterogeneity -e theory assumes that theinformation spreading is on large sparse uncorrelated andlocal tree-like weighted multiplex networks

In the edge-weight-based compartmental theory a nodein the cavity state [67] implies that it can only receive in-formation from but cannot transmit information to itsneighbors We denote θX(t)(X isin A B ) as the probabilitythat the information has not been passed to the S-stateneighbors alongside a randomly selected edge in layer X bytime t Furthermore in a weight multiplex network wedenote θX

w(t) as the probability that a node has not beeninformed through a randomly selected edge with weight w inlayer X by time t Following weight distribution gA(w) inlayer A and gB(w) in layer B we can obtain

θA(t) 1113944w

gA(w)θAw(t) (2)

θB(t) 1113944w

gB(w)θBw(t) (3)

An arbitrary S-state node i with degree vectorki

rarr (kA

i kBi ) receives mA and mB pieces of information by

time t with the probabilities

ϕAmA

kAi t1113872 1113873

kAi

mA

1113888 1113889θA(t)kA

i minus mA 1 minus θA(t)1113858 1113859mA

ϕBmB

kBi t1113872 1113873

kBi

mB

1113888 1113889θB(t)kB

i minus mB 1 minus θB(t)1113858 1113859mB

(4)

respectivelyAs mentioned in Section 2 an S-state node with degree

vector krarr

(kA kB) can adopt the information only if thecumulative numbers mA geTA and mB geTB occur simulta-neously inversely the S-state node having received mA andmB pieces of information by time t will remain in the S-statewith probability

s( krarr

t) 1 minus ρ0( 1113857 1 minus 1 minus 1113944

TAminus 1

mA0ϕA

mAk

A t1113872 1113873⎛⎝ ⎞⎠⎡⎢⎢⎣

times 1 minus 1113944

TBminus 1

mB0ϕB

mBk

B t1113872 1113873⎛⎝ ⎞⎠⎤⎥⎥⎦

(5)

We define the probability that a susceptible node withany degree in the layer X isin A B remaining in the S-state ifit receives mX pieces of information by time t as

ηX(t) 1113944kX

PX kX

1113872 1113873 1113944

TXminus 1

mX0ϕX

mXk

X t1113872 1113873 (6)

Furthermore the fraction of nodes in the S-state at time tcan be expressed as

S(t) 1113944

krarr

P( krarr

)s( krarr

t)

1 minus ρ0( 1113857 ηA + ηB minus ηAηB( 1113857

(7)

If we wish to obtain S(t) we should solve θX(t) whichdepends on θX

w(t) in a weighted network -erefore wefocus on θX

w(t) at first According to [57] in layer X theneighbor connected by an edge with weight w to a node inthe cavity state can be in the S-state A-state or R-statetherefore θX

w(t)(X isin A B ) comprises the following threeparts

θXw(t) ξX

Sw(t) + ξXAw(t) + ξX

Rw(t) (8)

where ξXSw(t) ξX

Aw(t) and ξXRw(t) indicate the probabilities

that a neighbor connected by an edge with weight w to anode in the cavity state remains in the S- A- and R-statesrespectively and has not transmitted information bytime t

If neighbor jwith degree vector kj

rarr (kA

j kBj ) is linked by

an edge of weight w to the node i in the cavity state in layerA(B) then neighbor j can only receive information fromother kA

j minus 1(kBj minus 1) edges in layer A(B) -erefore

neighbor j receives nA pieces of information in layer A withprobability ϕnA

(kAj minus 1 t) and with no constraint in layer B

receives nB pieces of information in layer B with probabilityϕnV

(kBj t) Subsequently neighbor j remains in the S-state in

layer A with probability

ΘA kj

rarr t1113874 1113875 1 minus 1 minus 1113944

TAminus 1

nA0ϕnA

kAj minus 1 t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

times 1 minus 1113944

TBminus 1

nB0ϕnB

kBj t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

(9)

Analogously neighbor j is connected with node i by anedge with weight w in layer B and remains in the S-state withprobability

ΘB kj

rarr t1113874 1113875 1 minus 1 minus 1113944

TA minus 1

nA0ϕnA

kAj t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

times 1 minus 1113944

TBminus 1

nB0ϕnB

kBj minus 1 t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

(10)

Accordingly given the joint degree distribution p( krarr

) inlayer X an edge with weight w connects to an S-stateneighbor j with probability

4 Complexity

ξXSw(t) 1 minus ρ0( 1113857

1113936kj

rarrkXj p kj

rarr1113874 1113875ΘX kj

rarr t1113874 1113875

langkXj rang

(11)

where the expression kXj p(kj

rarr)langkX

j rang denotes the probabilityan edge connecting to a neighbor with degree kX

j in layer XNext we proceed with the evolution of ξX

Aw(t) andξX

Rw(t) in the layer X isin A B If through an edge withweight w the information is successfully transmitted to theneighbor in layer X with probability λX

w the change in θXw(t)

is calculated as

dθXw(t)

dt minus λX

wξXAw(t) (12)

If neighbor j in the A-state has not transmitted in-formation to node i through an edge of weight w withprobability (1 minus λX

w) in layer X but enters into the R-statewith probability c at time t the value of ξX

Rw(t) will increaseand thus the dynamical differential value of ξX

Rw(t) can beobtained by

dξXRw(t)

dt c 1 minus λX

w1113872 1113873ξXAw(t) (13)

-e initial conditions are θXw(0) 1 and ξX

Rw(0) 0therefore combining equation (12) with (13) we can obtain

ξXRw(t)

c 1 minus λXw1113872 1113873 1 minus θX

w(t)1113960 1113961

λXw

(14)

After inserting equations (11) and (14) into equation (8)we can obtain

ξXAw(t) θX

w(t) minus 1 minus ρ0( 1113857

1113936kj

rarrkXj P kj

rarr1113874 1113875ΘX kj

rarr t1113874 1113875

langkXj rang

minusc 1 minus λX

w1113872 1113873 1 minus θXw(t)1113960 1113961

λXw

(15)

In equation (15) substituting ξXAw(t) with (1minus λX

w)

(dθXw(t)dt) according to equation (8) we further deduce

dθXw(t)

dt λX

w 1 minus ρ0( 1113857

1113936kj

rarrkXj P kj

rarr1113874 1113875ΘX kX

j t1113872 1113873

langkXj rang

+ c 1 minus λXw1113872 1113873

minus c +(1 minus c)λXw1113960 1113961θX

w(t)

(16)

When t⟶infin dθXw(t)dt 0 and we have the steady-

state value θXw(infin) as

θXw(infin)

λXw 1 minus ρ0( 11138571113936

kj

rarrkXj P kj

rarr1113874 1113875ΘX kj

rarrinfin1113874 1113875 +langkX

j rangc 1 minus λXw1113872 1113873

langkXj rang c +(1 minus c)λX

w1113960 1113961

(17)

Inserting θXw(infin) into equations (2) and (3) we thus

obtain

θX(infin) 1113944w

gX(w)θXw(infin)

fX θA(infin) θB(infin)( 1113857

(18)

For a convenient expression we rewriteθA(infin) fA(θA(infin) θA(infin)) as

θA FA θB( 1113857 (19)

and θB(infin) fB(θA(infin) θB(infin)) as

θB FB θA( 1113857 (20)

A discontinuous growth pattern will occur [68] whenequation (19) is tangent to equation (20) with θA lt 1 andθB lt 1 Moreover critical conditions can be acquired fromthe following equation

zfA θA(infin) θB(infin)( 1113857

zθB(infin)

zfB θA(infin) θB(infin)( 1113857

zθA(infin) 1 (21)

From the theoretical analysis above the general solutionsfor adoption thresholds TA and TB can be derived but thetheoretical analysis itself may not be easily comprehended-erefore an intuitive illustration exemplifies the generaltheory with a particular case ie TA TB 1 In this caseequations (9) and (10) are rewritten as

ΘA kj

rarr t1113874 1113875 θA(t)

kAj minus 1

+ θB(t)kB

j minus θA(t)kA

j minus 1θB(t)kB

j

ΘB kj

rarr t1113874 1113875 θA(t)

kAj + θB(t)

kBj minus 1

minus θA(t)kA

j θB(t)kB

j minus 1

(22)

Owing to independence we obtain P(kj

rarr) P(kA

j )P(kBj )

and substitute (23) and (24) into equation (17) as follows

θAw(infin)

λAw 1 minus ρ0( 1113857 H1

A θA(infin)( 1113857 + H0B θB(infin)( 1113857 minus H10

AB θA(infin) θB(infin)( 11138571113960 1113961

c +(1 minus c)λAw1113960 1113961

+c 1 minus λA

w1113872 1113873

c +(1 minus c)λAw1113960 1113961

(23)

θBw(infin)

λBw 1 minus ρ0( 1113857 H0

A θB(infin)( 1113857 + H1B θA(infin)( 1113857 minus H01

AB θA(infin) θB(infin)( 11138571113960 1113961

c +(1 minus c)λBw1113960 1113961

+c 1 minus λB

w1113872 1113873

c +(1 minus c)λBw1113960 1113961

(24)

Complexity 5

where H0X(x) 1113936kX

PX(kX)xkX represents the generationfunction of degree distribution PX(kX) X isin A B andH1

X(x) ((1113936kXkXP(kX)xkXminus 1)langkXrang) X isin A B repre-

sents the generation function of excess degree distributionPX(kX) Additionally the generation functions of degreedistribution P( k

rarr) are expressed as follows

H10AB(x y)

1HA0prime (1)

zH00AB(x y)

zx

H01AB(x y)

1HB0prime (1)

zH00AB(x y)

zy

H00AB(x y) 1113944

kAkB

P( krarr

)xkA y

kB

(25)

where krarr

(kA kB) Subsequently substituting equations(23) and (24) into equation (21) we can deduce the criticalconditions

4 Numerical Method

Based on ErdosndashRenyi (ER) [69] and the scale-free (SF) [62]random network model extensive numerical simulations onartificial two-layer networks were performed to verify thetheoretical analysis above Without loss of generality we setthe network size N 104 and mean-degree langkrangA

langkrangB langkrang 10 For convenience the information trans-mission unit probability in the two layers is set as βA βB

β Additionally the weight distribution of the network layerX isin A B follows gX(ω)simωminus αX

ω with ωmaxX simN1(αX

ω minus 1) andmean weight langωrangA langωrangB langωrang 8 At the steady state inwhich no node in the A-state exists we compute thespreading range by measuring the fraction of nodes in theR-state R(infin) -e simulation results are obtained by av-eraging over at least 104 independent dynamical realizationson 30 artificial networks

To obtain the critical condition from the simulations therelative variance χ [70] is applied in our numerical verifi-cation using the following method

χ NlangR(infin)2rang minus langR(infin)rang2

langR(infin)rang (26)

where lang rang is the ensemble average and langR(infin)rang andlangR(infin)2rang are the first and secondary moments of R(infin)respectively -e peaks of the curve of relative variance χversus the information transmission probability correspondto the critical points

5 Results and Discussion

In reality from the perspective of a degree distributionmodel layers in some multiplex social networks exhibit thehomogeneous creation mechanism such as MSN versusICQ but layers in other multiplex social networks mayexhibit heterogeneous creation mechanisms such as ICQ

versus Facebook -erefore from the aspects of homoge-neous and heterogeneous degree distributions of twolayers the effects of weight distribution heterogeneity oninformation spreading in multiplex social networks isdiscussed in this section in terms of the parameters inSection 4

51 ER-ER Weighted Multiplex Networks First we in-vestigate information spreading on a two-layered weightedER-ER network with Poisson degree distribution pA(kA)

eminus langkArang(langkArangkA kA) in layer A and pB(kB) eminus langkBrang (langkBrangkB kB) in layer B

We first explore the time evolutions of node densities inthree states (S-state A-state and R-state) under differentweight distribution exponents αA

ω and αBω as shown in

Figure 2 Under the same information transmission unitprobability β but different adoption thresholds and seedsizes Figure 2 shows the plots of curves with TA 1 TB

2 ρ0 003 in the top panel and TA 1 TB 3 ρ0 01 inthe bottom panel Regardless of the top or the bottom panelwith time t the size of S(t) decreases gradually from 1 minus ρ0 to0 A(t) changes from ρ0 to 0 eventually and R(t) indicatingthe final adoption size increases gradually from 0 to 1 at theend With the increase in the weight distribution exponentfrom αA

ω 3 αBω 25 in Figures 2(a) and 2(e) αA

ω 3 αBω

3 in Figures 2(b) and 2(f) and αAω 3 αB

ω 35 inFigures 2(c) and 2(g) the evolution time consumed de-creases from 19 to 13 in the top panel and 11 to 9 in thebottom panel Additionally the final adoption size R(infin)

augments at the same time t (see the comparison inFigures 2(d) and 2(h))-e phenomena occurring during thetime evolution suggest that increasing the weight distribu-tion exponent (reducing weight distribution heterogeneity)can facilitate information spreading in a weighted multiplexnetwork

Next given the adoption thresholds TATB and seed sizeρ0 we investigate the growth pattern of the final adoptionsize R(infin) versus information transmission unit probabilityβ In Figure 3(a) with TA 1 TB 1 and ρ0 0005 in-creasing the weight distribution exponent such as from αA

ω

3 αBω 25 to αA

ω 3 αBω 35 does not change the growth

pattern of the final adoption size R(infin) versus β but reducesthe critical information transmission unit probability βcwhich can be numerically captured by the peaks of relativevariance χ in Figure 3(e) with the same settings ofFigure 3(a) Figure 3(b) with different TA 1 TB

2 ρ0 003 exhibits the same pattern as Figure 3(a) withthe peaks of relative variance χ in Figure 3(f) Even enlargingthe seed size (ρ0 01) in Figure 3(c) with the same adoptionthreshold of Figure 3(b) the final adoption size R(infin) versusβ increases in the same pattern regardless of the increase inweight distribution Besides keeping the seed size ρ0 as thesame of Figure 3(c) (ρ0 01) Figure 3(d) with furtherincrease of the adoption threshold TA 1 TB 3 alsoshows similar phenomena as those in Figures 3(a)ndash3(c)verified by the peaks of χ in Figure 3(h) -e above phe-nomena suggest that increasing the weight distributionexponent (decreasing weight distribution heterogeneity) will

6 Complexity

S (t)A (t)R (t)

00

05

10R

(t)

2 4 6 8 10 12 14 16 180t

(a)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 12 140t

(b)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 120t

(c)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

2 4 6 8 10 12 14 16 180t

(d)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(e)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 100t

(f )

Figure 2 Continued

Complexity 7

S (t)A (t)R (t)

00

05

10R

(t)

1 2 3 4 5 6 7 8 90t

(g)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(h)

Figure 2 Time evolutions of node densities in three states (S-state A-state and R-state) on the ER-ER weighted multiplex network underdifferent weight distribution exponents αA

ω and αBω To unveil the effect of the threshold and seed size two groups of subgraphs with

TA 1 TB 2 ρ0 003 and TA 1 TB 3 ρ0 01 are shown in the top and bottom panels respectively Under the same informationtransmission unit probability β subgraphs (a) (b) and (c) from the top panel have similar parameters of αA

ω 3 αBω 25 αA

ω 3 αBω 3

and αAω 3 αB

ω 35 with (e) (f ) and (g) respectively Particularly subgraphs (d) and (h) emphasize the time evolution of the R-state nodedensity R(t) under different weight distribution exponents In both panels despite the different thresholds and seed sizes at the same timestep the density of nodes in the R-state under αA

ω 3 αBω 35 exceeds other densities under αA

ω 3 αBω 25 and αA

ω 3 αBω 3 -ese

phenomena suggest that large weight distribution exponents (small weight distribution heterogeneity) facilitate information spreading -ebasic parameters are N 104 langkrang 10 langωrang 8 c 10 and β 03

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

00 01 02β

03 04

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

Figure 3 Continued

8 Complexity

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(d)

00 01 02β

03

30

20

10

χ

0

(e)

00 01 02β

03 04

6

4

2

χ

0

(f )

00 01 02β

0300

02

04

χ

(g)

00 01 02β

0300

05

10

15

χ

(h)

Figure 3 Growth pattern of the final adoption sizeR(infin) versus information transmission unit probability β on the ER-ERweightedmultiplexnetwork under different weight distribution exponents αA

ω and αBω given adoption thresholds (TA TB) and initial fraction ρ0 of seeds-e final

adoption size R(infin) versus information transmission probability β with (a) TA 1 TB 1 ρ0 0005 (b) TA 1 TB 2 ρ0 003 (c)TA 1 TB 2 ρ0 01 and (d) TA 1 TB 3 ρ0 01 -e lines denoting theoretical solutions agree exceptionally well with the symbolsrepresenting numerical simulations Moreover in subgraphs (e) (f) (g) and (h) the relative variance χ in terms of the peaks numericallyindicates the critical spreading probabilities in (a) (b) (c) and (d) respectively showing that increasing the weight distribution exponent(decreasing weight distribution heterogeneity) can decrease the critical spreading probability -e phenomena above suggest that decreasingweight distribution heterogeneity facilitates information spreading but cannot alter the growth pattern of the final adoption size versusinformation transmission probability -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 9

not alter the growth pattern of R(infin) versus β in any pair ofadoption thresholds but can promote the outbreak of in-formation spreading Noticeably the theoretical solutions(lines) coincide well with the numerical simulations(symbols)

As shown in Figure 3 the initial seed size affects theinformation spreading dynamics We investigate the effectof seed size on information spreading on a weightedmultiplex social network as shown in Figure 4 InFigure 4(a) (TA 1 TB 1 αA

w 3 αBw 25) increasing

the seed size from ρ0 0005 to ρ0 001 not only changesthe growth pattern from discontinuous to continuous butalso decreases the critical probability to promote theoutbreak of information spreading Subsequently if theweight distribution exponent is enlarged such asαA

w 3 αBw 3 in Figure 4(d) and αA

w 3 αBw 35 in

Figure 4(g) increasing the seed size ρ0 still advances in-formation propagation Comparing Figure 4(a) 4(d) and4(g) vertically altering the initial fraction can change thegrowth pattern of R(infin) versus β but cannot change the factthat increasing the weight distribution exponent (reducingweight distribution heterogeneity) retains the growthpattern while facilitating the outbreak of informationspreading Moreover in a horizontal comparison althoughenlarging the adoption thresholds such as TA 1 TB 2in Figures 4(b) 4(e) and 4(h) or TA 1 TB 3 inFigure 4(c) 4(f ) and 4(i) renders information adoptionmore difficult it is clear that increasing ρ0 still transformsthe growth pattern of R(infin) from discontinuous to con-tinuous phase transition and promotes the outbreak ofinformation spreading as in the case of TA 1 TB 1Additionally changes in adoption thresholds and seed sizedo not qualitatively alter phenomena

For a more comprehensive study we obtained the resultsof the final adoption size on the (β ρ0) parameter plane asillustrated in Figure 5 Here we set the subgraphs in column1 with TA 1 TB 1 in column 2 with TA 1 TB 2 incolumn 3 with TA 1 TB 3 in row 1 withαA

w 3 αBw 25 in row 2 with αA

w 3 αBw 3 and in row 3

with αAw 3 αB

w 35 In the following we discuss thefindings based on Figure 5 in detail In Figure 5(a)(TA 1 TB 1 αA

w 3 αBw 25) a critical ρlowast0 0003 ex-

ists below which at any information transmission unitprobability β no outbreak of information spreading willoccur and above which outbreak and global adoption willoccur In a vertical comparison we retain the adoptionthresholds in Figures 5(a) 5(d) and 5(h) and further in-crease the weight distribution exponents in Figure 5(d)(αA

w 3 αBw 3) and Figure 5(h) (αA

w 3 αBw 35) -e

results indicate that the three subgraphs have the samecritical seed size ρlowast0 0003 below which increasing theweight distribution exponent (reducing the weight distri-bution heterogeneity) still cannot induce the outbreak ofinformation spreading but above which increasing theweight distribution exponent can accelerate the outbreak at asmaller information transmission unit probability βMoreover in a horizontal comparison we retain the weightdistribution exponent in Figures 5(a) 5(b) and 5(c) withαA

w 3 αBw 25 and increase the adoption thresholds in

Figure 5(b) (TA 1 TB 2) and Figure 5(c)(TA 1 TB 3) Under these circumstances we discov-ered that the critical seed size ρlowast0 is enlarged from 0003 to006 which implies increasing the difficulty in informationadoption Analogously other subgraphs exhibit similarfindings -e phenomena in Figure 5 suggest that a criticalseed size exists above which outbreak of informationspreading and global adoption will occur and the weightdistribution exponent (reducing the weight distributionheterogeneity) is essential for facilitating informationspreading

52 ER-SF Weighted Multiplex Networks For cases of het-erogeneous degree distributions we investigate informationspreading on a two-layered weighted ER-SF network withPoisson degree distribution pA(kA) eminus langkArang(langkArangkA kA) inlayer A and with power-law degree distribution pB(kB)

ζBkminus vB

B in layer B where ζB 11113936kBkminus vB

B and parameter vB

denotes the degree exponent of layer B Additionally we setthe minimum degree kmin

B 4 and maximum degreekmax

B N

radic

On the weighted ER-SF networks of distinct degreedistribution exponents the growth pattern is shown inFigure 6 with the common setting TA 1 TB 3 in allsubgraphs In Figure 6(a) with degree distribution exponentvB 21 we discovered that enlarging the weight distribu-tion exponent such as from αA

w 3 αBw 25 to

αAw 3 αB

w 35 can increase the final adoption size at thesame β and decrease the critical probability to promote theoutbreak of information spreading Moreover we proceed toincrease the degree distribution exponent (reduce the degreedistribution heterogeneity) in Figure 6(b) (vB 3) andFigure 6(c) (vB 4) Consequently from Figure 6(a) to 6(c)given αA

w and αBw the growth pattern changes from con-

tinuous to the discontinuous growth pattern However atany given vB changing the weight distribution exponent willnot alter the growth pattern of R(infin) -e phenomena inFigure 6 suggest that changing the degree distributionheterogeneity will not qualitatively alter the fact that re-ducing weight distribution heterogeneity can promote in-formation spreading

6 Conclusions

With the development of the Internet and mobile devicessocial networks have prevailed in peoplersquos lives Typically anindividual does not only use one social network but multiplesocial networks simultaneously Moreover the informationdoes not only diffuse in one social network but also spreadssimultaneously on several social networks eg informationspreads on Twitter and Facebook simultaneously Because ofdifferent extents of intimacy among friends colleagues andfamily members connections in social networks should bemodeled with different weights As such a multiplex socialnetwork should be a multilayer weighted network -ere-fore information spreading on a multiplex social networkshould be investigated on a multilayer weighted networkUnfortunately the mechanism of information spreading on

10 Complexity

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(a)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(b)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(c)

ρ0 = 0005ρ0 = 001ρ0 = 01

01 02 03 04 0500β

00

05

10

R (infin

)

(d)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(e)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(f )

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(g)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(h)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(i)

Figure 4 Effect of fraction of initial seeds on the growth pattern of the final adoption size R(infin) versus β on the ER-ER weighted multiplexnetwork under different weight distribution exponents αA

ω and αBω -e values of weight distribution exponent αA

ω of all subgraphs are 3 andthe subgraphs in each row have the same αB

ω ie subgraphs (a) (b) and (c) correspond to αBω 25 (d) (e) and (f) correspond to αB

ω 3and (g) (h) and (i) correspond to αB

ω 35 Additionally the subgraphs in each column have the same adoption threshold ie subgraphs(a) (d) and (g) are set with TA 1 TB 1 (b) (e) and (h) with TA 1 TB 2 and (c) (f ) and (i) with TA 1 TB 3 -e theoreticalsolutions (lines) coincide well with the numerical results (symbols) Given other parameters increasing the fraction of initial seeds willchange the growth pattern of R(infin) from discontinuous to continuous growth Moreover vertically increasing the weight distributionexponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability horizontally increasing theadoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability-e phenomena aboveconvey that changing the seed size can alter the growth pattern and decreasing the adoption threshold can facilitate information spreadingMost importantly changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distributionheterogeneity can facilitate information spreading -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 11

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

individual has not adopted the information -e A-stateindicates that an individual has adopted the information andcan transmit the information to neighbors -e R-statemeans that an individual has lost interest in the informationand stops diffusion

Furthermore the detailed information spreading processis as follows Initially randomly selecting a fraction ρ0 ofnodes as initial seeds (in the A-state) At each time step eachA-state node i attempts to transmit the information to itsS-state neighbor j simultaneously through an edge of weightwA

ij with probability λA(wAij) in layer A and through an edge

of weight wBij with probability λB(wB

ij) in layer B according toequation (1) Because of nonredundant information trans-mission once an edge between nodes i and j in layer A or Bsuccessfully delivers the information from i to j such an edgeis not allowed for repeated information delivery An adopted

node however can perform many attempts until it turnsinto the R-state If the S-state node j in the layer X isin A B

successfully receives one piece of information from theA-state neighbor node j its accumulative number mX

j ofinformation pieces in the layer X isin A B will increase byone ie mX

j ⟶ mXj + 1 -en node j will compare the

accumulative number mXj with its adoption threshold TX

In reality for confirming the legitimacy and credibility of theinformation an individual can adopt the information only ifheshe corroborates with the information in both sub-networks -erefore the susceptible node j can enter theA-state when mA

j geTA and mBj geTB co-occur in both layers

A and B otherwise it remains in the S-state Regardingsocial reinforcement because the accumulative number ofinformation pieces determines the information adoptionthe information spreading process is characterized by the

1

2

3

4

5

1

2

3

4

5

Susceptible

Adopted

Recovered

A

B

(a)

1

2

3

4

5

1

2

3

4

5

Susceptible

Adopted

Recovered

A

B

(b)

1

1

2

2

3

3

4

4

5

5

Susceptible

Adopted

Recovered

A

B

(c)

1

2

3

4

5

1

2

3

4

5

Susceptible

Adopted

Recovered

A

B

(d)

Figure 1 Illustration of social information spreading with transformation of three states (susceptible adopted and recovered) on aweighted multiplex network Two different independent social networks A and B comprising degrees of intimate relations represent twolayers of the weighted multiplex network in which different degrees of edge-widths denote different edge-weights

Complexity 3

non-Markovian effect At the same time step after informationtransmission the A-state node j will lose interest in the in-formation and enter the R-state with probability γ ie it willstop diffusing information in both layers A and B Once thenode changes into the R-state it will retreat from furtherinformation spreading Finally the disappearance of all A-statenodes signals the termination of the spreading dynamics

3 Edge-Weight-Based Compartmental Theory

To include the social reinforcement effect the mean-filedtheory [63] and an advanced approach [64] can be usedInspired by References [57 65 66] an edge-weight-basedcompartmental theory is conceived to analyze the spreadingprocess with theoretically strong correlation among thethree states of nodes in a multiplex the network featured bystrong weight heterogeneity -e theory assumes that theinformation spreading is on large sparse uncorrelated andlocal tree-like weighted multiplex networks

In the edge-weight-based compartmental theory a nodein the cavity state [67] implies that it can only receive in-formation from but cannot transmit information to itsneighbors We denote θX(t)(X isin A B ) as the probabilitythat the information has not been passed to the S-stateneighbors alongside a randomly selected edge in layer X bytime t Furthermore in a weight multiplex network wedenote θX

w(t) as the probability that a node has not beeninformed through a randomly selected edge with weight w inlayer X by time t Following weight distribution gA(w) inlayer A and gB(w) in layer B we can obtain

θA(t) 1113944w

gA(w)θAw(t) (2)

θB(t) 1113944w

gB(w)θBw(t) (3)

An arbitrary S-state node i with degree vectorki

rarr (kA

i kBi ) receives mA and mB pieces of information by

time t with the probabilities

ϕAmA

kAi t1113872 1113873

kAi

mA

1113888 1113889θA(t)kA

i minus mA 1 minus θA(t)1113858 1113859mA

ϕBmB

kBi t1113872 1113873

kBi

mB

1113888 1113889θB(t)kB

i minus mB 1 minus θB(t)1113858 1113859mB

(4)

respectivelyAs mentioned in Section 2 an S-state node with degree

vector krarr

(kA kB) can adopt the information only if thecumulative numbers mA geTA and mB geTB occur simulta-neously inversely the S-state node having received mA andmB pieces of information by time t will remain in the S-statewith probability

s( krarr

t) 1 minus ρ0( 1113857 1 minus 1 minus 1113944

TAminus 1

mA0ϕA

mAk

A t1113872 1113873⎛⎝ ⎞⎠⎡⎢⎢⎣

times 1 minus 1113944

TBminus 1

mB0ϕB

mBk

B t1113872 1113873⎛⎝ ⎞⎠⎤⎥⎥⎦

(5)

We define the probability that a susceptible node withany degree in the layer X isin A B remaining in the S-state ifit receives mX pieces of information by time t as

ηX(t) 1113944kX

PX kX

1113872 1113873 1113944

TXminus 1

mX0ϕX

mXk

X t1113872 1113873 (6)

Furthermore the fraction of nodes in the S-state at time tcan be expressed as

S(t) 1113944

krarr

P( krarr

)s( krarr

t)

1 minus ρ0( 1113857 ηA + ηB minus ηAηB( 1113857

(7)

If we wish to obtain S(t) we should solve θX(t) whichdepends on θX

w(t) in a weighted network -erefore wefocus on θX

w(t) at first According to [57] in layer X theneighbor connected by an edge with weight w to a node inthe cavity state can be in the S-state A-state or R-statetherefore θX

w(t)(X isin A B ) comprises the following threeparts

θXw(t) ξX

Sw(t) + ξXAw(t) + ξX

Rw(t) (8)

where ξXSw(t) ξX

Aw(t) and ξXRw(t) indicate the probabilities

that a neighbor connected by an edge with weight w to anode in the cavity state remains in the S- A- and R-statesrespectively and has not transmitted information bytime t

If neighbor jwith degree vector kj

rarr (kA

j kBj ) is linked by

an edge of weight w to the node i in the cavity state in layerA(B) then neighbor j can only receive information fromother kA

j minus 1(kBj minus 1) edges in layer A(B) -erefore

neighbor j receives nA pieces of information in layer A withprobability ϕnA

(kAj minus 1 t) and with no constraint in layer B

receives nB pieces of information in layer B with probabilityϕnV

(kBj t) Subsequently neighbor j remains in the S-state in

layer A with probability

ΘA kj

rarr t1113874 1113875 1 minus 1 minus 1113944

TAminus 1

nA0ϕnA

kAj minus 1 t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

times 1 minus 1113944

TBminus 1

nB0ϕnB

kBj t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

(9)

Analogously neighbor j is connected with node i by anedge with weight w in layer B and remains in the S-state withprobability

ΘB kj

rarr t1113874 1113875 1 minus 1 minus 1113944

TA minus 1

nA0ϕnA

kAj t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

times 1 minus 1113944

TBminus 1

nB0ϕnB

kBj minus 1 t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

(10)

Accordingly given the joint degree distribution p( krarr

) inlayer X an edge with weight w connects to an S-stateneighbor j with probability

4 Complexity

ξXSw(t) 1 minus ρ0( 1113857

1113936kj

rarrkXj p kj

rarr1113874 1113875ΘX kj

rarr t1113874 1113875

langkXj rang

(11)

where the expression kXj p(kj

rarr)langkX

j rang denotes the probabilityan edge connecting to a neighbor with degree kX

j in layer XNext we proceed with the evolution of ξX

Aw(t) andξX

Rw(t) in the layer X isin A B If through an edge withweight w the information is successfully transmitted to theneighbor in layer X with probability λX

w the change in θXw(t)

is calculated as

dθXw(t)

dt minus λX

wξXAw(t) (12)

If neighbor j in the A-state has not transmitted in-formation to node i through an edge of weight w withprobability (1 minus λX

w) in layer X but enters into the R-statewith probability c at time t the value of ξX

Rw(t) will increaseand thus the dynamical differential value of ξX

Rw(t) can beobtained by

dξXRw(t)

dt c 1 minus λX

w1113872 1113873ξXAw(t) (13)

-e initial conditions are θXw(0) 1 and ξX

Rw(0) 0therefore combining equation (12) with (13) we can obtain

ξXRw(t)

c 1 minus λXw1113872 1113873 1 minus θX

w(t)1113960 1113961

λXw

(14)

After inserting equations (11) and (14) into equation (8)we can obtain

ξXAw(t) θX

w(t) minus 1 minus ρ0( 1113857

1113936kj

rarrkXj P kj

rarr1113874 1113875ΘX kj

rarr t1113874 1113875

langkXj rang

minusc 1 minus λX

w1113872 1113873 1 minus θXw(t)1113960 1113961

λXw

(15)

In equation (15) substituting ξXAw(t) with (1minus λX

w)

(dθXw(t)dt) according to equation (8) we further deduce

dθXw(t)

dt λX

w 1 minus ρ0( 1113857

1113936kj

rarrkXj P kj

rarr1113874 1113875ΘX kX

j t1113872 1113873

langkXj rang

+ c 1 minus λXw1113872 1113873

minus c +(1 minus c)λXw1113960 1113961θX

w(t)

(16)

When t⟶infin dθXw(t)dt 0 and we have the steady-

state value θXw(infin) as

θXw(infin)

λXw 1 minus ρ0( 11138571113936

kj

rarrkXj P kj

rarr1113874 1113875ΘX kj

rarrinfin1113874 1113875 +langkX

j rangc 1 minus λXw1113872 1113873

langkXj rang c +(1 minus c)λX

w1113960 1113961

(17)

Inserting θXw(infin) into equations (2) and (3) we thus

obtain

θX(infin) 1113944w

gX(w)θXw(infin)

fX θA(infin) θB(infin)( 1113857

(18)

For a convenient expression we rewriteθA(infin) fA(θA(infin) θA(infin)) as

θA FA θB( 1113857 (19)

and θB(infin) fB(θA(infin) θB(infin)) as

θB FB θA( 1113857 (20)

A discontinuous growth pattern will occur [68] whenequation (19) is tangent to equation (20) with θA lt 1 andθB lt 1 Moreover critical conditions can be acquired fromthe following equation

zfA θA(infin) θB(infin)( 1113857

zθB(infin)

zfB θA(infin) θB(infin)( 1113857

zθA(infin) 1 (21)

From the theoretical analysis above the general solutionsfor adoption thresholds TA and TB can be derived but thetheoretical analysis itself may not be easily comprehended-erefore an intuitive illustration exemplifies the generaltheory with a particular case ie TA TB 1 In this caseequations (9) and (10) are rewritten as

ΘA kj

rarr t1113874 1113875 θA(t)

kAj minus 1

+ θB(t)kB

j minus θA(t)kA

j minus 1θB(t)kB

j

ΘB kj

rarr t1113874 1113875 θA(t)

kAj + θB(t)

kBj minus 1

minus θA(t)kA

j θB(t)kB

j minus 1

(22)

Owing to independence we obtain P(kj

rarr) P(kA

j )P(kBj )

and substitute (23) and (24) into equation (17) as follows

θAw(infin)

λAw 1 minus ρ0( 1113857 H1

A θA(infin)( 1113857 + H0B θB(infin)( 1113857 minus H10

AB θA(infin) θB(infin)( 11138571113960 1113961

c +(1 minus c)λAw1113960 1113961

+c 1 minus λA

w1113872 1113873

c +(1 minus c)λAw1113960 1113961

(23)

θBw(infin)

λBw 1 minus ρ0( 1113857 H0

A θB(infin)( 1113857 + H1B θA(infin)( 1113857 minus H01

AB θA(infin) θB(infin)( 11138571113960 1113961

c +(1 minus c)λBw1113960 1113961

+c 1 minus λB

w1113872 1113873

c +(1 minus c)λBw1113960 1113961

(24)

Complexity 5

where H0X(x) 1113936kX

PX(kX)xkX represents the generationfunction of degree distribution PX(kX) X isin A B andH1

X(x) ((1113936kXkXP(kX)xkXminus 1)langkXrang) X isin A B repre-

sents the generation function of excess degree distributionPX(kX) Additionally the generation functions of degreedistribution P( k

rarr) are expressed as follows

H10AB(x y)

1HA0prime (1)

zH00AB(x y)

zx

H01AB(x y)

1HB0prime (1)

zH00AB(x y)

zy

H00AB(x y) 1113944

kAkB

P( krarr

)xkA y

kB

(25)

where krarr

(kA kB) Subsequently substituting equations(23) and (24) into equation (21) we can deduce the criticalconditions

4 Numerical Method

Based on ErdosndashRenyi (ER) [69] and the scale-free (SF) [62]random network model extensive numerical simulations onartificial two-layer networks were performed to verify thetheoretical analysis above Without loss of generality we setthe network size N 104 and mean-degree langkrangA

langkrangB langkrang 10 For convenience the information trans-mission unit probability in the two layers is set as βA βB

β Additionally the weight distribution of the network layerX isin A B follows gX(ω)simωminus αX

ω with ωmaxX simN1(αX

ω minus 1) andmean weight langωrangA langωrangB langωrang 8 At the steady state inwhich no node in the A-state exists we compute thespreading range by measuring the fraction of nodes in theR-state R(infin) -e simulation results are obtained by av-eraging over at least 104 independent dynamical realizationson 30 artificial networks

To obtain the critical condition from the simulations therelative variance χ [70] is applied in our numerical verifi-cation using the following method

χ NlangR(infin)2rang minus langR(infin)rang2

langR(infin)rang (26)

where lang rang is the ensemble average and langR(infin)rang andlangR(infin)2rang are the first and secondary moments of R(infin)respectively -e peaks of the curve of relative variance χversus the information transmission probability correspondto the critical points

5 Results and Discussion

In reality from the perspective of a degree distributionmodel layers in some multiplex social networks exhibit thehomogeneous creation mechanism such as MSN versusICQ but layers in other multiplex social networks mayexhibit heterogeneous creation mechanisms such as ICQ

versus Facebook -erefore from the aspects of homoge-neous and heterogeneous degree distributions of twolayers the effects of weight distribution heterogeneity oninformation spreading in multiplex social networks isdiscussed in this section in terms of the parameters inSection 4

51 ER-ER Weighted Multiplex Networks First we in-vestigate information spreading on a two-layered weightedER-ER network with Poisson degree distribution pA(kA)

eminus langkArang(langkArangkA kA) in layer A and pB(kB) eminus langkBrang (langkBrangkB kB) in layer B

We first explore the time evolutions of node densities inthree states (S-state A-state and R-state) under differentweight distribution exponents αA

ω and αBω as shown in

Figure 2 Under the same information transmission unitprobability β but different adoption thresholds and seedsizes Figure 2 shows the plots of curves with TA 1 TB

2 ρ0 003 in the top panel and TA 1 TB 3 ρ0 01 inthe bottom panel Regardless of the top or the bottom panelwith time t the size of S(t) decreases gradually from 1 minus ρ0 to0 A(t) changes from ρ0 to 0 eventually and R(t) indicatingthe final adoption size increases gradually from 0 to 1 at theend With the increase in the weight distribution exponentfrom αA

ω 3 αBω 25 in Figures 2(a) and 2(e) αA

ω 3 αBω

3 in Figures 2(b) and 2(f) and αAω 3 αB

ω 35 inFigures 2(c) and 2(g) the evolution time consumed de-creases from 19 to 13 in the top panel and 11 to 9 in thebottom panel Additionally the final adoption size R(infin)

augments at the same time t (see the comparison inFigures 2(d) and 2(h))-e phenomena occurring during thetime evolution suggest that increasing the weight distribu-tion exponent (reducing weight distribution heterogeneity)can facilitate information spreading in a weighted multiplexnetwork

Next given the adoption thresholds TATB and seed sizeρ0 we investigate the growth pattern of the final adoptionsize R(infin) versus information transmission unit probabilityβ In Figure 3(a) with TA 1 TB 1 and ρ0 0005 in-creasing the weight distribution exponent such as from αA

ω

3 αBω 25 to αA

ω 3 αBω 35 does not change the growth

pattern of the final adoption size R(infin) versus β but reducesthe critical information transmission unit probability βcwhich can be numerically captured by the peaks of relativevariance χ in Figure 3(e) with the same settings ofFigure 3(a) Figure 3(b) with different TA 1 TB

2 ρ0 003 exhibits the same pattern as Figure 3(a) withthe peaks of relative variance χ in Figure 3(f) Even enlargingthe seed size (ρ0 01) in Figure 3(c) with the same adoptionthreshold of Figure 3(b) the final adoption size R(infin) versusβ increases in the same pattern regardless of the increase inweight distribution Besides keeping the seed size ρ0 as thesame of Figure 3(c) (ρ0 01) Figure 3(d) with furtherincrease of the adoption threshold TA 1 TB 3 alsoshows similar phenomena as those in Figures 3(a)ndash3(c)verified by the peaks of χ in Figure 3(h) -e above phe-nomena suggest that increasing the weight distributionexponent (decreasing weight distribution heterogeneity) will

6 Complexity

S (t)A (t)R (t)

00

05

10R

(t)

2 4 6 8 10 12 14 16 180t

(a)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 12 140t

(b)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 120t

(c)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

2 4 6 8 10 12 14 16 180t

(d)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(e)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 100t

(f )

Figure 2 Continued

Complexity 7

S (t)A (t)R (t)

00

05

10R

(t)

1 2 3 4 5 6 7 8 90t

(g)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(h)

Figure 2 Time evolutions of node densities in three states (S-state A-state and R-state) on the ER-ER weighted multiplex network underdifferent weight distribution exponents αA

ω and αBω To unveil the effect of the threshold and seed size two groups of subgraphs with

TA 1 TB 2 ρ0 003 and TA 1 TB 3 ρ0 01 are shown in the top and bottom panels respectively Under the same informationtransmission unit probability β subgraphs (a) (b) and (c) from the top panel have similar parameters of αA

ω 3 αBω 25 αA

ω 3 αBω 3

and αAω 3 αB

ω 35 with (e) (f ) and (g) respectively Particularly subgraphs (d) and (h) emphasize the time evolution of the R-state nodedensity R(t) under different weight distribution exponents In both panels despite the different thresholds and seed sizes at the same timestep the density of nodes in the R-state under αA

ω 3 αBω 35 exceeds other densities under αA

ω 3 αBω 25 and αA

ω 3 αBω 3 -ese

phenomena suggest that large weight distribution exponents (small weight distribution heterogeneity) facilitate information spreading -ebasic parameters are N 104 langkrang 10 langωrang 8 c 10 and β 03

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

00 01 02β

03 04

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

Figure 3 Continued

8 Complexity

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(d)

00 01 02β

03

30

20

10

χ

0

(e)

00 01 02β

03 04

6

4

2

χ

0

(f )

00 01 02β

0300

02

04

χ

(g)

00 01 02β

0300

05

10

15

χ

(h)

Figure 3 Growth pattern of the final adoption sizeR(infin) versus information transmission unit probability β on the ER-ERweightedmultiplexnetwork under different weight distribution exponents αA

ω and αBω given adoption thresholds (TA TB) and initial fraction ρ0 of seeds-e final

adoption size R(infin) versus information transmission probability β with (a) TA 1 TB 1 ρ0 0005 (b) TA 1 TB 2 ρ0 003 (c)TA 1 TB 2 ρ0 01 and (d) TA 1 TB 3 ρ0 01 -e lines denoting theoretical solutions agree exceptionally well with the symbolsrepresenting numerical simulations Moreover in subgraphs (e) (f) (g) and (h) the relative variance χ in terms of the peaks numericallyindicates the critical spreading probabilities in (a) (b) (c) and (d) respectively showing that increasing the weight distribution exponent(decreasing weight distribution heterogeneity) can decrease the critical spreading probability -e phenomena above suggest that decreasingweight distribution heterogeneity facilitates information spreading but cannot alter the growth pattern of the final adoption size versusinformation transmission probability -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 9

not alter the growth pattern of R(infin) versus β in any pair ofadoption thresholds but can promote the outbreak of in-formation spreading Noticeably the theoretical solutions(lines) coincide well with the numerical simulations(symbols)

As shown in Figure 3 the initial seed size affects theinformation spreading dynamics We investigate the effectof seed size on information spreading on a weightedmultiplex social network as shown in Figure 4 InFigure 4(a) (TA 1 TB 1 αA

w 3 αBw 25) increasing

the seed size from ρ0 0005 to ρ0 001 not only changesthe growth pattern from discontinuous to continuous butalso decreases the critical probability to promote theoutbreak of information spreading Subsequently if theweight distribution exponent is enlarged such asαA

w 3 αBw 3 in Figure 4(d) and αA

w 3 αBw 35 in

Figure 4(g) increasing the seed size ρ0 still advances in-formation propagation Comparing Figure 4(a) 4(d) and4(g) vertically altering the initial fraction can change thegrowth pattern of R(infin) versus β but cannot change the factthat increasing the weight distribution exponent (reducingweight distribution heterogeneity) retains the growthpattern while facilitating the outbreak of informationspreading Moreover in a horizontal comparison althoughenlarging the adoption thresholds such as TA 1 TB 2in Figures 4(b) 4(e) and 4(h) or TA 1 TB 3 inFigure 4(c) 4(f ) and 4(i) renders information adoptionmore difficult it is clear that increasing ρ0 still transformsthe growth pattern of R(infin) from discontinuous to con-tinuous phase transition and promotes the outbreak ofinformation spreading as in the case of TA 1 TB 1Additionally changes in adoption thresholds and seed sizedo not qualitatively alter phenomena

For a more comprehensive study we obtained the resultsof the final adoption size on the (β ρ0) parameter plane asillustrated in Figure 5 Here we set the subgraphs in column1 with TA 1 TB 1 in column 2 with TA 1 TB 2 incolumn 3 with TA 1 TB 3 in row 1 withαA

w 3 αBw 25 in row 2 with αA

w 3 αBw 3 and in row 3

with αAw 3 αB

w 35 In the following we discuss thefindings based on Figure 5 in detail In Figure 5(a)(TA 1 TB 1 αA

w 3 αBw 25) a critical ρlowast0 0003 ex-

ists below which at any information transmission unitprobability β no outbreak of information spreading willoccur and above which outbreak and global adoption willoccur In a vertical comparison we retain the adoptionthresholds in Figures 5(a) 5(d) and 5(h) and further in-crease the weight distribution exponents in Figure 5(d)(αA

w 3 αBw 3) and Figure 5(h) (αA

w 3 αBw 35) -e

results indicate that the three subgraphs have the samecritical seed size ρlowast0 0003 below which increasing theweight distribution exponent (reducing the weight distri-bution heterogeneity) still cannot induce the outbreak ofinformation spreading but above which increasing theweight distribution exponent can accelerate the outbreak at asmaller information transmission unit probability βMoreover in a horizontal comparison we retain the weightdistribution exponent in Figures 5(a) 5(b) and 5(c) withαA

w 3 αBw 25 and increase the adoption thresholds in

Figure 5(b) (TA 1 TB 2) and Figure 5(c)(TA 1 TB 3) Under these circumstances we discov-ered that the critical seed size ρlowast0 is enlarged from 0003 to006 which implies increasing the difficulty in informationadoption Analogously other subgraphs exhibit similarfindings -e phenomena in Figure 5 suggest that a criticalseed size exists above which outbreak of informationspreading and global adoption will occur and the weightdistribution exponent (reducing the weight distributionheterogeneity) is essential for facilitating informationspreading

52 ER-SF Weighted Multiplex Networks For cases of het-erogeneous degree distributions we investigate informationspreading on a two-layered weighted ER-SF network withPoisson degree distribution pA(kA) eminus langkArang(langkArangkA kA) inlayer A and with power-law degree distribution pB(kB)

ζBkminus vB

B in layer B where ζB 11113936kBkminus vB

B and parameter vB

denotes the degree exponent of layer B Additionally we setthe minimum degree kmin

B 4 and maximum degreekmax

B N

radic

On the weighted ER-SF networks of distinct degreedistribution exponents the growth pattern is shown inFigure 6 with the common setting TA 1 TB 3 in allsubgraphs In Figure 6(a) with degree distribution exponentvB 21 we discovered that enlarging the weight distribu-tion exponent such as from αA

w 3 αBw 25 to

αAw 3 αB

w 35 can increase the final adoption size at thesame β and decrease the critical probability to promote theoutbreak of information spreading Moreover we proceed toincrease the degree distribution exponent (reduce the degreedistribution heterogeneity) in Figure 6(b) (vB 3) andFigure 6(c) (vB 4) Consequently from Figure 6(a) to 6(c)given αA

w and αBw the growth pattern changes from con-

tinuous to the discontinuous growth pattern However atany given vB changing the weight distribution exponent willnot alter the growth pattern of R(infin) -e phenomena inFigure 6 suggest that changing the degree distributionheterogeneity will not qualitatively alter the fact that re-ducing weight distribution heterogeneity can promote in-formation spreading

6 Conclusions

With the development of the Internet and mobile devicessocial networks have prevailed in peoplersquos lives Typically anindividual does not only use one social network but multiplesocial networks simultaneously Moreover the informationdoes not only diffuse in one social network but also spreadssimultaneously on several social networks eg informationspreads on Twitter and Facebook simultaneously Because ofdifferent extents of intimacy among friends colleagues andfamily members connections in social networks should bemodeled with different weights As such a multiplex socialnetwork should be a multilayer weighted network -ere-fore information spreading on a multiplex social networkshould be investigated on a multilayer weighted networkUnfortunately the mechanism of information spreading on

10 Complexity

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(a)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(b)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(c)

ρ0 = 0005ρ0 = 001ρ0 = 01

01 02 03 04 0500β

00

05

10

R (infin

)

(d)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(e)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(f )

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(g)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(h)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(i)

Figure 4 Effect of fraction of initial seeds on the growth pattern of the final adoption size R(infin) versus β on the ER-ER weighted multiplexnetwork under different weight distribution exponents αA

ω and αBω -e values of weight distribution exponent αA

ω of all subgraphs are 3 andthe subgraphs in each row have the same αB

ω ie subgraphs (a) (b) and (c) correspond to αBω 25 (d) (e) and (f) correspond to αB

ω 3and (g) (h) and (i) correspond to αB

ω 35 Additionally the subgraphs in each column have the same adoption threshold ie subgraphs(a) (d) and (g) are set with TA 1 TB 1 (b) (e) and (h) with TA 1 TB 2 and (c) (f ) and (i) with TA 1 TB 3 -e theoreticalsolutions (lines) coincide well with the numerical results (symbols) Given other parameters increasing the fraction of initial seeds willchange the growth pattern of R(infin) from discontinuous to continuous growth Moreover vertically increasing the weight distributionexponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability horizontally increasing theadoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability-e phenomena aboveconvey that changing the seed size can alter the growth pattern and decreasing the adoption threshold can facilitate information spreadingMost importantly changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distributionheterogeneity can facilitate information spreading -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 11

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

non-Markovian effect At the same time step after informationtransmission the A-state node j will lose interest in the in-formation and enter the R-state with probability γ ie it willstop diffusing information in both layers A and B Once thenode changes into the R-state it will retreat from furtherinformation spreading Finally the disappearance of all A-statenodes signals the termination of the spreading dynamics

3 Edge-Weight-Based Compartmental Theory

To include the social reinforcement effect the mean-filedtheory [63] and an advanced approach [64] can be usedInspired by References [57 65 66] an edge-weight-basedcompartmental theory is conceived to analyze the spreadingprocess with theoretically strong correlation among thethree states of nodes in a multiplex the network featured bystrong weight heterogeneity -e theory assumes that theinformation spreading is on large sparse uncorrelated andlocal tree-like weighted multiplex networks

In the edge-weight-based compartmental theory a nodein the cavity state [67] implies that it can only receive in-formation from but cannot transmit information to itsneighbors We denote θX(t)(X isin A B ) as the probabilitythat the information has not been passed to the S-stateneighbors alongside a randomly selected edge in layer X bytime t Furthermore in a weight multiplex network wedenote θX

w(t) as the probability that a node has not beeninformed through a randomly selected edge with weight w inlayer X by time t Following weight distribution gA(w) inlayer A and gB(w) in layer B we can obtain

θA(t) 1113944w

gA(w)θAw(t) (2)

θB(t) 1113944w

gB(w)θBw(t) (3)

An arbitrary S-state node i with degree vectorki

rarr (kA

i kBi ) receives mA and mB pieces of information by

time t with the probabilities

ϕAmA

kAi t1113872 1113873

kAi

mA

1113888 1113889θA(t)kA

i minus mA 1 minus θA(t)1113858 1113859mA

ϕBmB

kBi t1113872 1113873

kBi

mB

1113888 1113889θB(t)kB

i minus mB 1 minus θB(t)1113858 1113859mB

(4)

respectivelyAs mentioned in Section 2 an S-state node with degree

vector krarr

(kA kB) can adopt the information only if thecumulative numbers mA geTA and mB geTB occur simulta-neously inversely the S-state node having received mA andmB pieces of information by time t will remain in the S-statewith probability

s( krarr

t) 1 minus ρ0( 1113857 1 minus 1 minus 1113944

TAminus 1

mA0ϕA

mAk

A t1113872 1113873⎛⎝ ⎞⎠⎡⎢⎢⎣

times 1 minus 1113944

TBminus 1

mB0ϕB

mBk

B t1113872 1113873⎛⎝ ⎞⎠⎤⎥⎥⎦

(5)

We define the probability that a susceptible node withany degree in the layer X isin A B remaining in the S-state ifit receives mX pieces of information by time t as

ηX(t) 1113944kX

PX kX

1113872 1113873 1113944

TXminus 1

mX0ϕX

mXk

X t1113872 1113873 (6)

Furthermore the fraction of nodes in the S-state at time tcan be expressed as

S(t) 1113944

krarr

P( krarr

)s( krarr

t)

1 minus ρ0( 1113857 ηA + ηB minus ηAηB( 1113857

(7)

If we wish to obtain S(t) we should solve θX(t) whichdepends on θX

w(t) in a weighted network -erefore wefocus on θX

w(t) at first According to [57] in layer X theneighbor connected by an edge with weight w to a node inthe cavity state can be in the S-state A-state or R-statetherefore θX

w(t)(X isin A B ) comprises the following threeparts

θXw(t) ξX

Sw(t) + ξXAw(t) + ξX

Rw(t) (8)

where ξXSw(t) ξX

Aw(t) and ξXRw(t) indicate the probabilities

that a neighbor connected by an edge with weight w to anode in the cavity state remains in the S- A- and R-statesrespectively and has not transmitted information bytime t

If neighbor jwith degree vector kj

rarr (kA

j kBj ) is linked by

an edge of weight w to the node i in the cavity state in layerA(B) then neighbor j can only receive information fromother kA

j minus 1(kBj minus 1) edges in layer A(B) -erefore

neighbor j receives nA pieces of information in layer A withprobability ϕnA

(kAj minus 1 t) and with no constraint in layer B

receives nB pieces of information in layer B with probabilityϕnV

(kBj t) Subsequently neighbor j remains in the S-state in

layer A with probability

ΘA kj

rarr t1113874 1113875 1 minus 1 minus 1113944

TAminus 1

nA0ϕnA

kAj minus 1 t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

times 1 minus 1113944

TBminus 1

nB0ϕnB

kBj t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

(9)

Analogously neighbor j is connected with node i by anedge with weight w in layer B and remains in the S-state withprobability

ΘB kj

rarr t1113874 1113875 1 minus 1 minus 1113944

TA minus 1

nA0ϕnA

kAj t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

times 1 minus 1113944

TBminus 1

nB0ϕnB

kBj minus 1 t1113872 1113873⎡⎢⎢⎣ ⎤⎥⎥⎦

(10)

Accordingly given the joint degree distribution p( krarr

) inlayer X an edge with weight w connects to an S-stateneighbor j with probability

4 Complexity

ξXSw(t) 1 minus ρ0( 1113857

1113936kj

rarrkXj p kj

rarr1113874 1113875ΘX kj

rarr t1113874 1113875

langkXj rang

(11)

where the expression kXj p(kj

rarr)langkX

j rang denotes the probabilityan edge connecting to a neighbor with degree kX

j in layer XNext we proceed with the evolution of ξX

Aw(t) andξX

Rw(t) in the layer X isin A B If through an edge withweight w the information is successfully transmitted to theneighbor in layer X with probability λX

w the change in θXw(t)

is calculated as

dθXw(t)

dt minus λX

wξXAw(t) (12)

If neighbor j in the A-state has not transmitted in-formation to node i through an edge of weight w withprobability (1 minus λX

w) in layer X but enters into the R-statewith probability c at time t the value of ξX

Rw(t) will increaseand thus the dynamical differential value of ξX

Rw(t) can beobtained by

dξXRw(t)

dt c 1 minus λX

w1113872 1113873ξXAw(t) (13)

-e initial conditions are θXw(0) 1 and ξX

Rw(0) 0therefore combining equation (12) with (13) we can obtain

ξXRw(t)

c 1 minus λXw1113872 1113873 1 minus θX

w(t)1113960 1113961

λXw

(14)

After inserting equations (11) and (14) into equation (8)we can obtain

ξXAw(t) θX

w(t) minus 1 minus ρ0( 1113857

1113936kj

rarrkXj P kj

rarr1113874 1113875ΘX kj

rarr t1113874 1113875

langkXj rang

minusc 1 minus λX

w1113872 1113873 1 minus θXw(t)1113960 1113961

λXw

(15)

In equation (15) substituting ξXAw(t) with (1minus λX

w)

(dθXw(t)dt) according to equation (8) we further deduce

dθXw(t)

dt λX

w 1 minus ρ0( 1113857

1113936kj

rarrkXj P kj

rarr1113874 1113875ΘX kX

j t1113872 1113873

langkXj rang

+ c 1 minus λXw1113872 1113873

minus c +(1 minus c)λXw1113960 1113961θX

w(t)

(16)

When t⟶infin dθXw(t)dt 0 and we have the steady-

state value θXw(infin) as

θXw(infin)

λXw 1 minus ρ0( 11138571113936

kj

rarrkXj P kj

rarr1113874 1113875ΘX kj

rarrinfin1113874 1113875 +langkX

j rangc 1 minus λXw1113872 1113873

langkXj rang c +(1 minus c)λX

w1113960 1113961

(17)

Inserting θXw(infin) into equations (2) and (3) we thus

obtain

θX(infin) 1113944w

gX(w)θXw(infin)

fX θA(infin) θB(infin)( 1113857

(18)

For a convenient expression we rewriteθA(infin) fA(θA(infin) θA(infin)) as

θA FA θB( 1113857 (19)

and θB(infin) fB(θA(infin) θB(infin)) as

θB FB θA( 1113857 (20)

A discontinuous growth pattern will occur [68] whenequation (19) is tangent to equation (20) with θA lt 1 andθB lt 1 Moreover critical conditions can be acquired fromthe following equation

zfA θA(infin) θB(infin)( 1113857

zθB(infin)

zfB θA(infin) θB(infin)( 1113857

zθA(infin) 1 (21)

From the theoretical analysis above the general solutionsfor adoption thresholds TA and TB can be derived but thetheoretical analysis itself may not be easily comprehended-erefore an intuitive illustration exemplifies the generaltheory with a particular case ie TA TB 1 In this caseequations (9) and (10) are rewritten as

ΘA kj

rarr t1113874 1113875 θA(t)

kAj minus 1

+ θB(t)kB

j minus θA(t)kA

j minus 1θB(t)kB

j

ΘB kj

rarr t1113874 1113875 θA(t)

kAj + θB(t)

kBj minus 1

minus θA(t)kA

j θB(t)kB

j minus 1

(22)

Owing to independence we obtain P(kj

rarr) P(kA

j )P(kBj )

and substitute (23) and (24) into equation (17) as follows

θAw(infin)

λAw 1 minus ρ0( 1113857 H1

A θA(infin)( 1113857 + H0B θB(infin)( 1113857 minus H10

AB θA(infin) θB(infin)( 11138571113960 1113961

c +(1 minus c)λAw1113960 1113961

+c 1 minus λA

w1113872 1113873

c +(1 minus c)λAw1113960 1113961

(23)

θBw(infin)

λBw 1 minus ρ0( 1113857 H0

A θB(infin)( 1113857 + H1B θA(infin)( 1113857 minus H01

AB θA(infin) θB(infin)( 11138571113960 1113961

c +(1 minus c)λBw1113960 1113961

+c 1 minus λB

w1113872 1113873

c +(1 minus c)λBw1113960 1113961

(24)

Complexity 5

where H0X(x) 1113936kX

PX(kX)xkX represents the generationfunction of degree distribution PX(kX) X isin A B andH1

X(x) ((1113936kXkXP(kX)xkXminus 1)langkXrang) X isin A B repre-

sents the generation function of excess degree distributionPX(kX) Additionally the generation functions of degreedistribution P( k

rarr) are expressed as follows

H10AB(x y)

1HA0prime (1)

zH00AB(x y)

zx

H01AB(x y)

1HB0prime (1)

zH00AB(x y)

zy

H00AB(x y) 1113944

kAkB

P( krarr

)xkA y

kB

(25)

where krarr

(kA kB) Subsequently substituting equations(23) and (24) into equation (21) we can deduce the criticalconditions

4 Numerical Method

Based on ErdosndashRenyi (ER) [69] and the scale-free (SF) [62]random network model extensive numerical simulations onartificial two-layer networks were performed to verify thetheoretical analysis above Without loss of generality we setthe network size N 104 and mean-degree langkrangA

langkrangB langkrang 10 For convenience the information trans-mission unit probability in the two layers is set as βA βB

β Additionally the weight distribution of the network layerX isin A B follows gX(ω)simωminus αX

ω with ωmaxX simN1(αX

ω minus 1) andmean weight langωrangA langωrangB langωrang 8 At the steady state inwhich no node in the A-state exists we compute thespreading range by measuring the fraction of nodes in theR-state R(infin) -e simulation results are obtained by av-eraging over at least 104 independent dynamical realizationson 30 artificial networks

To obtain the critical condition from the simulations therelative variance χ [70] is applied in our numerical verifi-cation using the following method

χ NlangR(infin)2rang minus langR(infin)rang2

langR(infin)rang (26)

where lang rang is the ensemble average and langR(infin)rang andlangR(infin)2rang are the first and secondary moments of R(infin)respectively -e peaks of the curve of relative variance χversus the information transmission probability correspondto the critical points

5 Results and Discussion

In reality from the perspective of a degree distributionmodel layers in some multiplex social networks exhibit thehomogeneous creation mechanism such as MSN versusICQ but layers in other multiplex social networks mayexhibit heterogeneous creation mechanisms such as ICQ

versus Facebook -erefore from the aspects of homoge-neous and heterogeneous degree distributions of twolayers the effects of weight distribution heterogeneity oninformation spreading in multiplex social networks isdiscussed in this section in terms of the parameters inSection 4

51 ER-ER Weighted Multiplex Networks First we in-vestigate information spreading on a two-layered weightedER-ER network with Poisson degree distribution pA(kA)

eminus langkArang(langkArangkA kA) in layer A and pB(kB) eminus langkBrang (langkBrangkB kB) in layer B

We first explore the time evolutions of node densities inthree states (S-state A-state and R-state) under differentweight distribution exponents αA

ω and αBω as shown in

Figure 2 Under the same information transmission unitprobability β but different adoption thresholds and seedsizes Figure 2 shows the plots of curves with TA 1 TB

2 ρ0 003 in the top panel and TA 1 TB 3 ρ0 01 inthe bottom panel Regardless of the top or the bottom panelwith time t the size of S(t) decreases gradually from 1 minus ρ0 to0 A(t) changes from ρ0 to 0 eventually and R(t) indicatingthe final adoption size increases gradually from 0 to 1 at theend With the increase in the weight distribution exponentfrom αA

ω 3 αBω 25 in Figures 2(a) and 2(e) αA

ω 3 αBω

3 in Figures 2(b) and 2(f) and αAω 3 αB

ω 35 inFigures 2(c) and 2(g) the evolution time consumed de-creases from 19 to 13 in the top panel and 11 to 9 in thebottom panel Additionally the final adoption size R(infin)

augments at the same time t (see the comparison inFigures 2(d) and 2(h))-e phenomena occurring during thetime evolution suggest that increasing the weight distribu-tion exponent (reducing weight distribution heterogeneity)can facilitate information spreading in a weighted multiplexnetwork

Next given the adoption thresholds TATB and seed sizeρ0 we investigate the growth pattern of the final adoptionsize R(infin) versus information transmission unit probabilityβ In Figure 3(a) with TA 1 TB 1 and ρ0 0005 in-creasing the weight distribution exponent such as from αA

ω

3 αBω 25 to αA

ω 3 αBω 35 does not change the growth

pattern of the final adoption size R(infin) versus β but reducesthe critical information transmission unit probability βcwhich can be numerically captured by the peaks of relativevariance χ in Figure 3(e) with the same settings ofFigure 3(a) Figure 3(b) with different TA 1 TB

2 ρ0 003 exhibits the same pattern as Figure 3(a) withthe peaks of relative variance χ in Figure 3(f) Even enlargingthe seed size (ρ0 01) in Figure 3(c) with the same adoptionthreshold of Figure 3(b) the final adoption size R(infin) versusβ increases in the same pattern regardless of the increase inweight distribution Besides keeping the seed size ρ0 as thesame of Figure 3(c) (ρ0 01) Figure 3(d) with furtherincrease of the adoption threshold TA 1 TB 3 alsoshows similar phenomena as those in Figures 3(a)ndash3(c)verified by the peaks of χ in Figure 3(h) -e above phe-nomena suggest that increasing the weight distributionexponent (decreasing weight distribution heterogeneity) will

6 Complexity

S (t)A (t)R (t)

00

05

10R

(t)

2 4 6 8 10 12 14 16 180t

(a)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 12 140t

(b)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 120t

(c)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

2 4 6 8 10 12 14 16 180t

(d)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(e)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 100t

(f )

Figure 2 Continued

Complexity 7

S (t)A (t)R (t)

00

05

10R

(t)

1 2 3 4 5 6 7 8 90t

(g)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(h)

Figure 2 Time evolutions of node densities in three states (S-state A-state and R-state) on the ER-ER weighted multiplex network underdifferent weight distribution exponents αA

ω and αBω To unveil the effect of the threshold and seed size two groups of subgraphs with

TA 1 TB 2 ρ0 003 and TA 1 TB 3 ρ0 01 are shown in the top and bottom panels respectively Under the same informationtransmission unit probability β subgraphs (a) (b) and (c) from the top panel have similar parameters of αA

ω 3 αBω 25 αA

ω 3 αBω 3

and αAω 3 αB

ω 35 with (e) (f ) and (g) respectively Particularly subgraphs (d) and (h) emphasize the time evolution of the R-state nodedensity R(t) under different weight distribution exponents In both panels despite the different thresholds and seed sizes at the same timestep the density of nodes in the R-state under αA

ω 3 αBω 35 exceeds other densities under αA

ω 3 αBω 25 and αA

ω 3 αBω 3 -ese

phenomena suggest that large weight distribution exponents (small weight distribution heterogeneity) facilitate information spreading -ebasic parameters are N 104 langkrang 10 langωrang 8 c 10 and β 03

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

00 01 02β

03 04

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

Figure 3 Continued

8 Complexity

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(d)

00 01 02β

03

30

20

10

χ

0

(e)

00 01 02β

03 04

6

4

2

χ

0

(f )

00 01 02β

0300

02

04

χ

(g)

00 01 02β

0300

05

10

15

χ

(h)

Figure 3 Growth pattern of the final adoption sizeR(infin) versus information transmission unit probability β on the ER-ERweightedmultiplexnetwork under different weight distribution exponents αA

ω and αBω given adoption thresholds (TA TB) and initial fraction ρ0 of seeds-e final

adoption size R(infin) versus information transmission probability β with (a) TA 1 TB 1 ρ0 0005 (b) TA 1 TB 2 ρ0 003 (c)TA 1 TB 2 ρ0 01 and (d) TA 1 TB 3 ρ0 01 -e lines denoting theoretical solutions agree exceptionally well with the symbolsrepresenting numerical simulations Moreover in subgraphs (e) (f) (g) and (h) the relative variance χ in terms of the peaks numericallyindicates the critical spreading probabilities in (a) (b) (c) and (d) respectively showing that increasing the weight distribution exponent(decreasing weight distribution heterogeneity) can decrease the critical spreading probability -e phenomena above suggest that decreasingweight distribution heterogeneity facilitates information spreading but cannot alter the growth pattern of the final adoption size versusinformation transmission probability -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 9

not alter the growth pattern of R(infin) versus β in any pair ofadoption thresholds but can promote the outbreak of in-formation spreading Noticeably the theoretical solutions(lines) coincide well with the numerical simulations(symbols)

As shown in Figure 3 the initial seed size affects theinformation spreading dynamics We investigate the effectof seed size on information spreading on a weightedmultiplex social network as shown in Figure 4 InFigure 4(a) (TA 1 TB 1 αA

w 3 αBw 25) increasing

the seed size from ρ0 0005 to ρ0 001 not only changesthe growth pattern from discontinuous to continuous butalso decreases the critical probability to promote theoutbreak of information spreading Subsequently if theweight distribution exponent is enlarged such asαA

w 3 αBw 3 in Figure 4(d) and αA

w 3 αBw 35 in

Figure 4(g) increasing the seed size ρ0 still advances in-formation propagation Comparing Figure 4(a) 4(d) and4(g) vertically altering the initial fraction can change thegrowth pattern of R(infin) versus β but cannot change the factthat increasing the weight distribution exponent (reducingweight distribution heterogeneity) retains the growthpattern while facilitating the outbreak of informationspreading Moreover in a horizontal comparison althoughenlarging the adoption thresholds such as TA 1 TB 2in Figures 4(b) 4(e) and 4(h) or TA 1 TB 3 inFigure 4(c) 4(f ) and 4(i) renders information adoptionmore difficult it is clear that increasing ρ0 still transformsthe growth pattern of R(infin) from discontinuous to con-tinuous phase transition and promotes the outbreak ofinformation spreading as in the case of TA 1 TB 1Additionally changes in adoption thresholds and seed sizedo not qualitatively alter phenomena

For a more comprehensive study we obtained the resultsof the final adoption size on the (β ρ0) parameter plane asillustrated in Figure 5 Here we set the subgraphs in column1 with TA 1 TB 1 in column 2 with TA 1 TB 2 incolumn 3 with TA 1 TB 3 in row 1 withαA

w 3 αBw 25 in row 2 with αA

w 3 αBw 3 and in row 3

with αAw 3 αB

w 35 In the following we discuss thefindings based on Figure 5 in detail In Figure 5(a)(TA 1 TB 1 αA

w 3 αBw 25) a critical ρlowast0 0003 ex-

ists below which at any information transmission unitprobability β no outbreak of information spreading willoccur and above which outbreak and global adoption willoccur In a vertical comparison we retain the adoptionthresholds in Figures 5(a) 5(d) and 5(h) and further in-crease the weight distribution exponents in Figure 5(d)(αA

w 3 αBw 3) and Figure 5(h) (αA

w 3 αBw 35) -e

results indicate that the three subgraphs have the samecritical seed size ρlowast0 0003 below which increasing theweight distribution exponent (reducing the weight distri-bution heterogeneity) still cannot induce the outbreak ofinformation spreading but above which increasing theweight distribution exponent can accelerate the outbreak at asmaller information transmission unit probability βMoreover in a horizontal comparison we retain the weightdistribution exponent in Figures 5(a) 5(b) and 5(c) withαA

w 3 αBw 25 and increase the adoption thresholds in

Figure 5(b) (TA 1 TB 2) and Figure 5(c)(TA 1 TB 3) Under these circumstances we discov-ered that the critical seed size ρlowast0 is enlarged from 0003 to006 which implies increasing the difficulty in informationadoption Analogously other subgraphs exhibit similarfindings -e phenomena in Figure 5 suggest that a criticalseed size exists above which outbreak of informationspreading and global adoption will occur and the weightdistribution exponent (reducing the weight distributionheterogeneity) is essential for facilitating informationspreading

52 ER-SF Weighted Multiplex Networks For cases of het-erogeneous degree distributions we investigate informationspreading on a two-layered weighted ER-SF network withPoisson degree distribution pA(kA) eminus langkArang(langkArangkA kA) inlayer A and with power-law degree distribution pB(kB)

ζBkminus vB

B in layer B where ζB 11113936kBkminus vB

B and parameter vB

denotes the degree exponent of layer B Additionally we setthe minimum degree kmin

B 4 and maximum degreekmax

B N

radic

On the weighted ER-SF networks of distinct degreedistribution exponents the growth pattern is shown inFigure 6 with the common setting TA 1 TB 3 in allsubgraphs In Figure 6(a) with degree distribution exponentvB 21 we discovered that enlarging the weight distribu-tion exponent such as from αA

w 3 αBw 25 to

αAw 3 αB

w 35 can increase the final adoption size at thesame β and decrease the critical probability to promote theoutbreak of information spreading Moreover we proceed toincrease the degree distribution exponent (reduce the degreedistribution heterogeneity) in Figure 6(b) (vB 3) andFigure 6(c) (vB 4) Consequently from Figure 6(a) to 6(c)given αA

w and αBw the growth pattern changes from con-

tinuous to the discontinuous growth pattern However atany given vB changing the weight distribution exponent willnot alter the growth pattern of R(infin) -e phenomena inFigure 6 suggest that changing the degree distributionheterogeneity will not qualitatively alter the fact that re-ducing weight distribution heterogeneity can promote in-formation spreading

6 Conclusions

With the development of the Internet and mobile devicessocial networks have prevailed in peoplersquos lives Typically anindividual does not only use one social network but multiplesocial networks simultaneously Moreover the informationdoes not only diffuse in one social network but also spreadssimultaneously on several social networks eg informationspreads on Twitter and Facebook simultaneously Because ofdifferent extents of intimacy among friends colleagues andfamily members connections in social networks should bemodeled with different weights As such a multiplex socialnetwork should be a multilayer weighted network -ere-fore information spreading on a multiplex social networkshould be investigated on a multilayer weighted networkUnfortunately the mechanism of information spreading on

10 Complexity

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(a)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(b)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(c)

ρ0 = 0005ρ0 = 001ρ0 = 01

01 02 03 04 0500β

00

05

10

R (infin

)

(d)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(e)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(f )

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(g)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(h)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(i)

Figure 4 Effect of fraction of initial seeds on the growth pattern of the final adoption size R(infin) versus β on the ER-ER weighted multiplexnetwork under different weight distribution exponents αA

ω and αBω -e values of weight distribution exponent αA

ω of all subgraphs are 3 andthe subgraphs in each row have the same αB

ω ie subgraphs (a) (b) and (c) correspond to αBω 25 (d) (e) and (f) correspond to αB

ω 3and (g) (h) and (i) correspond to αB

ω 35 Additionally the subgraphs in each column have the same adoption threshold ie subgraphs(a) (d) and (g) are set with TA 1 TB 1 (b) (e) and (h) with TA 1 TB 2 and (c) (f ) and (i) with TA 1 TB 3 -e theoreticalsolutions (lines) coincide well with the numerical results (symbols) Given other parameters increasing the fraction of initial seeds willchange the growth pattern of R(infin) from discontinuous to continuous growth Moreover vertically increasing the weight distributionexponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability horizontally increasing theadoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability-e phenomena aboveconvey that changing the seed size can alter the growth pattern and decreasing the adoption threshold can facilitate information spreadingMost importantly changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distributionheterogeneity can facilitate information spreading -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 11

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

ξXSw(t) 1 minus ρ0( 1113857

1113936kj

rarrkXj p kj

rarr1113874 1113875ΘX kj

rarr t1113874 1113875

langkXj rang

(11)

where the expression kXj p(kj

rarr)langkX

j rang denotes the probabilityan edge connecting to a neighbor with degree kX

j in layer XNext we proceed with the evolution of ξX

Aw(t) andξX

Rw(t) in the layer X isin A B If through an edge withweight w the information is successfully transmitted to theneighbor in layer X with probability λX

w the change in θXw(t)

is calculated as

dθXw(t)

dt minus λX

wξXAw(t) (12)

If neighbor j in the A-state has not transmitted in-formation to node i through an edge of weight w withprobability (1 minus λX

w) in layer X but enters into the R-statewith probability c at time t the value of ξX

Rw(t) will increaseand thus the dynamical differential value of ξX

Rw(t) can beobtained by

dξXRw(t)

dt c 1 minus λX

w1113872 1113873ξXAw(t) (13)

-e initial conditions are θXw(0) 1 and ξX

Rw(0) 0therefore combining equation (12) with (13) we can obtain

ξXRw(t)

c 1 minus λXw1113872 1113873 1 minus θX

w(t)1113960 1113961

λXw

(14)

After inserting equations (11) and (14) into equation (8)we can obtain

ξXAw(t) θX

w(t) minus 1 minus ρ0( 1113857

1113936kj

rarrkXj P kj

rarr1113874 1113875ΘX kj

rarr t1113874 1113875

langkXj rang

minusc 1 minus λX

w1113872 1113873 1 minus θXw(t)1113960 1113961

λXw

(15)

In equation (15) substituting ξXAw(t) with (1minus λX

w)

(dθXw(t)dt) according to equation (8) we further deduce

dθXw(t)

dt λX

w 1 minus ρ0( 1113857

1113936kj

rarrkXj P kj

rarr1113874 1113875ΘX kX

j t1113872 1113873

langkXj rang

+ c 1 minus λXw1113872 1113873

minus c +(1 minus c)λXw1113960 1113961θX

w(t)

(16)

When t⟶infin dθXw(t)dt 0 and we have the steady-

state value θXw(infin) as

θXw(infin)

λXw 1 minus ρ0( 11138571113936

kj

rarrkXj P kj

rarr1113874 1113875ΘX kj

rarrinfin1113874 1113875 +langkX

j rangc 1 minus λXw1113872 1113873

langkXj rang c +(1 minus c)λX

w1113960 1113961

(17)

Inserting θXw(infin) into equations (2) and (3) we thus

obtain

θX(infin) 1113944w

gX(w)θXw(infin)

fX θA(infin) θB(infin)( 1113857

(18)

For a convenient expression we rewriteθA(infin) fA(θA(infin) θA(infin)) as

θA FA θB( 1113857 (19)

and θB(infin) fB(θA(infin) θB(infin)) as

θB FB θA( 1113857 (20)

A discontinuous growth pattern will occur [68] whenequation (19) is tangent to equation (20) with θA lt 1 andθB lt 1 Moreover critical conditions can be acquired fromthe following equation

zfA θA(infin) θB(infin)( 1113857

zθB(infin)

zfB θA(infin) θB(infin)( 1113857

zθA(infin) 1 (21)

From the theoretical analysis above the general solutionsfor adoption thresholds TA and TB can be derived but thetheoretical analysis itself may not be easily comprehended-erefore an intuitive illustration exemplifies the generaltheory with a particular case ie TA TB 1 In this caseequations (9) and (10) are rewritten as

ΘA kj

rarr t1113874 1113875 θA(t)

kAj minus 1

+ θB(t)kB

j minus θA(t)kA

j minus 1θB(t)kB

j

ΘB kj

rarr t1113874 1113875 θA(t)

kAj + θB(t)

kBj minus 1

minus θA(t)kA

j θB(t)kB

j minus 1

(22)

Owing to independence we obtain P(kj

rarr) P(kA

j )P(kBj )

and substitute (23) and (24) into equation (17) as follows

θAw(infin)

λAw 1 minus ρ0( 1113857 H1

A θA(infin)( 1113857 + H0B θB(infin)( 1113857 minus H10

AB θA(infin) θB(infin)( 11138571113960 1113961

c +(1 minus c)λAw1113960 1113961

+c 1 minus λA

w1113872 1113873

c +(1 minus c)λAw1113960 1113961

(23)

θBw(infin)

λBw 1 minus ρ0( 1113857 H0

A θB(infin)( 1113857 + H1B θA(infin)( 1113857 minus H01

AB θA(infin) θB(infin)( 11138571113960 1113961

c +(1 minus c)λBw1113960 1113961

+c 1 minus λB

w1113872 1113873

c +(1 minus c)λBw1113960 1113961

(24)

Complexity 5

where H0X(x) 1113936kX

PX(kX)xkX represents the generationfunction of degree distribution PX(kX) X isin A B andH1

X(x) ((1113936kXkXP(kX)xkXminus 1)langkXrang) X isin A B repre-

sents the generation function of excess degree distributionPX(kX) Additionally the generation functions of degreedistribution P( k

rarr) are expressed as follows

H10AB(x y)

1HA0prime (1)

zH00AB(x y)

zx

H01AB(x y)

1HB0prime (1)

zH00AB(x y)

zy

H00AB(x y) 1113944

kAkB

P( krarr

)xkA y

kB

(25)

where krarr

(kA kB) Subsequently substituting equations(23) and (24) into equation (21) we can deduce the criticalconditions

4 Numerical Method

Based on ErdosndashRenyi (ER) [69] and the scale-free (SF) [62]random network model extensive numerical simulations onartificial two-layer networks were performed to verify thetheoretical analysis above Without loss of generality we setthe network size N 104 and mean-degree langkrangA

langkrangB langkrang 10 For convenience the information trans-mission unit probability in the two layers is set as βA βB

β Additionally the weight distribution of the network layerX isin A B follows gX(ω)simωminus αX

ω with ωmaxX simN1(αX

ω minus 1) andmean weight langωrangA langωrangB langωrang 8 At the steady state inwhich no node in the A-state exists we compute thespreading range by measuring the fraction of nodes in theR-state R(infin) -e simulation results are obtained by av-eraging over at least 104 independent dynamical realizationson 30 artificial networks

To obtain the critical condition from the simulations therelative variance χ [70] is applied in our numerical verifi-cation using the following method

χ NlangR(infin)2rang minus langR(infin)rang2

langR(infin)rang (26)

where lang rang is the ensemble average and langR(infin)rang andlangR(infin)2rang are the first and secondary moments of R(infin)respectively -e peaks of the curve of relative variance χversus the information transmission probability correspondto the critical points

5 Results and Discussion

In reality from the perspective of a degree distributionmodel layers in some multiplex social networks exhibit thehomogeneous creation mechanism such as MSN versusICQ but layers in other multiplex social networks mayexhibit heterogeneous creation mechanisms such as ICQ

versus Facebook -erefore from the aspects of homoge-neous and heterogeneous degree distributions of twolayers the effects of weight distribution heterogeneity oninformation spreading in multiplex social networks isdiscussed in this section in terms of the parameters inSection 4

51 ER-ER Weighted Multiplex Networks First we in-vestigate information spreading on a two-layered weightedER-ER network with Poisson degree distribution pA(kA)

eminus langkArang(langkArangkA kA) in layer A and pB(kB) eminus langkBrang (langkBrangkB kB) in layer B

We first explore the time evolutions of node densities inthree states (S-state A-state and R-state) under differentweight distribution exponents αA

ω and αBω as shown in

Figure 2 Under the same information transmission unitprobability β but different adoption thresholds and seedsizes Figure 2 shows the plots of curves with TA 1 TB

2 ρ0 003 in the top panel and TA 1 TB 3 ρ0 01 inthe bottom panel Regardless of the top or the bottom panelwith time t the size of S(t) decreases gradually from 1 minus ρ0 to0 A(t) changes from ρ0 to 0 eventually and R(t) indicatingthe final adoption size increases gradually from 0 to 1 at theend With the increase in the weight distribution exponentfrom αA

ω 3 αBω 25 in Figures 2(a) and 2(e) αA

ω 3 αBω

3 in Figures 2(b) and 2(f) and αAω 3 αB

ω 35 inFigures 2(c) and 2(g) the evolution time consumed de-creases from 19 to 13 in the top panel and 11 to 9 in thebottom panel Additionally the final adoption size R(infin)

augments at the same time t (see the comparison inFigures 2(d) and 2(h))-e phenomena occurring during thetime evolution suggest that increasing the weight distribu-tion exponent (reducing weight distribution heterogeneity)can facilitate information spreading in a weighted multiplexnetwork

Next given the adoption thresholds TATB and seed sizeρ0 we investigate the growth pattern of the final adoptionsize R(infin) versus information transmission unit probabilityβ In Figure 3(a) with TA 1 TB 1 and ρ0 0005 in-creasing the weight distribution exponent such as from αA

ω

3 αBω 25 to αA

ω 3 αBω 35 does not change the growth

pattern of the final adoption size R(infin) versus β but reducesthe critical information transmission unit probability βcwhich can be numerically captured by the peaks of relativevariance χ in Figure 3(e) with the same settings ofFigure 3(a) Figure 3(b) with different TA 1 TB

2 ρ0 003 exhibits the same pattern as Figure 3(a) withthe peaks of relative variance χ in Figure 3(f) Even enlargingthe seed size (ρ0 01) in Figure 3(c) with the same adoptionthreshold of Figure 3(b) the final adoption size R(infin) versusβ increases in the same pattern regardless of the increase inweight distribution Besides keeping the seed size ρ0 as thesame of Figure 3(c) (ρ0 01) Figure 3(d) with furtherincrease of the adoption threshold TA 1 TB 3 alsoshows similar phenomena as those in Figures 3(a)ndash3(c)verified by the peaks of χ in Figure 3(h) -e above phe-nomena suggest that increasing the weight distributionexponent (decreasing weight distribution heterogeneity) will

6 Complexity

S (t)A (t)R (t)

00

05

10R

(t)

2 4 6 8 10 12 14 16 180t

(a)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 12 140t

(b)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 120t

(c)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

2 4 6 8 10 12 14 16 180t

(d)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(e)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 100t

(f )

Figure 2 Continued

Complexity 7

S (t)A (t)R (t)

00

05

10R

(t)

1 2 3 4 5 6 7 8 90t

(g)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(h)

Figure 2 Time evolutions of node densities in three states (S-state A-state and R-state) on the ER-ER weighted multiplex network underdifferent weight distribution exponents αA

ω and αBω To unveil the effect of the threshold and seed size two groups of subgraphs with

TA 1 TB 2 ρ0 003 and TA 1 TB 3 ρ0 01 are shown in the top and bottom panels respectively Under the same informationtransmission unit probability β subgraphs (a) (b) and (c) from the top panel have similar parameters of αA

ω 3 αBω 25 αA

ω 3 αBω 3

and αAω 3 αB

ω 35 with (e) (f ) and (g) respectively Particularly subgraphs (d) and (h) emphasize the time evolution of the R-state nodedensity R(t) under different weight distribution exponents In both panels despite the different thresholds and seed sizes at the same timestep the density of nodes in the R-state under αA

ω 3 αBω 35 exceeds other densities under αA

ω 3 αBω 25 and αA

ω 3 αBω 3 -ese

phenomena suggest that large weight distribution exponents (small weight distribution heterogeneity) facilitate information spreading -ebasic parameters are N 104 langkrang 10 langωrang 8 c 10 and β 03

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

00 01 02β

03 04

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

Figure 3 Continued

8 Complexity

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(d)

00 01 02β

03

30

20

10

χ

0

(e)

00 01 02β

03 04

6

4

2

χ

0

(f )

00 01 02β

0300

02

04

χ

(g)

00 01 02β

0300

05

10

15

χ

(h)

Figure 3 Growth pattern of the final adoption sizeR(infin) versus information transmission unit probability β on the ER-ERweightedmultiplexnetwork under different weight distribution exponents αA

ω and αBω given adoption thresholds (TA TB) and initial fraction ρ0 of seeds-e final

adoption size R(infin) versus information transmission probability β with (a) TA 1 TB 1 ρ0 0005 (b) TA 1 TB 2 ρ0 003 (c)TA 1 TB 2 ρ0 01 and (d) TA 1 TB 3 ρ0 01 -e lines denoting theoretical solutions agree exceptionally well with the symbolsrepresenting numerical simulations Moreover in subgraphs (e) (f) (g) and (h) the relative variance χ in terms of the peaks numericallyindicates the critical spreading probabilities in (a) (b) (c) and (d) respectively showing that increasing the weight distribution exponent(decreasing weight distribution heterogeneity) can decrease the critical spreading probability -e phenomena above suggest that decreasingweight distribution heterogeneity facilitates information spreading but cannot alter the growth pattern of the final adoption size versusinformation transmission probability -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 9

not alter the growth pattern of R(infin) versus β in any pair ofadoption thresholds but can promote the outbreak of in-formation spreading Noticeably the theoretical solutions(lines) coincide well with the numerical simulations(symbols)

As shown in Figure 3 the initial seed size affects theinformation spreading dynamics We investigate the effectof seed size on information spreading on a weightedmultiplex social network as shown in Figure 4 InFigure 4(a) (TA 1 TB 1 αA

w 3 αBw 25) increasing

the seed size from ρ0 0005 to ρ0 001 not only changesthe growth pattern from discontinuous to continuous butalso decreases the critical probability to promote theoutbreak of information spreading Subsequently if theweight distribution exponent is enlarged such asαA

w 3 αBw 3 in Figure 4(d) and αA

w 3 αBw 35 in

Figure 4(g) increasing the seed size ρ0 still advances in-formation propagation Comparing Figure 4(a) 4(d) and4(g) vertically altering the initial fraction can change thegrowth pattern of R(infin) versus β but cannot change the factthat increasing the weight distribution exponent (reducingweight distribution heterogeneity) retains the growthpattern while facilitating the outbreak of informationspreading Moreover in a horizontal comparison althoughenlarging the adoption thresholds such as TA 1 TB 2in Figures 4(b) 4(e) and 4(h) or TA 1 TB 3 inFigure 4(c) 4(f ) and 4(i) renders information adoptionmore difficult it is clear that increasing ρ0 still transformsthe growth pattern of R(infin) from discontinuous to con-tinuous phase transition and promotes the outbreak ofinformation spreading as in the case of TA 1 TB 1Additionally changes in adoption thresholds and seed sizedo not qualitatively alter phenomena

For a more comprehensive study we obtained the resultsof the final adoption size on the (β ρ0) parameter plane asillustrated in Figure 5 Here we set the subgraphs in column1 with TA 1 TB 1 in column 2 with TA 1 TB 2 incolumn 3 with TA 1 TB 3 in row 1 withαA

w 3 αBw 25 in row 2 with αA

w 3 αBw 3 and in row 3

with αAw 3 αB

w 35 In the following we discuss thefindings based on Figure 5 in detail In Figure 5(a)(TA 1 TB 1 αA

w 3 αBw 25) a critical ρlowast0 0003 ex-

ists below which at any information transmission unitprobability β no outbreak of information spreading willoccur and above which outbreak and global adoption willoccur In a vertical comparison we retain the adoptionthresholds in Figures 5(a) 5(d) and 5(h) and further in-crease the weight distribution exponents in Figure 5(d)(αA

w 3 αBw 3) and Figure 5(h) (αA

w 3 αBw 35) -e

results indicate that the three subgraphs have the samecritical seed size ρlowast0 0003 below which increasing theweight distribution exponent (reducing the weight distri-bution heterogeneity) still cannot induce the outbreak ofinformation spreading but above which increasing theweight distribution exponent can accelerate the outbreak at asmaller information transmission unit probability βMoreover in a horizontal comparison we retain the weightdistribution exponent in Figures 5(a) 5(b) and 5(c) withαA

w 3 αBw 25 and increase the adoption thresholds in

Figure 5(b) (TA 1 TB 2) and Figure 5(c)(TA 1 TB 3) Under these circumstances we discov-ered that the critical seed size ρlowast0 is enlarged from 0003 to006 which implies increasing the difficulty in informationadoption Analogously other subgraphs exhibit similarfindings -e phenomena in Figure 5 suggest that a criticalseed size exists above which outbreak of informationspreading and global adoption will occur and the weightdistribution exponent (reducing the weight distributionheterogeneity) is essential for facilitating informationspreading

52 ER-SF Weighted Multiplex Networks For cases of het-erogeneous degree distributions we investigate informationspreading on a two-layered weighted ER-SF network withPoisson degree distribution pA(kA) eminus langkArang(langkArangkA kA) inlayer A and with power-law degree distribution pB(kB)

ζBkminus vB

B in layer B where ζB 11113936kBkminus vB

B and parameter vB

denotes the degree exponent of layer B Additionally we setthe minimum degree kmin

B 4 and maximum degreekmax

B N

radic

On the weighted ER-SF networks of distinct degreedistribution exponents the growth pattern is shown inFigure 6 with the common setting TA 1 TB 3 in allsubgraphs In Figure 6(a) with degree distribution exponentvB 21 we discovered that enlarging the weight distribu-tion exponent such as from αA

w 3 αBw 25 to

αAw 3 αB

w 35 can increase the final adoption size at thesame β and decrease the critical probability to promote theoutbreak of information spreading Moreover we proceed toincrease the degree distribution exponent (reduce the degreedistribution heterogeneity) in Figure 6(b) (vB 3) andFigure 6(c) (vB 4) Consequently from Figure 6(a) to 6(c)given αA

w and αBw the growth pattern changes from con-

tinuous to the discontinuous growth pattern However atany given vB changing the weight distribution exponent willnot alter the growth pattern of R(infin) -e phenomena inFigure 6 suggest that changing the degree distributionheterogeneity will not qualitatively alter the fact that re-ducing weight distribution heterogeneity can promote in-formation spreading

6 Conclusions

With the development of the Internet and mobile devicessocial networks have prevailed in peoplersquos lives Typically anindividual does not only use one social network but multiplesocial networks simultaneously Moreover the informationdoes not only diffuse in one social network but also spreadssimultaneously on several social networks eg informationspreads on Twitter and Facebook simultaneously Because ofdifferent extents of intimacy among friends colleagues andfamily members connections in social networks should bemodeled with different weights As such a multiplex socialnetwork should be a multilayer weighted network -ere-fore information spreading on a multiplex social networkshould be investigated on a multilayer weighted networkUnfortunately the mechanism of information spreading on

10 Complexity

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(a)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(b)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(c)

ρ0 = 0005ρ0 = 001ρ0 = 01

01 02 03 04 0500β

00

05

10

R (infin

)

(d)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(e)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(f )

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(g)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(h)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(i)

Figure 4 Effect of fraction of initial seeds on the growth pattern of the final adoption size R(infin) versus β on the ER-ER weighted multiplexnetwork under different weight distribution exponents αA

ω and αBω -e values of weight distribution exponent αA

ω of all subgraphs are 3 andthe subgraphs in each row have the same αB

ω ie subgraphs (a) (b) and (c) correspond to αBω 25 (d) (e) and (f) correspond to αB

ω 3and (g) (h) and (i) correspond to αB

ω 35 Additionally the subgraphs in each column have the same adoption threshold ie subgraphs(a) (d) and (g) are set with TA 1 TB 1 (b) (e) and (h) with TA 1 TB 2 and (c) (f ) and (i) with TA 1 TB 3 -e theoreticalsolutions (lines) coincide well with the numerical results (symbols) Given other parameters increasing the fraction of initial seeds willchange the growth pattern of R(infin) from discontinuous to continuous growth Moreover vertically increasing the weight distributionexponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability horizontally increasing theadoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability-e phenomena aboveconvey that changing the seed size can alter the growth pattern and decreasing the adoption threshold can facilitate information spreadingMost importantly changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distributionheterogeneity can facilitate information spreading -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 11

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

where H0X(x) 1113936kX

PX(kX)xkX represents the generationfunction of degree distribution PX(kX) X isin A B andH1

X(x) ((1113936kXkXP(kX)xkXminus 1)langkXrang) X isin A B repre-

sents the generation function of excess degree distributionPX(kX) Additionally the generation functions of degreedistribution P( k

rarr) are expressed as follows

H10AB(x y)

1HA0prime (1)

zH00AB(x y)

zx

H01AB(x y)

1HB0prime (1)

zH00AB(x y)

zy

H00AB(x y) 1113944

kAkB

P( krarr

)xkA y

kB

(25)

where krarr

(kA kB) Subsequently substituting equations(23) and (24) into equation (21) we can deduce the criticalconditions

4 Numerical Method

Based on ErdosndashRenyi (ER) [69] and the scale-free (SF) [62]random network model extensive numerical simulations onartificial two-layer networks were performed to verify thetheoretical analysis above Without loss of generality we setthe network size N 104 and mean-degree langkrangA

langkrangB langkrang 10 For convenience the information trans-mission unit probability in the two layers is set as βA βB

β Additionally the weight distribution of the network layerX isin A B follows gX(ω)simωminus αX

ω with ωmaxX simN1(αX

ω minus 1) andmean weight langωrangA langωrangB langωrang 8 At the steady state inwhich no node in the A-state exists we compute thespreading range by measuring the fraction of nodes in theR-state R(infin) -e simulation results are obtained by av-eraging over at least 104 independent dynamical realizationson 30 artificial networks

To obtain the critical condition from the simulations therelative variance χ [70] is applied in our numerical verifi-cation using the following method

χ NlangR(infin)2rang minus langR(infin)rang2

langR(infin)rang (26)

where lang rang is the ensemble average and langR(infin)rang andlangR(infin)2rang are the first and secondary moments of R(infin)respectively -e peaks of the curve of relative variance χversus the information transmission probability correspondto the critical points

5 Results and Discussion

In reality from the perspective of a degree distributionmodel layers in some multiplex social networks exhibit thehomogeneous creation mechanism such as MSN versusICQ but layers in other multiplex social networks mayexhibit heterogeneous creation mechanisms such as ICQ

versus Facebook -erefore from the aspects of homoge-neous and heterogeneous degree distributions of twolayers the effects of weight distribution heterogeneity oninformation spreading in multiplex social networks isdiscussed in this section in terms of the parameters inSection 4

51 ER-ER Weighted Multiplex Networks First we in-vestigate information spreading on a two-layered weightedER-ER network with Poisson degree distribution pA(kA)

eminus langkArang(langkArangkA kA) in layer A and pB(kB) eminus langkBrang (langkBrangkB kB) in layer B

We first explore the time evolutions of node densities inthree states (S-state A-state and R-state) under differentweight distribution exponents αA

ω and αBω as shown in

Figure 2 Under the same information transmission unitprobability β but different adoption thresholds and seedsizes Figure 2 shows the plots of curves with TA 1 TB

2 ρ0 003 in the top panel and TA 1 TB 3 ρ0 01 inthe bottom panel Regardless of the top or the bottom panelwith time t the size of S(t) decreases gradually from 1 minus ρ0 to0 A(t) changes from ρ0 to 0 eventually and R(t) indicatingthe final adoption size increases gradually from 0 to 1 at theend With the increase in the weight distribution exponentfrom αA

ω 3 αBω 25 in Figures 2(a) and 2(e) αA

ω 3 αBω

3 in Figures 2(b) and 2(f) and αAω 3 αB

ω 35 inFigures 2(c) and 2(g) the evolution time consumed de-creases from 19 to 13 in the top panel and 11 to 9 in thebottom panel Additionally the final adoption size R(infin)

augments at the same time t (see the comparison inFigures 2(d) and 2(h))-e phenomena occurring during thetime evolution suggest that increasing the weight distribu-tion exponent (reducing weight distribution heterogeneity)can facilitate information spreading in a weighted multiplexnetwork

Next given the adoption thresholds TATB and seed sizeρ0 we investigate the growth pattern of the final adoptionsize R(infin) versus information transmission unit probabilityβ In Figure 3(a) with TA 1 TB 1 and ρ0 0005 in-creasing the weight distribution exponent such as from αA

ω

3 αBω 25 to αA

ω 3 αBω 35 does not change the growth

pattern of the final adoption size R(infin) versus β but reducesthe critical information transmission unit probability βcwhich can be numerically captured by the peaks of relativevariance χ in Figure 3(e) with the same settings ofFigure 3(a) Figure 3(b) with different TA 1 TB

2 ρ0 003 exhibits the same pattern as Figure 3(a) withthe peaks of relative variance χ in Figure 3(f) Even enlargingthe seed size (ρ0 01) in Figure 3(c) with the same adoptionthreshold of Figure 3(b) the final adoption size R(infin) versusβ increases in the same pattern regardless of the increase inweight distribution Besides keeping the seed size ρ0 as thesame of Figure 3(c) (ρ0 01) Figure 3(d) with furtherincrease of the adoption threshold TA 1 TB 3 alsoshows similar phenomena as those in Figures 3(a)ndash3(c)verified by the peaks of χ in Figure 3(h) -e above phe-nomena suggest that increasing the weight distributionexponent (decreasing weight distribution heterogeneity) will

6 Complexity

S (t)A (t)R (t)

00

05

10R

(t)

2 4 6 8 10 12 14 16 180t

(a)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 12 140t

(b)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 120t

(c)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

2 4 6 8 10 12 14 16 180t

(d)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(e)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 100t

(f )

Figure 2 Continued

Complexity 7

S (t)A (t)R (t)

00

05

10R

(t)

1 2 3 4 5 6 7 8 90t

(g)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(h)

Figure 2 Time evolutions of node densities in three states (S-state A-state and R-state) on the ER-ER weighted multiplex network underdifferent weight distribution exponents αA

ω and αBω To unveil the effect of the threshold and seed size two groups of subgraphs with

TA 1 TB 2 ρ0 003 and TA 1 TB 3 ρ0 01 are shown in the top and bottom panels respectively Under the same informationtransmission unit probability β subgraphs (a) (b) and (c) from the top panel have similar parameters of αA

ω 3 αBω 25 αA

ω 3 αBω 3

and αAω 3 αB

ω 35 with (e) (f ) and (g) respectively Particularly subgraphs (d) and (h) emphasize the time evolution of the R-state nodedensity R(t) under different weight distribution exponents In both panels despite the different thresholds and seed sizes at the same timestep the density of nodes in the R-state under αA

ω 3 αBω 35 exceeds other densities under αA

ω 3 αBω 25 and αA

ω 3 αBω 3 -ese

phenomena suggest that large weight distribution exponents (small weight distribution heterogeneity) facilitate information spreading -ebasic parameters are N 104 langkrang 10 langωrang 8 c 10 and β 03

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

00 01 02β

03 04

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

Figure 3 Continued

8 Complexity

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(d)

00 01 02β

03

30

20

10

χ

0

(e)

00 01 02β

03 04

6

4

2

χ

0

(f )

00 01 02β

0300

02

04

χ

(g)

00 01 02β

0300

05

10

15

χ

(h)

Figure 3 Growth pattern of the final adoption sizeR(infin) versus information transmission unit probability β on the ER-ERweightedmultiplexnetwork under different weight distribution exponents αA

ω and αBω given adoption thresholds (TA TB) and initial fraction ρ0 of seeds-e final

adoption size R(infin) versus information transmission probability β with (a) TA 1 TB 1 ρ0 0005 (b) TA 1 TB 2 ρ0 003 (c)TA 1 TB 2 ρ0 01 and (d) TA 1 TB 3 ρ0 01 -e lines denoting theoretical solutions agree exceptionally well with the symbolsrepresenting numerical simulations Moreover in subgraphs (e) (f) (g) and (h) the relative variance χ in terms of the peaks numericallyindicates the critical spreading probabilities in (a) (b) (c) and (d) respectively showing that increasing the weight distribution exponent(decreasing weight distribution heterogeneity) can decrease the critical spreading probability -e phenomena above suggest that decreasingweight distribution heterogeneity facilitates information spreading but cannot alter the growth pattern of the final adoption size versusinformation transmission probability -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 9

not alter the growth pattern of R(infin) versus β in any pair ofadoption thresholds but can promote the outbreak of in-formation spreading Noticeably the theoretical solutions(lines) coincide well with the numerical simulations(symbols)

As shown in Figure 3 the initial seed size affects theinformation spreading dynamics We investigate the effectof seed size on information spreading on a weightedmultiplex social network as shown in Figure 4 InFigure 4(a) (TA 1 TB 1 αA

w 3 αBw 25) increasing

the seed size from ρ0 0005 to ρ0 001 not only changesthe growth pattern from discontinuous to continuous butalso decreases the critical probability to promote theoutbreak of information spreading Subsequently if theweight distribution exponent is enlarged such asαA

w 3 αBw 3 in Figure 4(d) and αA

w 3 αBw 35 in

Figure 4(g) increasing the seed size ρ0 still advances in-formation propagation Comparing Figure 4(a) 4(d) and4(g) vertically altering the initial fraction can change thegrowth pattern of R(infin) versus β but cannot change the factthat increasing the weight distribution exponent (reducingweight distribution heterogeneity) retains the growthpattern while facilitating the outbreak of informationspreading Moreover in a horizontal comparison althoughenlarging the adoption thresholds such as TA 1 TB 2in Figures 4(b) 4(e) and 4(h) or TA 1 TB 3 inFigure 4(c) 4(f ) and 4(i) renders information adoptionmore difficult it is clear that increasing ρ0 still transformsthe growth pattern of R(infin) from discontinuous to con-tinuous phase transition and promotes the outbreak ofinformation spreading as in the case of TA 1 TB 1Additionally changes in adoption thresholds and seed sizedo not qualitatively alter phenomena

For a more comprehensive study we obtained the resultsof the final adoption size on the (β ρ0) parameter plane asillustrated in Figure 5 Here we set the subgraphs in column1 with TA 1 TB 1 in column 2 with TA 1 TB 2 incolumn 3 with TA 1 TB 3 in row 1 withαA

w 3 αBw 25 in row 2 with αA

w 3 αBw 3 and in row 3

with αAw 3 αB

w 35 In the following we discuss thefindings based on Figure 5 in detail In Figure 5(a)(TA 1 TB 1 αA

w 3 αBw 25) a critical ρlowast0 0003 ex-

ists below which at any information transmission unitprobability β no outbreak of information spreading willoccur and above which outbreak and global adoption willoccur In a vertical comparison we retain the adoptionthresholds in Figures 5(a) 5(d) and 5(h) and further in-crease the weight distribution exponents in Figure 5(d)(αA

w 3 αBw 3) and Figure 5(h) (αA

w 3 αBw 35) -e

results indicate that the three subgraphs have the samecritical seed size ρlowast0 0003 below which increasing theweight distribution exponent (reducing the weight distri-bution heterogeneity) still cannot induce the outbreak ofinformation spreading but above which increasing theweight distribution exponent can accelerate the outbreak at asmaller information transmission unit probability βMoreover in a horizontal comparison we retain the weightdistribution exponent in Figures 5(a) 5(b) and 5(c) withαA

w 3 αBw 25 and increase the adoption thresholds in

Figure 5(b) (TA 1 TB 2) and Figure 5(c)(TA 1 TB 3) Under these circumstances we discov-ered that the critical seed size ρlowast0 is enlarged from 0003 to006 which implies increasing the difficulty in informationadoption Analogously other subgraphs exhibit similarfindings -e phenomena in Figure 5 suggest that a criticalseed size exists above which outbreak of informationspreading and global adoption will occur and the weightdistribution exponent (reducing the weight distributionheterogeneity) is essential for facilitating informationspreading

52 ER-SF Weighted Multiplex Networks For cases of het-erogeneous degree distributions we investigate informationspreading on a two-layered weighted ER-SF network withPoisson degree distribution pA(kA) eminus langkArang(langkArangkA kA) inlayer A and with power-law degree distribution pB(kB)

ζBkminus vB

B in layer B where ζB 11113936kBkminus vB

B and parameter vB

denotes the degree exponent of layer B Additionally we setthe minimum degree kmin

B 4 and maximum degreekmax

B N

radic

On the weighted ER-SF networks of distinct degreedistribution exponents the growth pattern is shown inFigure 6 with the common setting TA 1 TB 3 in allsubgraphs In Figure 6(a) with degree distribution exponentvB 21 we discovered that enlarging the weight distribu-tion exponent such as from αA

w 3 αBw 25 to

αAw 3 αB

w 35 can increase the final adoption size at thesame β and decrease the critical probability to promote theoutbreak of information spreading Moreover we proceed toincrease the degree distribution exponent (reduce the degreedistribution heterogeneity) in Figure 6(b) (vB 3) andFigure 6(c) (vB 4) Consequently from Figure 6(a) to 6(c)given αA

w and αBw the growth pattern changes from con-

tinuous to the discontinuous growth pattern However atany given vB changing the weight distribution exponent willnot alter the growth pattern of R(infin) -e phenomena inFigure 6 suggest that changing the degree distributionheterogeneity will not qualitatively alter the fact that re-ducing weight distribution heterogeneity can promote in-formation spreading

6 Conclusions

With the development of the Internet and mobile devicessocial networks have prevailed in peoplersquos lives Typically anindividual does not only use one social network but multiplesocial networks simultaneously Moreover the informationdoes not only diffuse in one social network but also spreadssimultaneously on several social networks eg informationspreads on Twitter and Facebook simultaneously Because ofdifferent extents of intimacy among friends colleagues andfamily members connections in social networks should bemodeled with different weights As such a multiplex socialnetwork should be a multilayer weighted network -ere-fore information spreading on a multiplex social networkshould be investigated on a multilayer weighted networkUnfortunately the mechanism of information spreading on

10 Complexity

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(a)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(b)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(c)

ρ0 = 0005ρ0 = 001ρ0 = 01

01 02 03 04 0500β

00

05

10

R (infin

)

(d)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(e)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(f )

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(g)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(h)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(i)

Figure 4 Effect of fraction of initial seeds on the growth pattern of the final adoption size R(infin) versus β on the ER-ER weighted multiplexnetwork under different weight distribution exponents αA

ω and αBω -e values of weight distribution exponent αA

ω of all subgraphs are 3 andthe subgraphs in each row have the same αB

ω ie subgraphs (a) (b) and (c) correspond to αBω 25 (d) (e) and (f) correspond to αB

ω 3and (g) (h) and (i) correspond to αB

ω 35 Additionally the subgraphs in each column have the same adoption threshold ie subgraphs(a) (d) and (g) are set with TA 1 TB 1 (b) (e) and (h) with TA 1 TB 2 and (c) (f ) and (i) with TA 1 TB 3 -e theoreticalsolutions (lines) coincide well with the numerical results (symbols) Given other parameters increasing the fraction of initial seeds willchange the growth pattern of R(infin) from discontinuous to continuous growth Moreover vertically increasing the weight distributionexponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability horizontally increasing theadoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability-e phenomena aboveconvey that changing the seed size can alter the growth pattern and decreasing the adoption threshold can facilitate information spreadingMost importantly changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distributionheterogeneity can facilitate information spreading -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 11

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

S (t)A (t)R (t)

00

05

10R

(t)

2 4 6 8 10 12 14 16 180t

(a)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 12 140t

(b)

S (t)A (t)R (t)

00

05

10

R (t)

2 4 6 8 10 120t

(c)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

2 4 6 8 10 12 14 16 180t

(d)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(e)

S (t)A (t)R (t)

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 100t

(f )

Figure 2 Continued

Complexity 7

S (t)A (t)R (t)

00

05

10R

(t)

1 2 3 4 5 6 7 8 90t

(g)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(h)

Figure 2 Time evolutions of node densities in three states (S-state A-state and R-state) on the ER-ER weighted multiplex network underdifferent weight distribution exponents αA

ω and αBω To unveil the effect of the threshold and seed size two groups of subgraphs with

TA 1 TB 2 ρ0 003 and TA 1 TB 3 ρ0 01 are shown in the top and bottom panels respectively Under the same informationtransmission unit probability β subgraphs (a) (b) and (c) from the top panel have similar parameters of αA

ω 3 αBω 25 αA

ω 3 αBω 3

and αAω 3 αB

ω 35 with (e) (f ) and (g) respectively Particularly subgraphs (d) and (h) emphasize the time evolution of the R-state nodedensity R(t) under different weight distribution exponents In both panels despite the different thresholds and seed sizes at the same timestep the density of nodes in the R-state under αA

ω 3 αBω 35 exceeds other densities under αA

ω 3 αBω 25 and αA

ω 3 αBω 3 -ese

phenomena suggest that large weight distribution exponents (small weight distribution heterogeneity) facilitate information spreading -ebasic parameters are N 104 langkrang 10 langωrang 8 c 10 and β 03

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

00 01 02β

03 04

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

Figure 3 Continued

8 Complexity

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(d)

00 01 02β

03

30

20

10

χ

0

(e)

00 01 02β

03 04

6

4

2

χ

0

(f )

00 01 02β

0300

02

04

χ

(g)

00 01 02β

0300

05

10

15

χ

(h)

Figure 3 Growth pattern of the final adoption sizeR(infin) versus information transmission unit probability β on the ER-ERweightedmultiplexnetwork under different weight distribution exponents αA

ω and αBω given adoption thresholds (TA TB) and initial fraction ρ0 of seeds-e final

adoption size R(infin) versus information transmission probability β with (a) TA 1 TB 1 ρ0 0005 (b) TA 1 TB 2 ρ0 003 (c)TA 1 TB 2 ρ0 01 and (d) TA 1 TB 3 ρ0 01 -e lines denoting theoretical solutions agree exceptionally well with the symbolsrepresenting numerical simulations Moreover in subgraphs (e) (f) (g) and (h) the relative variance χ in terms of the peaks numericallyindicates the critical spreading probabilities in (a) (b) (c) and (d) respectively showing that increasing the weight distribution exponent(decreasing weight distribution heterogeneity) can decrease the critical spreading probability -e phenomena above suggest that decreasingweight distribution heterogeneity facilitates information spreading but cannot alter the growth pattern of the final adoption size versusinformation transmission probability -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 9

not alter the growth pattern of R(infin) versus β in any pair ofadoption thresholds but can promote the outbreak of in-formation spreading Noticeably the theoretical solutions(lines) coincide well with the numerical simulations(symbols)

As shown in Figure 3 the initial seed size affects theinformation spreading dynamics We investigate the effectof seed size on information spreading on a weightedmultiplex social network as shown in Figure 4 InFigure 4(a) (TA 1 TB 1 αA

w 3 αBw 25) increasing

the seed size from ρ0 0005 to ρ0 001 not only changesthe growth pattern from discontinuous to continuous butalso decreases the critical probability to promote theoutbreak of information spreading Subsequently if theweight distribution exponent is enlarged such asαA

w 3 αBw 3 in Figure 4(d) and αA

w 3 αBw 35 in

Figure 4(g) increasing the seed size ρ0 still advances in-formation propagation Comparing Figure 4(a) 4(d) and4(g) vertically altering the initial fraction can change thegrowth pattern of R(infin) versus β but cannot change the factthat increasing the weight distribution exponent (reducingweight distribution heterogeneity) retains the growthpattern while facilitating the outbreak of informationspreading Moreover in a horizontal comparison althoughenlarging the adoption thresholds such as TA 1 TB 2in Figures 4(b) 4(e) and 4(h) or TA 1 TB 3 inFigure 4(c) 4(f ) and 4(i) renders information adoptionmore difficult it is clear that increasing ρ0 still transformsthe growth pattern of R(infin) from discontinuous to con-tinuous phase transition and promotes the outbreak ofinformation spreading as in the case of TA 1 TB 1Additionally changes in adoption thresholds and seed sizedo not qualitatively alter phenomena

For a more comprehensive study we obtained the resultsof the final adoption size on the (β ρ0) parameter plane asillustrated in Figure 5 Here we set the subgraphs in column1 with TA 1 TB 1 in column 2 with TA 1 TB 2 incolumn 3 with TA 1 TB 3 in row 1 withαA

w 3 αBw 25 in row 2 with αA

w 3 αBw 3 and in row 3

with αAw 3 αB

w 35 In the following we discuss thefindings based on Figure 5 in detail In Figure 5(a)(TA 1 TB 1 αA

w 3 αBw 25) a critical ρlowast0 0003 ex-

ists below which at any information transmission unitprobability β no outbreak of information spreading willoccur and above which outbreak and global adoption willoccur In a vertical comparison we retain the adoptionthresholds in Figures 5(a) 5(d) and 5(h) and further in-crease the weight distribution exponents in Figure 5(d)(αA

w 3 αBw 3) and Figure 5(h) (αA

w 3 αBw 35) -e

results indicate that the three subgraphs have the samecritical seed size ρlowast0 0003 below which increasing theweight distribution exponent (reducing the weight distri-bution heterogeneity) still cannot induce the outbreak ofinformation spreading but above which increasing theweight distribution exponent can accelerate the outbreak at asmaller information transmission unit probability βMoreover in a horizontal comparison we retain the weightdistribution exponent in Figures 5(a) 5(b) and 5(c) withαA

w 3 αBw 25 and increase the adoption thresholds in

Figure 5(b) (TA 1 TB 2) and Figure 5(c)(TA 1 TB 3) Under these circumstances we discov-ered that the critical seed size ρlowast0 is enlarged from 0003 to006 which implies increasing the difficulty in informationadoption Analogously other subgraphs exhibit similarfindings -e phenomena in Figure 5 suggest that a criticalseed size exists above which outbreak of informationspreading and global adoption will occur and the weightdistribution exponent (reducing the weight distributionheterogeneity) is essential for facilitating informationspreading

52 ER-SF Weighted Multiplex Networks For cases of het-erogeneous degree distributions we investigate informationspreading on a two-layered weighted ER-SF network withPoisson degree distribution pA(kA) eminus langkArang(langkArangkA kA) inlayer A and with power-law degree distribution pB(kB)

ζBkminus vB

B in layer B where ζB 11113936kBkminus vB

B and parameter vB

denotes the degree exponent of layer B Additionally we setthe minimum degree kmin

B 4 and maximum degreekmax

B N

radic

On the weighted ER-SF networks of distinct degreedistribution exponents the growth pattern is shown inFigure 6 with the common setting TA 1 TB 3 in allsubgraphs In Figure 6(a) with degree distribution exponentvB 21 we discovered that enlarging the weight distribu-tion exponent such as from αA

w 3 αBw 25 to

αAw 3 αB

w 35 can increase the final adoption size at thesame β and decrease the critical probability to promote theoutbreak of information spreading Moreover we proceed toincrease the degree distribution exponent (reduce the degreedistribution heterogeneity) in Figure 6(b) (vB 3) andFigure 6(c) (vB 4) Consequently from Figure 6(a) to 6(c)given αA

w and αBw the growth pattern changes from con-

tinuous to the discontinuous growth pattern However atany given vB changing the weight distribution exponent willnot alter the growth pattern of R(infin) -e phenomena inFigure 6 suggest that changing the degree distributionheterogeneity will not qualitatively alter the fact that re-ducing weight distribution heterogeneity can promote in-formation spreading

6 Conclusions

With the development of the Internet and mobile devicessocial networks have prevailed in peoplersquos lives Typically anindividual does not only use one social network but multiplesocial networks simultaneously Moreover the informationdoes not only diffuse in one social network but also spreadssimultaneously on several social networks eg informationspreads on Twitter and Facebook simultaneously Because ofdifferent extents of intimacy among friends colleagues andfamily members connections in social networks should bemodeled with different weights As such a multiplex socialnetwork should be a multilayer weighted network -ere-fore information spreading on a multiplex social networkshould be investigated on a multilayer weighted networkUnfortunately the mechanism of information spreading on

10 Complexity

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(a)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(b)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(c)

ρ0 = 0005ρ0 = 001ρ0 = 01

01 02 03 04 0500β

00

05

10

R (infin

)

(d)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(e)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(f )

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(g)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(h)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(i)

Figure 4 Effect of fraction of initial seeds on the growth pattern of the final adoption size R(infin) versus β on the ER-ER weighted multiplexnetwork under different weight distribution exponents αA

ω and αBω -e values of weight distribution exponent αA

ω of all subgraphs are 3 andthe subgraphs in each row have the same αB

ω ie subgraphs (a) (b) and (c) correspond to αBω 25 (d) (e) and (f) correspond to αB

ω 3and (g) (h) and (i) correspond to αB

ω 35 Additionally the subgraphs in each column have the same adoption threshold ie subgraphs(a) (d) and (g) are set with TA 1 TB 1 (b) (e) and (h) with TA 1 TB 2 and (c) (f ) and (i) with TA 1 TB 3 -e theoreticalsolutions (lines) coincide well with the numerical results (symbols) Given other parameters increasing the fraction of initial seeds willchange the growth pattern of R(infin) from discontinuous to continuous growth Moreover vertically increasing the weight distributionexponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability horizontally increasing theadoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability-e phenomena aboveconvey that changing the seed size can alter the growth pattern and decreasing the adoption threshold can facilitate information spreadingMost importantly changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distributionheterogeneity can facilitate information spreading -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 11

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

S (t)A (t)R (t)

00

05

10R

(t)

1 2 3 4 5 6 7 8 90t

(g)

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

00

05

10

R (t)

1 2 3 4 5 6 7 8 9 10 110t

(h)

Figure 2 Time evolutions of node densities in three states (S-state A-state and R-state) on the ER-ER weighted multiplex network underdifferent weight distribution exponents αA

ω and αBω To unveil the effect of the threshold and seed size two groups of subgraphs with

TA 1 TB 2 ρ0 003 and TA 1 TB 3 ρ0 01 are shown in the top and bottom panels respectively Under the same informationtransmission unit probability β subgraphs (a) (b) and (c) from the top panel have similar parameters of αA

ω 3 αBω 25 αA

ω 3 αBω 3

and αAω 3 αB

ω 35 with (e) (f ) and (g) respectively Particularly subgraphs (d) and (h) emphasize the time evolution of the R-state nodedensity R(t) under different weight distribution exponents In both panels despite the different thresholds and seed sizes at the same timestep the density of nodes in the R-state under αA

ω 3 αBω 35 exceeds other densities under αA

ω 3 αBω 25 and αA

ω 3 αBω 3 -ese

phenomena suggest that large weight distribution exponents (small weight distribution heterogeneity) facilitate information spreading -ebasic parameters are N 104 langkrang 10 langωrang 8 c 10 and β 03

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

00 01 02β

03 04

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

Figure 3 Continued

8 Complexity

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(d)

00 01 02β

03

30

20

10

χ

0

(e)

00 01 02β

03 04

6

4

2

χ

0

(f )

00 01 02β

0300

02

04

χ

(g)

00 01 02β

0300

05

10

15

χ

(h)

Figure 3 Growth pattern of the final adoption sizeR(infin) versus information transmission unit probability β on the ER-ERweightedmultiplexnetwork under different weight distribution exponents αA

ω and αBω given adoption thresholds (TA TB) and initial fraction ρ0 of seeds-e final

adoption size R(infin) versus information transmission probability β with (a) TA 1 TB 1 ρ0 0005 (b) TA 1 TB 2 ρ0 003 (c)TA 1 TB 2 ρ0 01 and (d) TA 1 TB 3 ρ0 01 -e lines denoting theoretical solutions agree exceptionally well with the symbolsrepresenting numerical simulations Moreover in subgraphs (e) (f) (g) and (h) the relative variance χ in terms of the peaks numericallyindicates the critical spreading probabilities in (a) (b) (c) and (d) respectively showing that increasing the weight distribution exponent(decreasing weight distribution heterogeneity) can decrease the critical spreading probability -e phenomena above suggest that decreasingweight distribution heterogeneity facilitates information spreading but cannot alter the growth pattern of the final adoption size versusinformation transmission probability -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 9

not alter the growth pattern of R(infin) versus β in any pair ofadoption thresholds but can promote the outbreak of in-formation spreading Noticeably the theoretical solutions(lines) coincide well with the numerical simulations(symbols)

As shown in Figure 3 the initial seed size affects theinformation spreading dynamics We investigate the effectof seed size on information spreading on a weightedmultiplex social network as shown in Figure 4 InFigure 4(a) (TA 1 TB 1 αA

w 3 αBw 25) increasing

the seed size from ρ0 0005 to ρ0 001 not only changesthe growth pattern from discontinuous to continuous butalso decreases the critical probability to promote theoutbreak of information spreading Subsequently if theweight distribution exponent is enlarged such asαA

w 3 αBw 3 in Figure 4(d) and αA

w 3 αBw 35 in

Figure 4(g) increasing the seed size ρ0 still advances in-formation propagation Comparing Figure 4(a) 4(d) and4(g) vertically altering the initial fraction can change thegrowth pattern of R(infin) versus β but cannot change the factthat increasing the weight distribution exponent (reducingweight distribution heterogeneity) retains the growthpattern while facilitating the outbreak of informationspreading Moreover in a horizontal comparison althoughenlarging the adoption thresholds such as TA 1 TB 2in Figures 4(b) 4(e) and 4(h) or TA 1 TB 3 inFigure 4(c) 4(f ) and 4(i) renders information adoptionmore difficult it is clear that increasing ρ0 still transformsthe growth pattern of R(infin) from discontinuous to con-tinuous phase transition and promotes the outbreak ofinformation spreading as in the case of TA 1 TB 1Additionally changes in adoption thresholds and seed sizedo not qualitatively alter phenomena

For a more comprehensive study we obtained the resultsof the final adoption size on the (β ρ0) parameter plane asillustrated in Figure 5 Here we set the subgraphs in column1 with TA 1 TB 1 in column 2 with TA 1 TB 2 incolumn 3 with TA 1 TB 3 in row 1 withαA

w 3 αBw 25 in row 2 with αA

w 3 αBw 3 and in row 3

with αAw 3 αB

w 35 In the following we discuss thefindings based on Figure 5 in detail In Figure 5(a)(TA 1 TB 1 αA

w 3 αBw 25) a critical ρlowast0 0003 ex-

ists below which at any information transmission unitprobability β no outbreak of information spreading willoccur and above which outbreak and global adoption willoccur In a vertical comparison we retain the adoptionthresholds in Figures 5(a) 5(d) and 5(h) and further in-crease the weight distribution exponents in Figure 5(d)(αA

w 3 αBw 3) and Figure 5(h) (αA

w 3 αBw 35) -e

results indicate that the three subgraphs have the samecritical seed size ρlowast0 0003 below which increasing theweight distribution exponent (reducing the weight distri-bution heterogeneity) still cannot induce the outbreak ofinformation spreading but above which increasing theweight distribution exponent can accelerate the outbreak at asmaller information transmission unit probability βMoreover in a horizontal comparison we retain the weightdistribution exponent in Figures 5(a) 5(b) and 5(c) withαA

w 3 αBw 25 and increase the adoption thresholds in

Figure 5(b) (TA 1 TB 2) and Figure 5(c)(TA 1 TB 3) Under these circumstances we discov-ered that the critical seed size ρlowast0 is enlarged from 0003 to006 which implies increasing the difficulty in informationadoption Analogously other subgraphs exhibit similarfindings -e phenomena in Figure 5 suggest that a criticalseed size exists above which outbreak of informationspreading and global adoption will occur and the weightdistribution exponent (reducing the weight distributionheterogeneity) is essential for facilitating informationspreading

52 ER-SF Weighted Multiplex Networks For cases of het-erogeneous degree distributions we investigate informationspreading on a two-layered weighted ER-SF network withPoisson degree distribution pA(kA) eminus langkArang(langkArangkA kA) inlayer A and with power-law degree distribution pB(kB)

ζBkminus vB

B in layer B where ζB 11113936kBkminus vB

B and parameter vB

denotes the degree exponent of layer B Additionally we setthe minimum degree kmin

B 4 and maximum degreekmax

B N

radic

On the weighted ER-SF networks of distinct degreedistribution exponents the growth pattern is shown inFigure 6 with the common setting TA 1 TB 3 in allsubgraphs In Figure 6(a) with degree distribution exponentvB 21 we discovered that enlarging the weight distribu-tion exponent such as from αA

w 3 αBw 25 to

αAw 3 αB

w 35 can increase the final adoption size at thesame β and decrease the critical probability to promote theoutbreak of information spreading Moreover we proceed toincrease the degree distribution exponent (reduce the degreedistribution heterogeneity) in Figure 6(b) (vB 3) andFigure 6(c) (vB 4) Consequently from Figure 6(a) to 6(c)given αA

w and αBw the growth pattern changes from con-

tinuous to the discontinuous growth pattern However atany given vB changing the weight distribution exponent willnot alter the growth pattern of R(infin) -e phenomena inFigure 6 suggest that changing the degree distributionheterogeneity will not qualitatively alter the fact that re-ducing weight distribution heterogeneity can promote in-formation spreading

6 Conclusions

With the development of the Internet and mobile devicessocial networks have prevailed in peoplersquos lives Typically anindividual does not only use one social network but multiplesocial networks simultaneously Moreover the informationdoes not only diffuse in one social network but also spreadssimultaneously on several social networks eg informationspreads on Twitter and Facebook simultaneously Because ofdifferent extents of intimacy among friends colleagues andfamily members connections in social networks should bemodeled with different weights As such a multiplex socialnetwork should be a multilayer weighted network -ere-fore information spreading on a multiplex social networkshould be investigated on a multilayer weighted networkUnfortunately the mechanism of information spreading on

10 Complexity

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(a)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(b)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(c)

ρ0 = 0005ρ0 = 001ρ0 = 01

01 02 03 04 0500β

00

05

10

R (infin

)

(d)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(e)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(f )

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(g)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(h)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(i)

Figure 4 Effect of fraction of initial seeds on the growth pattern of the final adoption size R(infin) versus β on the ER-ER weighted multiplexnetwork under different weight distribution exponents αA

ω and αBω -e values of weight distribution exponent αA

ω of all subgraphs are 3 andthe subgraphs in each row have the same αB

ω ie subgraphs (a) (b) and (c) correspond to αBω 25 (d) (e) and (f) correspond to αB

ω 3and (g) (h) and (i) correspond to αB

ω 35 Additionally the subgraphs in each column have the same adoption threshold ie subgraphs(a) (d) and (g) are set with TA 1 TB 1 (b) (e) and (h) with TA 1 TB 2 and (c) (f ) and (i) with TA 1 TB 3 -e theoreticalsolutions (lines) coincide well with the numerical results (symbols) Given other parameters increasing the fraction of initial seeds willchange the growth pattern of R(infin) from discontinuous to continuous growth Moreover vertically increasing the weight distributionexponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability horizontally increasing theadoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability-e phenomena aboveconvey that changing the seed size can alter the growth pattern and decreasing the adoption threshold can facilitate information spreadingMost importantly changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distributionheterogeneity can facilitate information spreading -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 11

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

00

05

10

R (infin

)

00 01 02β

03

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(d)

00 01 02β

03

30

20

10

χ

0

(e)

00 01 02β

03 04

6

4

2

χ

0

(f )

00 01 02β

0300

02

04

χ

(g)

00 01 02β

0300

05

10

15

χ

(h)

Figure 3 Growth pattern of the final adoption sizeR(infin) versus information transmission unit probability β on the ER-ERweightedmultiplexnetwork under different weight distribution exponents αA

ω and αBω given adoption thresholds (TA TB) and initial fraction ρ0 of seeds-e final

adoption size R(infin) versus information transmission probability β with (a) TA 1 TB 1 ρ0 0005 (b) TA 1 TB 2 ρ0 003 (c)TA 1 TB 2 ρ0 01 and (d) TA 1 TB 3 ρ0 01 -e lines denoting theoretical solutions agree exceptionally well with the symbolsrepresenting numerical simulations Moreover in subgraphs (e) (f) (g) and (h) the relative variance χ in terms of the peaks numericallyindicates the critical spreading probabilities in (a) (b) (c) and (d) respectively showing that increasing the weight distribution exponent(decreasing weight distribution heterogeneity) can decrease the critical spreading probability -e phenomena above suggest that decreasingweight distribution heterogeneity facilitates information spreading but cannot alter the growth pattern of the final adoption size versusinformation transmission probability -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 9

not alter the growth pattern of R(infin) versus β in any pair ofadoption thresholds but can promote the outbreak of in-formation spreading Noticeably the theoretical solutions(lines) coincide well with the numerical simulations(symbols)

As shown in Figure 3 the initial seed size affects theinformation spreading dynamics We investigate the effectof seed size on information spreading on a weightedmultiplex social network as shown in Figure 4 InFigure 4(a) (TA 1 TB 1 αA

w 3 αBw 25) increasing

the seed size from ρ0 0005 to ρ0 001 not only changesthe growth pattern from discontinuous to continuous butalso decreases the critical probability to promote theoutbreak of information spreading Subsequently if theweight distribution exponent is enlarged such asαA

w 3 αBw 3 in Figure 4(d) and αA

w 3 αBw 35 in

Figure 4(g) increasing the seed size ρ0 still advances in-formation propagation Comparing Figure 4(a) 4(d) and4(g) vertically altering the initial fraction can change thegrowth pattern of R(infin) versus β but cannot change the factthat increasing the weight distribution exponent (reducingweight distribution heterogeneity) retains the growthpattern while facilitating the outbreak of informationspreading Moreover in a horizontal comparison althoughenlarging the adoption thresholds such as TA 1 TB 2in Figures 4(b) 4(e) and 4(h) or TA 1 TB 3 inFigure 4(c) 4(f ) and 4(i) renders information adoptionmore difficult it is clear that increasing ρ0 still transformsthe growth pattern of R(infin) from discontinuous to con-tinuous phase transition and promotes the outbreak ofinformation spreading as in the case of TA 1 TB 1Additionally changes in adoption thresholds and seed sizedo not qualitatively alter phenomena

For a more comprehensive study we obtained the resultsof the final adoption size on the (β ρ0) parameter plane asillustrated in Figure 5 Here we set the subgraphs in column1 with TA 1 TB 1 in column 2 with TA 1 TB 2 incolumn 3 with TA 1 TB 3 in row 1 withαA

w 3 αBw 25 in row 2 with αA

w 3 αBw 3 and in row 3

with αAw 3 αB

w 35 In the following we discuss thefindings based on Figure 5 in detail In Figure 5(a)(TA 1 TB 1 αA

w 3 αBw 25) a critical ρlowast0 0003 ex-

ists below which at any information transmission unitprobability β no outbreak of information spreading willoccur and above which outbreak and global adoption willoccur In a vertical comparison we retain the adoptionthresholds in Figures 5(a) 5(d) and 5(h) and further in-crease the weight distribution exponents in Figure 5(d)(αA

w 3 αBw 3) and Figure 5(h) (αA

w 3 αBw 35) -e

results indicate that the three subgraphs have the samecritical seed size ρlowast0 0003 below which increasing theweight distribution exponent (reducing the weight distri-bution heterogeneity) still cannot induce the outbreak ofinformation spreading but above which increasing theweight distribution exponent can accelerate the outbreak at asmaller information transmission unit probability βMoreover in a horizontal comparison we retain the weightdistribution exponent in Figures 5(a) 5(b) and 5(c) withαA

w 3 αBw 25 and increase the adoption thresholds in

Figure 5(b) (TA 1 TB 2) and Figure 5(c)(TA 1 TB 3) Under these circumstances we discov-ered that the critical seed size ρlowast0 is enlarged from 0003 to006 which implies increasing the difficulty in informationadoption Analogously other subgraphs exhibit similarfindings -e phenomena in Figure 5 suggest that a criticalseed size exists above which outbreak of informationspreading and global adoption will occur and the weightdistribution exponent (reducing the weight distributionheterogeneity) is essential for facilitating informationspreading

52 ER-SF Weighted Multiplex Networks For cases of het-erogeneous degree distributions we investigate informationspreading on a two-layered weighted ER-SF network withPoisson degree distribution pA(kA) eminus langkArang(langkArangkA kA) inlayer A and with power-law degree distribution pB(kB)

ζBkminus vB

B in layer B where ζB 11113936kBkminus vB

B and parameter vB

denotes the degree exponent of layer B Additionally we setthe minimum degree kmin

B 4 and maximum degreekmax

B N

radic

On the weighted ER-SF networks of distinct degreedistribution exponents the growth pattern is shown inFigure 6 with the common setting TA 1 TB 3 in allsubgraphs In Figure 6(a) with degree distribution exponentvB 21 we discovered that enlarging the weight distribu-tion exponent such as from αA

w 3 αBw 25 to

αAw 3 αB

w 35 can increase the final adoption size at thesame β and decrease the critical probability to promote theoutbreak of information spreading Moreover we proceed toincrease the degree distribution exponent (reduce the degreedistribution heterogeneity) in Figure 6(b) (vB 3) andFigure 6(c) (vB 4) Consequently from Figure 6(a) to 6(c)given αA

w and αBw the growth pattern changes from con-

tinuous to the discontinuous growth pattern However atany given vB changing the weight distribution exponent willnot alter the growth pattern of R(infin) -e phenomena inFigure 6 suggest that changing the degree distributionheterogeneity will not qualitatively alter the fact that re-ducing weight distribution heterogeneity can promote in-formation spreading

6 Conclusions

With the development of the Internet and mobile devicessocial networks have prevailed in peoplersquos lives Typically anindividual does not only use one social network but multiplesocial networks simultaneously Moreover the informationdoes not only diffuse in one social network but also spreadssimultaneously on several social networks eg informationspreads on Twitter and Facebook simultaneously Because ofdifferent extents of intimacy among friends colleagues andfamily members connections in social networks should bemodeled with different weights As such a multiplex socialnetwork should be a multilayer weighted network -ere-fore information spreading on a multiplex social networkshould be investigated on a multilayer weighted networkUnfortunately the mechanism of information spreading on

10 Complexity

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(a)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(b)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(c)

ρ0 = 0005ρ0 = 001ρ0 = 01

01 02 03 04 0500β

00

05

10

R (infin

)

(d)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(e)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(f )

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(g)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(h)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(i)

Figure 4 Effect of fraction of initial seeds on the growth pattern of the final adoption size R(infin) versus β on the ER-ER weighted multiplexnetwork under different weight distribution exponents αA

ω and αBω -e values of weight distribution exponent αA

ω of all subgraphs are 3 andthe subgraphs in each row have the same αB

ω ie subgraphs (a) (b) and (c) correspond to αBω 25 (d) (e) and (f) correspond to αB

ω 3and (g) (h) and (i) correspond to αB

ω 35 Additionally the subgraphs in each column have the same adoption threshold ie subgraphs(a) (d) and (g) are set with TA 1 TB 1 (b) (e) and (h) with TA 1 TB 2 and (c) (f ) and (i) with TA 1 TB 3 -e theoreticalsolutions (lines) coincide well with the numerical results (symbols) Given other parameters increasing the fraction of initial seeds willchange the growth pattern of R(infin) from discontinuous to continuous growth Moreover vertically increasing the weight distributionexponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability horizontally increasing theadoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability-e phenomena aboveconvey that changing the seed size can alter the growth pattern and decreasing the adoption threshold can facilitate information spreadingMost importantly changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distributionheterogeneity can facilitate information spreading -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 11

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

not alter the growth pattern of R(infin) versus β in any pair ofadoption thresholds but can promote the outbreak of in-formation spreading Noticeably the theoretical solutions(lines) coincide well with the numerical simulations(symbols)

As shown in Figure 3 the initial seed size affects theinformation spreading dynamics We investigate the effectof seed size on information spreading on a weightedmultiplex social network as shown in Figure 4 InFigure 4(a) (TA 1 TB 1 αA

w 3 αBw 25) increasing

the seed size from ρ0 0005 to ρ0 001 not only changesthe growth pattern from discontinuous to continuous butalso decreases the critical probability to promote theoutbreak of information spreading Subsequently if theweight distribution exponent is enlarged such asαA

w 3 αBw 3 in Figure 4(d) and αA

w 3 αBw 35 in

Figure 4(g) increasing the seed size ρ0 still advances in-formation propagation Comparing Figure 4(a) 4(d) and4(g) vertically altering the initial fraction can change thegrowth pattern of R(infin) versus β but cannot change the factthat increasing the weight distribution exponent (reducingweight distribution heterogeneity) retains the growthpattern while facilitating the outbreak of informationspreading Moreover in a horizontal comparison althoughenlarging the adoption thresholds such as TA 1 TB 2in Figures 4(b) 4(e) and 4(h) or TA 1 TB 3 inFigure 4(c) 4(f ) and 4(i) renders information adoptionmore difficult it is clear that increasing ρ0 still transformsthe growth pattern of R(infin) from discontinuous to con-tinuous phase transition and promotes the outbreak ofinformation spreading as in the case of TA 1 TB 1Additionally changes in adoption thresholds and seed sizedo not qualitatively alter phenomena

For a more comprehensive study we obtained the resultsof the final adoption size on the (β ρ0) parameter plane asillustrated in Figure 5 Here we set the subgraphs in column1 with TA 1 TB 1 in column 2 with TA 1 TB 2 incolumn 3 with TA 1 TB 3 in row 1 withαA

w 3 αBw 25 in row 2 with αA

w 3 αBw 3 and in row 3

with αAw 3 αB

w 35 In the following we discuss thefindings based on Figure 5 in detail In Figure 5(a)(TA 1 TB 1 αA

w 3 αBw 25) a critical ρlowast0 0003 ex-

ists below which at any information transmission unitprobability β no outbreak of information spreading willoccur and above which outbreak and global adoption willoccur In a vertical comparison we retain the adoptionthresholds in Figures 5(a) 5(d) and 5(h) and further in-crease the weight distribution exponents in Figure 5(d)(αA

w 3 αBw 3) and Figure 5(h) (αA

w 3 αBw 35) -e

results indicate that the three subgraphs have the samecritical seed size ρlowast0 0003 below which increasing theweight distribution exponent (reducing the weight distri-bution heterogeneity) still cannot induce the outbreak ofinformation spreading but above which increasing theweight distribution exponent can accelerate the outbreak at asmaller information transmission unit probability βMoreover in a horizontal comparison we retain the weightdistribution exponent in Figures 5(a) 5(b) and 5(c) withαA

w 3 αBw 25 and increase the adoption thresholds in

Figure 5(b) (TA 1 TB 2) and Figure 5(c)(TA 1 TB 3) Under these circumstances we discov-ered that the critical seed size ρlowast0 is enlarged from 0003 to006 which implies increasing the difficulty in informationadoption Analogously other subgraphs exhibit similarfindings -e phenomena in Figure 5 suggest that a criticalseed size exists above which outbreak of informationspreading and global adoption will occur and the weightdistribution exponent (reducing the weight distributionheterogeneity) is essential for facilitating informationspreading

52 ER-SF Weighted Multiplex Networks For cases of het-erogeneous degree distributions we investigate informationspreading on a two-layered weighted ER-SF network withPoisson degree distribution pA(kA) eminus langkArang(langkArangkA kA) inlayer A and with power-law degree distribution pB(kB)

ζBkminus vB

B in layer B where ζB 11113936kBkminus vB

B and parameter vB

denotes the degree exponent of layer B Additionally we setthe minimum degree kmin

B 4 and maximum degreekmax

B N

radic

On the weighted ER-SF networks of distinct degreedistribution exponents the growth pattern is shown inFigure 6 with the common setting TA 1 TB 3 in allsubgraphs In Figure 6(a) with degree distribution exponentvB 21 we discovered that enlarging the weight distribu-tion exponent such as from αA

w 3 αBw 25 to

αAw 3 αB

w 35 can increase the final adoption size at thesame β and decrease the critical probability to promote theoutbreak of information spreading Moreover we proceed toincrease the degree distribution exponent (reduce the degreedistribution heterogeneity) in Figure 6(b) (vB 3) andFigure 6(c) (vB 4) Consequently from Figure 6(a) to 6(c)given αA

w and αBw the growth pattern changes from con-

tinuous to the discontinuous growth pattern However atany given vB changing the weight distribution exponent willnot alter the growth pattern of R(infin) -e phenomena inFigure 6 suggest that changing the degree distributionheterogeneity will not qualitatively alter the fact that re-ducing weight distribution heterogeneity can promote in-formation spreading

6 Conclusions

With the development of the Internet and mobile devicessocial networks have prevailed in peoplersquos lives Typically anindividual does not only use one social network but multiplesocial networks simultaneously Moreover the informationdoes not only diffuse in one social network but also spreadssimultaneously on several social networks eg informationspreads on Twitter and Facebook simultaneously Because ofdifferent extents of intimacy among friends colleagues andfamily members connections in social networks should bemodeled with different weights As such a multiplex socialnetwork should be a multilayer weighted network -ere-fore information spreading on a multiplex social networkshould be investigated on a multilayer weighted networkUnfortunately the mechanism of information spreading on

10 Complexity

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(a)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(b)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(c)

ρ0 = 0005ρ0 = 001ρ0 = 01

01 02 03 04 0500β

00

05

10

R (infin

)

(d)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(e)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(f )

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(g)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(h)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(i)

Figure 4 Effect of fraction of initial seeds on the growth pattern of the final adoption size R(infin) versus β on the ER-ER weighted multiplexnetwork under different weight distribution exponents αA

ω and αBω -e values of weight distribution exponent αA

ω of all subgraphs are 3 andthe subgraphs in each row have the same αB

ω ie subgraphs (a) (b) and (c) correspond to αBω 25 (d) (e) and (f) correspond to αB

ω 3and (g) (h) and (i) correspond to αB

ω 35 Additionally the subgraphs in each column have the same adoption threshold ie subgraphs(a) (d) and (g) are set with TA 1 TB 1 (b) (e) and (h) with TA 1 TB 2 and (c) (f ) and (i) with TA 1 TB 3 -e theoreticalsolutions (lines) coincide well with the numerical results (symbols) Given other parameters increasing the fraction of initial seeds willchange the growth pattern of R(infin) from discontinuous to continuous growth Moreover vertically increasing the weight distributionexponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability horizontally increasing theadoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability-e phenomena aboveconvey that changing the seed size can alter the growth pattern and decreasing the adoption threshold can facilitate information spreadingMost importantly changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distributionheterogeneity can facilitate information spreading -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 11

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(a)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(b)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(c)

ρ0 = 0005ρ0 = 001ρ0 = 01

01 02 03 04 0500β

00

05

10

R (infin

)

(d)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(e)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(f )

ρ0 = 0005ρ0 = 001ρ0 = 01

00

05

10

R (infin

)

01 02 03 04 0500β

(g)

ρ0 = 003ρ0 = 01ρ0 = 02

00

05

10

R (infin

)

01 02 03 04 0500β

(h)

ρ0 = 006ρ0 = 01ρ0 = 03

00

05

10

R (infin

)

01 02 03 04 0500β

(i)

Figure 4 Effect of fraction of initial seeds on the growth pattern of the final adoption size R(infin) versus β on the ER-ER weighted multiplexnetwork under different weight distribution exponents αA

ω and αBω -e values of weight distribution exponent αA

ω of all subgraphs are 3 andthe subgraphs in each row have the same αB

ω ie subgraphs (a) (b) and (c) correspond to αBω 25 (d) (e) and (f) correspond to αB

ω 3and (g) (h) and (i) correspond to αB

ω 35 Additionally the subgraphs in each column have the same adoption threshold ie subgraphs(a) (d) and (g) are set with TA 1 TB 1 (b) (e) and (h) with TA 1 TB 2 and (c) (f ) and (i) with TA 1 TB 3 -e theoreticalsolutions (lines) coincide well with the numerical results (symbols) Given other parameters increasing the fraction of initial seeds willchange the growth pattern of R(infin) from discontinuous to continuous growth Moreover vertically increasing the weight distributionexponents (decreasing weight distribution heterogeneity) can decrease the critical transmission probability horizontally increasing theadoption threshold (increasing the difficulty of adoption) can contrarily enlarge the critical transmission probability-e phenomena aboveconvey that changing the seed size can alter the growth pattern and decreasing the adoption threshold can facilitate information spreadingMost importantly changing the seed size of the adoption threshold cannot qualitatively alter the fact that decreasing the weight distributionheterogeneity can facilitate information spreading -e basic parameters are N 104 langkrang 10 langωrang 8 and c 10

Complexity 11

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

a weighted multiplex network has not been investigatedcomprehensively

Besides social information spreading possesses the char-acteristics of social reinforcement derived from the memory ofnonredundant information Herein in terms of social re-inforcement and weight distribution we first proposed ageneral information spreading model on a weighted multiplexnetwork to depict the dynamics of information spreading on atwo-layered weighted network Subsequently we conceived ageneralized edge-weight-based compartmental theory to ana-lyze the mechanism of information spreading on a weightedmultiplex network with the effect of the memory of non-redundant information Based on the numerical method

extensive simulations confirmed that the theoretical pre-dictions coincided well with the simulation results

Combining numerical simulations with theoretical an-alyses we discovered that under any adoption threshold oftwo layers reducing weight distribution heterogeneity didnot alter the growth pattern of the final adoption size butcould accelerate information spreading and promote theoutbreak of information spreading Furthermore we dis-covered that a critical seed size existed below which nooutbreak of information spreading occurred and abovewhich reducing weight distribution heterogeneity wasmeaningful to the facilitation of information spreadingAdditionally we discovered that changing degree

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(a)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(b)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(c)

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(d)

ρ0lowast = 003

0

02

04

06

08

1ρ 0

0

02

04

06

08

1

05 10β

(e)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(f )

ρ0lowast = 0003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(g)

ρ0lowast = 003

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(h)

ρ0lowast = 006

0

02

04

06

08

1

ρ 0

0

02

04

06

08

1

05 10β

(i)

Figure 5 Illustrations of the growth pattern of R(infin) on the (β ρ0) parameter plane on the ER-ER weighted multiplex network Accordingto the theoretical method the final adoption size R(infin) versus the parameter pair (β ρ0) was obtained -e results are plotted in subgraphs(a) (b) and (c) from row 1 with TA 1 TB 1 (d) (e) and (f) from row 2 with TA 1 TB 2 and (g) (h) and (i) from row 3 withTA 1 TB 3 -e subgraphs have weight distribution exponents of αA

ω 3 αBω 25 in column 1 αA

ω 3 αBω 3 in column 2 and αA

ω 3αBω 35 in column 3 All subgraphs show that a critical seed size ρlowast0 exists under which no global adoption of information exists and over

which a global adoption becomes possible Additionally the changes in the subgraphs from the first column to the third confirm thatincrease the weight distribution exponents (decreasing weight distribution heterogeneity) can promote information spreading -e basicparameters are N 104 langkrang 10 langωrang 8 ρ0 0001 and c 10

12 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

distribution heterogeneity could transform the growthpattern but did not qualitatively affect the promotion ofreducing weight distribution heterogeneity to informationspreading Above all increasing the weight distributionexponent can decrease critical transmission probabilitywithout the influence of the degree distribution exponent

In this study we unveiled the effect of weighted mul-tiplex network structures on information spreading -isstudy can inspire further studies related to the design of

optimized strategies to control information spreading onmultiplex social networks Additionally more accuratetheories to depict the social contagion on real two-layercoupled networks should be further explored

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

00

05

10R

(infin)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(a)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(b)

00

05

10

R (infin

)

0010 05β

αAw = 3 αB

w = 25

αAw = 3 αB

w = 3

αAw = 3 αB

w = 35

(c)

Figure 6 Effect of degree distribution heterogeneity on the final adoption size R(infin) versus β on the ER-SF weighted multiplex networkwith different degree distribution exponents vB Subgraphs (a) (vB 21) (b) (vB 3) and (c) (vB 4) illustrate the growth patterns ofR(infin) versus β under different weight distribution exponents (αA

ω 3 αBω 25) (αB

ω 3 αBω 3) and (αA

ω 3 αBω 35) respectively From

subgraphs (a)ndash(c) increasing the degree distribution exponent does not alter the fact that reducing the weight distribution heterogeneity(increasing the weight distribution exponent) can facilitate the spread of social behavior

Complexity 13

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 61602048 and 61673086) and theFundamental Research Funds of the Central Universities

References

[1] G-Q Zhang G-Q Zhang Q-F Yang S-Q Cheng andT Zhou ldquoEvolution of the Internet and its coresrdquo NewJournal of Physics vol 10 no 12 Article ID 123027 2008

[2] S Anand M Venkataraman K P Subbalakshmi andR Chandramouli ldquoSpatio-temporal analysis of passive con-sumption in internet mediardquo IEEE Transactions on Knowledgeand Data Engineering vol 27 no 10 pp 2839ndash2850 2015

[3] T OrsquoReilly ldquoWhat is web 20 design patterns and businessmodels for the next generation of softwarerdquo 2007 httpwwworeillycompubaweb2archivewhat-is-web-20html

[4] G Goggin Cell Phone Culture Mobile Technology in EverydayLife Routledge Abingdon UK 2006

[5] Y Dong N V Chawla J Tang and Y Yang ldquoUser modelingon demographic attributes in big mobile social networksrdquoACM Transactions on Information Systems vol 35 no 4pp 1ndash33 2017

[6] A Macarthy 500 Social Media Marketing Tips EssentialAdvice Hints and Strategy for Business Facebook TwitterPinterest Google+ YouTube Instagram LinkedIn and MoreCreateSpace Independent Publishing Platform Scotts ValleyCA USA 2018

[7] S F Waterloo S E Baumgartner J Peter andP M Valkenburg ldquoNorms of online expressions of emotioncomparing facebook twitter instagram and whatsapprdquo NewMedia amp Society vol 20 no 5 pp 1813ndash1831 2018

[8] J Cao Z Bu Y Wang H Yang J Jiang and H-J LildquoDetecting prosumer-community groups in smart grids fromthe multiagent perspectiverdquo IEEE Transactions on SystemsMan and Cybernetics Systems vol 49 no 8 pp 1652ndash16642019

[9] M D Vicario F Zollo G Caldarelli A Scala andW Quattrociocchi ldquoMapping social dynamics on facebookthe brexit debaterdquo Social Networks vol 50 pp 6ndash16 2017

[10] B J Jansen M Zhang K Sobel and A Chowdury ldquoTwitterpower tweets as electronic word of mouthrdquo Journal of theAmerican Society for Information Science and Technologyvol 60 no 11 pp 2169ndash2188 2009

[11] D Yu N Chen and X Ran ldquoComputational modeling ofWeibo user influence based on information interactive net-workrdquo Online Information Review vol 40 no 7 pp 867ndash8812016

[12] C Ju and W Tao ldquoRelationship strength estimation based onWechat Friends Circlerdquo Neurocomputing vol 253 pp 15ndash232017

[13] D Li S Zhang X Sun H Zhou S Li and X Li ldquoModelinginformation diffusion over social networks for temporaldynamic predictionrdquo IEEE Transactions on Knowledge andData Engineering vol 29 no 9 pp 1985ndash1997 2017

[14] L Wu Y Ge Q Liu et al ldquoModeling the evolution of usersrsquopreferences and social links in social networking servicesrdquoIEEE Transactions on Knowledge and Data Engineeringvol 29 no 6 pp 1240ndash1253 2017

[15] S Vosoughi D Roy and S Aral ldquo-e spread of true and falsenews onlinerdquo Science vol 359 no 6380 pp 1146ndash1151 2018

[16] X Zhu H Tian X Chen W Wang and S Cai ldquoHetero-geneous behavioral adoption in multiplex networksrdquo NewJournal of Physics vol 20 no 12 Article ID 125002 2018

[17] S V Jin ldquoldquoCelebrity 20 and beyondrdquo Effects of Facebookprofile sources on social networking advertisingrdquo Computersin Human Behavior vol 79 pp 154ndash168 2018

[18] A Song Y Liu Z Wu M Zhai and J Luo ldquoA local randomwalk model for complex networks based on discriminativefeature combinationsrdquo Expert Systems with Applicationsvol 118 pp 329ndash339 2019

[19] D Centola ldquoAn experimental study of homophily in theadoption of health behaviorrdquo Science vol 334 no 6060pp 1269ndash1272 2011

[20] Z Bu Y Wang H-J Li J Jiang Z Wu and J Cao ldquoLinkprediction in temporal networks integrating survival analysisand game theoryrdquo Information Sciences vol 498 pp 41ndash612019

[21] H P Young ldquo-e dynamics of social innovationrdquo Proceedingsof the National Academy of Sciences vol 108 no 4pp 21285ndash21291 2011

[22] M E J Newman S Forrest and J Balthrop ldquoEmail networksand the spread of computer virusesrdquo Physical Review Evol 66 no 3 Article ID 035101 2002

[23] Z Wang C T Bauch S Bhattacharyya et al ldquoStatisticalphysics of vaccinationrdquo Physics Reports vol 664 pp 1ndash1132016

[24] Z Wang Y Moreno S Boccaletti and M Perc ldquoVaccinationand epidemics in networked populationsndashan introductionrdquoChaos Solitons amp Fractals vol 103 pp 177ndash183 2017

[25] Y Zhang S Zhou Z Zhang J Guan and S Zhou ldquoRumorevolution in social networksrdquo Physical Review E vol 87 no 32013

[26] A Banerjee A G Chandrasekhar E Duflo andM O Jackson ldquo-e diffusion of microfinancerdquo Sciencevol 341 no 6144 Article ID 1236498 2013

[27] S Jiang H Fan and M Xia ldquoCredit risk contagion based onasymmetric information associationrdquo Complexity vol 2018Article ID 2929157 11 pages 2018

[28] D J Watts and P S Dodds ldquoInfluentials networks andpublic opinion formationrdquo Journal of Consumer Researchvol 34 no 4 pp 441ndash458 2007

[29] X Liu C Liu and X Zeng ldquoOnline social network emergencypublic event information propagation and nonlinear math-ematical modelingrdquo Complexity vol 2017 Article ID5857372 7 pages 2017

[30] M Gutierrez-Roig O Sagarra A Oltra et al ldquoActive andreactive behaviour in human mobility the influence of at-traction points on pedestriansrdquo Royal Society Open Sciencevol 3 no 7 Article ID 160177 2016

[31] C Liu X-X Zhan Z-K Zhang G-Q Sun and P M HuildquoHow events determine spreading patterns informationtransmission via internal and external influences on socialnetworksrdquo New Journal of Physics vol 17 no 11 Article ID113045 2015

[32] Z-K Zhang C Liu X-X Zhan X Lu C-X Zhang andY-C Zhang ldquoDynamics of information diffusion and itsapplications on complex networksrdquo Physics Reports vol 651pp 1ndash34 2016

[33] M E J Newman ldquo-e structure and function of complexnetworksrdquo SIAM Review vol 45 no 2 pp 167ndash256 2003

[34] M NewmanNetworks Oxford University Press Oxford UK2018

14 Complexity

[35] C Castellano S Fortunato and V Loreto ldquoStatistical physicsof social dynamicsrdquo Reviews of Modern Physics vol 81 no 2pp 591ndash646 2009

[36] J P Gleeson and D J Cahalane ldquoSeed size strongly affectscascades on random networksrdquo Physical Review E vol 75no 5 Article ID 056103 2007

[37] D E Whitney ldquoDynamic theory of cascades on finite clus-tered random networks with a threshold rulerdquo Physical Re-view E vol 82 no 6 Article ID 066110 2010

[38] A Nematzadeh E Ferrara A Flammini and Y AhnldquoOptimal network modularity for information diffusionrdquoPhysical Review Letters vol 113 no 25 Article ID 0887012014

[39] M Liu W Wang Y Liu M Tang S Cai and H ZhangldquoSocial contagions on time-varying community networksrdquoPhysical Review E vol 95 no 5 2017

[40] Q H Liu W Wang M Tang T Zhou and Y Lai ldquoExplosivespreading on complex networks the role of synergyrdquo PhysicalReview E vol 95 no 4 Article ID 042320 2017

[41] W Wang Q Liu J Liang Y Hu and T Zhou ldquoCoevolutionSpreading in Complex Networksrdquo Physics and Societyvol 820 pp 1ndash51 2019

[42] W Wang M Tang P Shu and Z Wang ldquoDynamics of socialcontagions with heterogeneous adoption thresholds cross-over phenomena in phase transitionrdquo New Journal of Physicsvol 18 no 1 Article ID 013029 2016

[43] P S Dodds K D Harris and C M Danforth ldquoLimitedimitation contagion on random networks chaos universalityand unpredictabilityrdquo Physical Review Letters vol 110 no 15Article ID 158701 2013

[44] X Zhu W Wang S Cai and H E Stanley ldquoOptimal imi-tation capacity and crossover phenomenon in the dynamics ofsocial contagionsrdquo Journal of Statistical Mechanics Keoryand Experiment vol 2018 no 6 Article ID 063405 2018

[45] X Zhu W Wang S Cai and H E Stanley ldquoDynamics ofsocial contagions with local trend imitationrdquo Scientific Re-ports vol 8 no 1 p 7335 2018

[46] Z Wang A Szolnoki and M Perc ldquoOptimal in-terdependence between networks for the evolution of co-operationrdquo Scientific Reports vol 3 no 1 p 2470 2013

[47] Z Wang A Szolnoki and M Perc ldquoSelf-organization to-wards optimally interdependent networks by means of co-evolutionrdquo New Journal of Physics vol 16 no 3 Article ID033041 2014

[48] E Cozzo M Kivela M De Domenico et al ldquoStructure oftriadic relations in multiplex networksrdquo New Journal ofPhysics vol 17 no 7 Article ID 073029 2015

[49] W Wang M Tang H E Stanley and L A BraunsteinldquoSocial contagions with communication channel alternationonmultiplex networksrdquo Physical Review E vol 98 no 6 2018

[50] R Amato N E Kouvaris M S Miguel and A Dıaz-GuileraldquoOpinion competition dynamics on multiplex networksrdquoNew Journal of Physics vol 19 no 12 Article ID 123019 2017

[51] W Li S Tang W Fang Q Guo X Zhang and Z ZhengldquoHow multiple social networks affect user awareness theinformation diffusion process in multiplex networksrdquo Phys-ical Review E vol 92 no 4 Article ID 042810 2015

[52] X Chen R Wang M Tang S Cai H E Stanley andL A Braunstein ldquoSuppressing epidemic spreading in mul-tiplex networks with social-supportrdquo New Journal of Physicsvol 20 no 1 Article ID 013007 2018

[53] R Amato A Diaz-Guilera and K-K Kleineberg ldquoInterplaybetween social influence and competitive strategical games in

multiplex networksrdquo Scientific Reports vol 7 no 1 p 70872017

[54] R J Requejo and A Diaz-Guilera ldquoReplicator dynamics withdiffusion on multiplex networksrdquo Physical Review E vol 94no 2 Article ID 022301 2016

[55] M Zheng L Lu and M Zhao ldquoSpreading in online socialnetworks the role of social reinforcementrdquo Physical Review Evol 88 no 1 Article ID 012818 2013

[56] E M Tur P Zeppini and K Frenken ldquoDiffusion with socialreinforcement the role of individual preferencesrdquo PhysicalReview E vol 97 no 2 Article ID 002300 2018

[57] W Wang M Tang H-F Zhang and Y-C Lai ldquoDynamics ofsocial contagions with memory of nonredundant in-formationrdquo Physical Review E vol 92 no 1 Article ID012820 2015

[58] S Boccaletti V Latora Y Moreno M Chavez andD Hwang ldquoComplex networks structure and dynamicsrdquoPhysics Reports vol 424 no 4-5 pp 175ndash308 2006

[59] Y Sun C Liu C-X Zhang and Z-K Zhang ldquoEpidemicspreading on weighted complex networksrdquo Physics Letters Avol 378 no 7-8 pp 635ndash640 2014

[60] A Barrat M Barthelemy R Pastor-Satorras andA Vespignani ldquo-e architecture of complex weighted net-worksrdquo Proceedings of the National Academy of Sciencesvol 101 no 11 pp 3747ndash3752 2004

[61] Y ZhuWWang M Tang and Y Ahn ldquoSocial contagions onweighted networksrdquo Physical Review E vol 96 no 1 2017

[62] M Catanzaro M Boguntildea and R Pastor-Satorras ldquoGener-ation of uncorrelated random scale-free networksrdquo PhysicalReview E vol 71 no 2 Article ID 027103 2004

[63] R Pastor-Satorras and A Vespignani ldquoEpidemic spreading inscale-free networksrdquo Physical Review Letters vol 86 no 14pp 3200ndash3203 2001

[64] X-X Zhan C Liu G Zhou et al ldquoCoupling dynamics ofepidemic spreading and information diffusion on complexnetworksrdquo Applied Mathematics and Computation vol 332pp 437ndash448 2018

[65] J C Miller and E M Volz ldquoModel hierarchies in edge-basedcompartmental modeling for infectious disease spreadrdquoJournal of Mathematical Biology vol 67 no 4 pp 869ndash8992013

[66] J C Miller and E M Volz ldquoIncorporating disease andpopulation structure into models of sir disease in contactnetworksrdquo PLoS One vol 8 no 8 Article ID e69162 2013

[67] K Brian andM E J Newman ldquoMessage passing approach forgeneral epidemic modelsrdquo Physical Review E vol 82 no 1Article ID 016101 2010

[68] X Yuan Y Hu H E Stanley and S Havlin ldquoEradicatingcatastrophic collapse in interdependent networks via rein-forced nodesrdquo Proceedings of the National Academy of Sci-ences vol 114 no 13 pp 3311ndash3315 2017

[69] P Erdos and A Renyi ldquoOn random graphs irdquo PublicationesMathematicae vol 4 pp 3286ndash3291 1959

[70] P Shu Q-H Liu S Wang and W Wang ldquoSocial contagionson interconnected networks of heterogeneous populationsrdquoChaos An Interdisciplinary Journal of Nonlinear Sciencevol 28 no 11 Article ID 113114 2018

Complexity 15

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom