Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 781281 14 pageshttpdxdoiorg1011552013781281

Research ArticleCommunity Core Evolution in Mobile Social Networks

Hao Xu Weidong Xiao Daquan Tang Jiuyang Tang and Zhenwen Wang

Key Laboratory for Information System Technology National University of Defense Technology Changsha 410073 China

Correspondence should be addressed to Hao Xu xuhaonudteducn

Received 30 June 2013 Accepted 12 August 2013

Academic Editors F J Cabrerizo and F Giannini

Copyright copy 2013 Hao Xu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Community detection in social networks attracts a lot of attention in the recent years Existing methods always depict therelationship of two nodes using the temporary connection However these temporary connections cannot be fully recognized as thereal relationships when the history connections among nodes are considered For example a casual visit in Facebook cannot be seenas an establishment of friendship Hence our question is the following how to cluster the real friends in mobile social networks Inthis paper we study the problem of detecting the stable community core in mobile social networks The cumulative stable contactis proposed to depict the relationship among nodes The whole process is divided into timestamps Nodes and their connectionscan be added or removed at each timestamp and historical contacts are considered when detecting the community core Alsocommunity cores can be tracked through the incremental computing which can help to recognize the evolving of communitystructure Empirical studies on real-world social networks demonstrate that our proposed method can effectively detect stablecommunity cores in mobile social networks

1 Introduction

In the recent years the way of communication among peoplehas experienced a dramatic change Thanks to the devel-opment of mobile communication technology the relativegeographical topology among passengers can be caught eas-ily Hence clustering people in such mobile social networkswhich can be further used in information recommendationand other social services attracts more and more concerns

There are a lot of literatures concerned with the commu-nity detection in social networks including static approachand dynamic approach Nodes are usually depicted as peoplein the real world and links are always denoted to be thecontacts among nodes The static approach focuses on highaggregation of nodes which have the same features [1 2]While the dynamic approach divides the networkrsquos evolvingprocess into lots of timestamps they not only pay attentionto clustering nodes in the network but are also concernedwith the computational complexity at each timestamp [3 4]At each timestamp the computational complexity dependson change of links rather than all links in the network Itis very important when analyzing evolution of the networkstructure especially with multiple timestamps Howeverfew of these methods consider the stability of communities

between two timestamps Intuitively in our real world therelationship among people will not change sharply That isto say for a giving link 119897

119886119887(119905) which denotes a link between

nodes 119886 and 119887 at time 119905 119897119886119887

(119905) cannot depict the relationshipbetween nodes 119886 and 119887 exactly The study in [5] considersthe stability in community detection but it tries to obtain acommunity partition under the stablemodularity rather thanthe stable contact

Moreover [6] pointed out that intercontact time amongpeople follows the power-law distribution which means thefollowing (1) we spend most of our time contacting with theldquocommunityrdquo people (2) there are still a lot of temporarycontacts taking place between ldquostrangersrdquo If all of the linksare considered when detecting community some temporarylinks among ldquostrangersrdquo will influence the effect In order toeliminate the negative influence only familiar links should beconcerned And our question is the following how to find thereal friends having those familiar links with each other in amobile social network

The biggest feature of mobile social network is thatnodes and links are always changing The studies in [37] have classified all of the situations that occur at eachtimestamp into several events including node addremoveand link (edge) addremove And their experiment results

2 The Scientific World Journal

demonstrate that discretization of the continuous time isa useful way to model the evolution of the network Inthis paper the discretization of the continuous time is stilladopted when modeling the evolution process However theprominent difference of our method compared with others isthe discrimination between familiar link and temporary link

Based on previous works [8] the number of familiar linksis higher than that of temporary links In other words peoplewho come from the same community have higher contactfrequency than those who come from different communitiesAnd the changing frequency of familiar link is lower than thatof temporary link That is to say people who come from thesame community always keep the relatively stable contactwhile contacts among people who come from differentcommunities seem to be uncertain

Based on the discussion above we use ldquocommunity corerdquoto solve our problem Unlike our definition the conceptdenoted by [5] is based on the nondeterministic communitydetection algorithm Generally speaking a community corein mobile social networks is the subset of a community Forlinks in the same community core few changes will occurbetween consecutive timestamps

The study in [9] extracts the core structure of social net-works using (120572 120573) community The authors have discoveredthe core structure by lots of overlapping (120572 120573) communitiesHowever the proposed static heuristic algorithm did notconsider the dynamic features of social networks The studyin [5] analyzed the community core in evolving networks Byrecursive computation stable community cores are detectedand well tracked However the instability of communitiesin [5] is caused by the instability of the nondeterministicdetecting algorithm which does not suit the community coredetection under the instability of links at each timestamp

In this paper we propose a novel approach for communitycore detection in mobile social networks The main idea ofour approach is to find a partition based on stable links in agiving network To the best of our knowledge we are the firstto find the relatively stable community core using the historycumulative contact in mobile social networks and the first tofind the power-law distribution of these contactsrsquo changingbetween consecutive timestamps In order to recognize thechange of community core at each timestamp the trackingmechanism is also concerned

The rest of this paper is organized as follows We intro-duce the preliminaries used in this paper in Section 2 InSection 3 we discuss the characteristics of cumulative stablecontact Then we present our community core detection andtracking algorithm separately in Section 4 We evaluate ouralgorithms in Section 5 and we finally conclude the workwith Section 6

2 Preliminary

In this section we present the notion and themobile networkmodel that we will use throughout the paper

Definition 1 (mobile social network) A mobile social net-work is denoted as 119866 = ⟨119864 119881⟩ where 119881 is the vertex setand 119864 sube 119881 times 119881 is the link set

The nodes and links of a mobile social network willchange according to different timestamps A mobile socialnetwork at 119879 = 119905 is denoted as 119866(119905) = ⟨119864(119905) 119881(119905)⟩ Here119881(119905) is the set of nodes that appears at 119879 = 119905 and 119864(119905) is theset of links that appears at the same timestamp 119881(119905) sube 119881and 119864(119905) sube 119864

Topologies of a certain mobile social network are alwayschanging due to the time variation which is the most dif-ference compared with static networks Like previous workswe treat the continuous time as a sequence of timestampsFurthermore nodes and links may be different from theconsecutive timestamps Hence we use the following fourevents to describe the evolution of network node add noderemove link add and link remove

Definition 2 (cumulative stable contact (CSC)) There is aCSC between two nodes V

119894and V

119895if and only if their history

contact duration is higher than a threshold (we will discussthis threshold in the following section)

As mentioned before the temporary link cannot depictthe relationship between two nodes in the mobile socialnetwork Inversely two nodes that disconnect at119879 = 119905 cannotdemonstrate that they are irrelevant Considering the historyconnection among nodes we use cumulative contact to judgethe stability of links

Definition 3 (community core set) It is denoted as CRS =

CO0CO1 CO

119898 where CO

119894is a community core and

is a subset of 119866 ⋃119894=0

CO119894(119905) sube 119866(119905)

The community core at119879 = 119905 is a partition about the givennetwork the same as the concept ldquocommunityrdquo in previousworks A community set is defined as a partition of a givennetwork CS = 119862

0 1198621 119862

119898 However we only focus on

detecting the ldquouseful linksrdquo rather than carry on a clusterprocess Hence some of the nodes and links may not beincluded in CRS(119905) even if they appear at 119879 = 119905 Thereforethe biggest difference between community and communitycore is ⋃

119894=0CO119894(119905) sube ⋃

119895=0119862119895(119905)

3 Cumulative Stable Link

In this section we study the characteristics of CSC Firsta well-known mobile social network is introduced Then astable link extraction method is proposed to find the CSCFinally we discuss the distribution of the change of CSC(CCSC)

31 Dataset Due to the increasing concern about mobilesocial networks various datasets about peoplersquos behavior arecollected by researchers The studies in [8 10] have collectedthe traces information about attendants of INFOCOM06and SIGCOMM09 separately The features of these datasetsare as follows (1) These datasets include not only thecontact information but also the attributes of attendants TheSIGCOMM09 has 76 attendants and the INFOCOM06 has78 attendants (2) Both of these datasets contain several daystraces information more than 300000 timestamps can beused to describe the evolution of networks

The Scientific World Journal 3

SIGCOMM09 collects the traces information amongattendants in SIGCOMM 2009 The dataset not only recordsthe contact time of each device pair but also includes theprofile of each attendant such as country city instituteand interest The contact information is recorded in theform of lttimestamp user id seen user id gt which meansthe scanning time the scanning device and the discovereddevice Hence the cumulative contact pair at each timestampis easier to obtain

Similar to SIGCOMM09 INFOCOM06 collects the con-tact traces among attendants in INFOCOM 2006 Eachparticipant was asked to fill in a questionnaire that includedname nationality affiliation country and other items Thecontact information is also well refined by the author so thatin the form of ltuser id seen (user id) start time end time gt which means scanning device discovered device andtheir duration contact time

32 Stable Link Extraction Both the SIGCOMM09 andINFOCOM06 contain connection duration between eachpair of nodesWe use a contactmatrixM denoting the contactamong nodes 119898

119894119895(119905) is the cumulative contact duration

between V119894and V

119895from 119879 = 0 to 119879 = 119905

The study in [8] has studied the correlation betweenregularity and familiarity on Cambridge students and it isobserved that most of contacts among nodes reveal a shortduration while few of them have long duration which isdenoted as ldquocommunityrdquo In this paper we use M1015840 = [119898

1015840

119894119895]

to denote whether V119894and V

119895have a contact duration higher

than a threshold 120575sdotmax(119898119894119895)Then we cluster the attendants

into several groups by their friendship graphs which areextracted from the two datasets as follows

1198981015840

119894119895=

119898119894119895 119898119894119895ge 120575 sdotmax (119898)

0 119898119894119895lt 120575 sdotmax (119898)

(1)

33 Distribution of CCSC We first construct a contactduration matrix M = [119898

119894119895] where 119898

119894119895presents the his-

tory contact duration between V119894and V

119895during the whole

lifetime of the network (INFOCOM06 119879 = [6207 340927]SIGCOMM09 119879 = [21 349811]) Without loss of generalitywe denote X and Y as two consecutive M and then computethe change of history contact (CHC) which is depicted as thedistance of X and Y using

Distance (XY) = sum

119895

sum

119894

119904 (119894 119895)

119904 (119894 119895) = 0 119883

119894119895= 119884119894119895

1 119883119894119895

= 119884119894119895

(2)

The distribution of the distance is plotted in the log-logscale (Figure 1) The power-law distribution of intercontacttime in the mobile social network is fully discussed inthe existing literatures However the change of contactsin consecutive timestamps does not follow the power-lawdistribution Then we use M1015840 where 119898

1015840

119894119895denotes whether a

link between V119894and V

119895is a CSC or not and we compute the

distance of two consecutive M1015840 then the CCSC at different

timestamps is obtained (Figure 1(a)) The distribution ofCCSC is plotted in Figure 1(b) using log-log scale It is clearthat the CCSC extracted from SIGCOMM09 follows thepower-law distribution ranging from 2 changes to 6 changesIn INFOCOM06 when the changes range from 2 to 7 theCCSC also follows the power-law distribution The diversityof distribution between history contact duration and changeof cumulative stable contact might be caused by removal ofthe temporary contact Considering two people in the realworld the more familiar the more stable their relationship isMoreover people denoted as ldquofamiliarrdquo have longer contactduration and contact time which has been proven in previousworks According to the discussion above the CSC removesthe temporary links among nodes

4 Community Core Evolution

In this section we first introduce the community coredetection algorithm and then discuss the community coretracking mechanism in mobile social networks

41 Community Core Detection Let us first discuss the topol-ogy change of the network which is constantly updated bynodes and links changing through different timestamps Theincreasing nodes or links can be decomposed as a sequenceof node or link insertions while the decreasing nodes orlinks can be decomposed as a sequence of node or linkremovals We define four events that may cause the evolutionof network node add node remove link add and linkremove However in the definition above the communitycore is based on links between two nodes Hence a singlenode without associated CSC cannot exist in the communitycore Then we refine the events as shown in Figure 2

Link Add The cumulative contact between V119894and V119895is higher

than the current threshold then link 119890119894119895

that is associatedwith two nodes V

119894and V119895is added to a community core CO

Both V119894and V119895will be added to CO even if any of them did

not belong to CO

Link Remove The cumulative contact between V119894and V

119895

is lower than the current threshold then link 119890119894119895

that isassociated with two nodes V

119894and V

119895is removed from a

community core CO If V119894or V119895has no link associated with

other nodes in CO then the corresponding node will beremoved from CO

The basic operation procedure of community core detec-tion at one timestamp is described in Figures 3 and 4 Weextract the decreasing and increasing links at each timestampand then use LRS and LAS to denote the decreasing link setand the increasing link set separately

Giving a certain community core partition the LinkRemove will result in deleting existing community cores(Figure 3) while Link Add will result in expanding the com-munity core or inserting a new community core (Figure 4)The procedure can be divided into two phases Firstly wetreat the Link Remove set and refine the existing communitycore through the existence of connected path If there is nopath connected the link will be removed from CO Secondly

4 The Scientific World Journal

Contact variation

Prob

abili

ty

INFOCOM06SIGCOMM09

100

10minus2

10minus4

10minus6

100 101 102

(a)

Contact variation

Prob

abili

ty

INFOCOM06SIGCOMM09

6 changes

7 changes

100

10minus2

10minus4

10minus6

100 101 102

(b)

Figure 1 (a) Distribution of history contact duration (b) Distribution of CCSC

a b

c

d

e

f

g

med lt 120575 lowast max(m998400)

(a)

ab

cd

e

f

g

mbc gt 120575 lowast max(m998400)

(b)

Figure 2 Two events cause the structure variation of network (a) A link between nodes 119887 and 119888 is added into the network because 1198981015840

119887119888gt

120575 lowastmax(1198981015840) (b) A link between nodes 119889 and 119890 is removed from the network because 1198981015840

119889119890lt 120575 lowastmax(1198981015840)

Select linkij in LRS Yes LRS ne 120601 No End

i and j in the same core No Remove linkij in LRS

Split the core intotwo parts

No

There is a pathfrom i to j

Yes

No

No

YesYes

Yes

i or j not in CRS998400(t + 1)NoRemove linkij inCRS998400(t + 1)

Remove node not inCRS998400(t + 1)

CRS998400(t + 1) larrminus CRS(t)

m998400ij lt 120575 times max(m998400)

i and j in CRS998400(t + 1)

Figure 3 Basic operation procedure of link remove

The Scientific World Journal 5

Select linkij in LAS Yes

Yes

LAS ne 120601 No End

No Remove linkij in LASThe one neighbor nodejoins into the other core No

i and j have morethan one neighbor

NoNo

i and j in the same core Yes

Yes

Add linkij in CRS(t + 1)Create a new

community core

i in COp and j in COq Yes

Yes

i and j have onlyone neighbor

Merge the twocommunity cores

CRS(t + 1) larrminus CRS998400(t + 1)

Update M998400(t + 1)

m998400ij gt 120575 times max(m998400)

(COpq in CRS998400(t + 1))

Figure 4 Basic operation procedure of link add

a

c b

a

c

b

e

d

a c

b

a

cb

e

da

c

be

d

a

cb e

d

a

c

b e

d

a

c

b

e

d

a

c

b

d

a

c b

d

a c

bd

a

c

b

DeathBirth Merging

Splitting Growth Contraction

e

d

Figure 5 Six events provide variation of community cores The red soliddash line denotes the link addremove separately and differentcolors mean different community cores

by updating the max contact duration addition link and itsnodes will be added into the existing community core or anew community core

42 Evolution of Community Core In order to study theevolution process of community core we should track thecommunity core at each timestamp How to distinguish twocommunity cores in the consecutive timestamps is the biggestproblem about tracking

421 Model According to Definition 3 the evolution ofcommunity core can be presented by CRS

0CRS1 CRS

119896

For a community core at 119879 = 119905 CRS(119905) = CO0CO1

CO119898 and 119871(119905) = 119897

0 1198971 119897119899 is denoted as the label set

of community cores identifying a community core at 119879 = 119905

uniquely In the previous literatures there can be seen a broadconsensus on the basic events that can be used to describe theevolution of dynamic communities [3 7 11] We extend andspecify these events as shown in Figure 5

6 The Scientific World Journal

0 05 1 15 2 25 30

10

20

35times105

(a)

0 05 1 15 2 25 30

150

300

450

35times105

(b)

0 05 1 15 2 25 30

10

20

35times105

(c)

0 05 1 15 2 25 30

50

100

35times105

(d)

0 05 1 15 2 25 30

10

20

35times105

(e)

0 05 1 15 2 25 30

80

160

35times105

(f)

0 05 1 15 2 25 30

20

40

60

Link numberNode numberNode degree

35times105

(g)

0 05 1 15 2 25 3 350

100

200

300

Link numberNode numberNode degree

times105

(h)Figure 6 Description of two datasets SIGCOMM09 records the discrete contact time among nodes while INFOCOM06 records thecontinuous contact time (a) Total link number (SIGCOMM09) (b) total link number (INFOCOM06) (c) link add (SIGCOMM09)(d) link add (INFOCOM06) (e) link remove (SIGCOMM09) (f) link remove (INFOCOM06) (g) network feature (SIGCOMM09) and(h) network feature (INFOCOM06)

Birth It is the emergence of a new community core at 119879 =

119905 + 1 and there is no corresponding community core inCRS(119905) A community core COnew which is labeled as 119897new iscreated in CRS(119905 + 1) at 119879 = 119905 + 1 while COnew notin CRS(119905)That is for any 119897new isin 119871(119905 + 1) there is 119897new notin 119871(119905)

Death It is the disappearance of a community core COold at119879 = 119905 + 1 as COold exists in CRS(119905) at 119879 = 119905 There is nocorresponding community core in CRS(119905 + 1) If we use 119897olddenoting core label of COold then the death of a communitycoreCOold means 119897old isin 119871(119905) and for any 119897

119904isin 119871(119905+1) 119897

119904= 119897old

Merging It means that two community cores merge into onecommunity core at 119879 = 119905 + 1

Splitting A community core is divided into two separatecommunity cores

Growth A node joins into a community core And this willresult in no changes between 119871(119905) and 119871(119905 + 1)

Contraction A nodemoves out of a community coreThis alsowill result in no changes between 119871(119905) and 119871(119905 + 1)

422 Tracking Community Cores In the context of themodeldescribed above the most important thing is to identifythe changes of community cores Three questions should beanswered (1) Where does the core come from (2) Whathappened to community core after updating (3)Where is thecore going

The study in [12] is concerned with tracking communitiesbased on detected communities using the Jaccard coeffi-cient Differing from this method we track these cores byenhancing our algorithm When the algorithm runs thetracking process works simultaneously which is triggered bythe variation of links

According to our algorithm there exist only contactfrequency between two core nodes lower than 120575 sdot max(119898

119894119895)

and there is no other path connecting these two nodes inthe original core the splitting process will be triggered

The Scientific World Journal 7

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(a)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(b)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(c)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(d)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(e)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(f)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(g)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(h)

Figure 7 Number of changes in 0-1 matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 = 0204 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

Hence V119894and V

119895have their isolate core sets separately We

use node ID as the core label when creating a new communitycore Initially each node will be tagged a community label asitself The benefits are as follows Firstly the finite namespaceof core label will not cause muddle of naming Secondlycommunity core can be tracked easier when a communitycore is created again The tracking algorithm uses nodeID as initial community core ID if the node is extractedto a community core and when a node is removed fromcommunity core set the node ID is restored

Basic operation procedure of tracking is given as follows

Switch (events)Case Birth (V

119894and V119895form a new core)

119871 (119905 + 1) larr997888 119897new 119897new = 119894 or 119895 (3)

Case Death (CO119894isin CRS(119905) but CO

119894notin CRS(119905 + 1))

Every node in CO119894labeled as its node ID

CaseMerging (CO119894and CO

119895merge into on core)

Nodes with fewer core members change theirIDs to the other core

Case Splitting (V119894and V119895have no path to connect)

Delete 119897old in 119871(119905 + 1) and 119871(119905 + 1) larr 119894 119895 V119894and

V119895change label as their node ID

Case Growth (V119894join into an existing community

core)

V119894changes label as the community core

Case Contraction (V119894moves out of community core)

V119894changes label as its node ID

5 Evaluation

In this section we discuss the evolution of community corewhich is detected and tracked by our algorithm COPRA[13] is a well-known efficient algorithm of fast communitydetection and we choose COPRA to compare with our algo-rithm

51 Contact Variation In order to reveal the communitycore evolution briefly we first show the contact variationof both SIGCOMM09 and INFOCOM06 under the wholelifetime (Figure 6) Due to the difference of data preprocess-ing SIGCOMM09 records the discrete contact time amongnodes while INFOCOM06 records the continuous contacttime According to the description of SIGCOMM09 eachdevice performs a periodic Bluetooth device discovery every120 plusmn 1024 seconds (randomized) for 1024 seconds Whena device finds others it records the current time Howeverthis mechanism can only record one contact time in a periodfor the same pair which will miss some contact informationHence the total link change link add and link remove ofSIGCOMM09 almost have no differences

52 Change of 0-1 Contact Matrix According to M1015840 the 0-1 contact matrix B = [119887

119894119895] can be obtained M1015840 changes

when the time increases and the maximum contact durationamong nodes may be changed which results in the variationof B The change of B during the whole collection procedureis plotted in Figure 7 Consider the following

119887119894119895=

1 1198981015840

119894119895ge 120575 sdotmax (119898

1015840

)

0 1198981015840

119894119895lt 120575 sdotmax (119898

1015840

)

(4)

8 The Scientific World Journal

Nod

e cou

nt70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

Nod

e cou

nt

80

60

40

20

0

(b)

Aver

age n

ode c

ount

80

70

60

50

40

30

20

10

0

120575

01 03 05 07 09

SIGCOMM09INFOCOM06

(c)

Figure 8 Number of nodes in community core (a) Variation in number of nodes in community core in SIGCOMM09 (b) Variation innumber of nodes in community core in INFOCOM06 (c) Average number of nodes in community core in the two datasets under different120575

As shown in Figure 7 most of the changes occur atthe beginning of collection and they are then graduallydiminished over timeThe number of changes reduces by theincreasing of 120575 Meanwhile in SIGCOMM09 the maximumchanges of 0-1 contact matrix under 120575 = 02 04 06 08 are8 6 4 and 2 respectively while in INFOCOM06 the valuesare 12 4 2 and 2 With 120575 increasing the maximum changesof 0-1 contact matrix are reduced

53 Selected Node Count One of the biggest differencesbetween community and community core is the number ofclustered nodes In other words the community core of amobile social network is the subset of the whole communityAccording to previous works in [8] few of nodes can be

classified as ldquofamiliar strangersrdquo and ldquofriendsrdquo and nodes inthe ldquocommunityrdquo are even less Hence only the node pairswhich have high contact frequency can be selected to thecommunity core

Intuitively improving 120575 will result in fewer selectedcontacts and nodes which is illustrated in Figures 8(a) and8(b) Let us first consider SIGCOMM09 when time goesby the selected nodes become more and more stable Inthe first day of data collection the selected node changesdramatically especially under the low 120575 Then the stableduration of selected nodes prolonged and the selected nodesno longer have change after about 2lowast10

5 seconds Comparedwith different 120575 the lower 120575 will result in a higher selectednodes which is meaningless for community cores (when

The Scientific World Journal 9

Nod

e cou

nt35

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Nod

e cou

nt

80

60

40

20

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 9 Number of nodes in community detected by COPRA (a) Node number at each timestamp in communities (SIGCOMM09)(b) Number of node changes in community between two consecutive timestamps (SIGCOMM09) (c) Node number at each timestamp incommunities (SIGCOMM09) (d) Number of node changes in community between two consecutive timestamps (INFOCOM06)

120575 = 02 during the data collection the maximum selectednodes in SIGCOMM09 are 61 and the maximum selectednodes in INFOCOM06 are 71) while the higher the 120575 is thefewer nodes are selected to construct the community core Abrief comparison of average selected nodes under different120575 is depicted in Figure 8(c) The same as discussed abovethe average of selected nodes decreases when 120575 increasesAlthough the selected nodes increase with the decreasingof 120575 the variation of selected nodes in INFOCOM06 stillhas little difference Unlike SIGCOMM09 the selected nodesin INFOCOM06 change frequently even at the end of thedata collection This phenomenon can reflect the fact thatin SIGCOMM09 the ldquofriendsrdquo of attendants are relativelystable and participants usually contact with familiar people

while in INFOCOM06 contacts among strangers are morefrequent than in SIGCOMM09 hence the selected nodechanges more often Nevertheless the variation of selectednodes in SIGCOMM09 and INFOCOM06 is relatively stableafter 2lowast10

5 seconds which is important according to featuresof community core

The number of nodes in communities detected byCOPRA is depicted in Figure 9Different fromour algorithmthe number of nodes detected by COPRA is highly unstableAnd the number of changes between two consecutive times-tamps is also very large compared with our algorithm

54 Number of Community Cores A community core set iscomposed of selected nodes which divides a mobile social

10 The Scientific World Journal

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(a)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(b)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(c)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(d)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(e)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(f)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(g)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(h)

Figure 10 Number of changes in core matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 =02 04 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

network into pieces of closely connected fragments Hencethe community core number in a community set determinesthe fragmentation degree of this mobile social network Afterrunning our algorithm the community core count in eachtimestamp under different 120575 is obtained (Figure 10)

As depicted in Figure 11(a) after about 2 lowast 105 seconds

the community count under each different 120575 becomes fixedwhich is the same as the selected nodes However the corenumber in INFOCOM06 is more complicated Like thevariation of selected node the number of community corefluctuates with time changes see Figure 11(b)

As mentioned before this is due to the frequent contactamong strangers In other words attendants in SGICOMM09have a more fixed social circle than in INFOCOM06 Thelast but the most important feature of community core countis that the core number is not monotonically changing withchangesWe can see in Figure 11(c) that themaximumaveragecore count (asymp771) in SIGCOMM09 appears when 120575 = 02After a reduction to 120575 = 04 the average core count risesapproximate to 487 when 120575 = 5 The same phenomenonappears in INFOCOM06 the core count reaches the maxvalue at 120575 = 4 (asymp931) Then it declines approximate to 84at 120575 = 5 and grows approximate to 905 at 120575 = 6 Thereason for this situation can be explained by two sides Onone hand the higher will filter out more contacts whichmakes more fragments of the mobile network and increasesthe number of community cores On the other hand somenetwork fragments with low contact frequency will be movedout of the community core set entirely which results in thereduction of cores Hence the number of cores fluctuatesunder the different

Stability is very important to the community core Com-pared with Figures 11 and 12 the community core is much

more stable than traditional community detection algorithmThe number of communities detected by COPRA is depictedin Figure 12 which reveals high instability And the numberof changes between two consecutive timestamps is also verylarge compared with our algorithm

55 Change of Core Matrix In order to evaluate the changeof community core set between consecutive timestamps weuse the distance function discussed in Section 33 to illustratethe difference between consecutive community core sets

The distance function requires that the twomatrices havethe same line and row number However the nodes and linksin the community core set between consecutive timestampsare different According to the tracking algorithm nodes inthemobile social networks are assigned an initial communitycore label which is equal to their node ID And the finitenamespace of core label ensures that the label will notbe beyond the scope of node ID Then we construct acommunity core matrix CM = [119888119898

119894119895] where

119888119898119894119895=

node ID 119894 119895 notin CRS

core ID 119894 119895 isin CRS(5)

Note that although node ID is considered in computationit will not influence the final result of distance

The distance under different 120575 is depicted in Figure 10It can be observed that the majority of consecutive corematrices have no changes Only few of them have 1 dif-ferent element and the variation of 2 elements is evenless (Figure 13) With 120575 increasing the variation of core

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

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2 The Scientific World Journal

demonstrate that discretization of the continuous time isa useful way to model the evolution of the network Inthis paper the discretization of the continuous time is stilladopted when modeling the evolution process However theprominent difference of our method compared with others isthe discrimination between familiar link and temporary link

Based on previous works [8] the number of familiar linksis higher than that of temporary links In other words peoplewho come from the same community have higher contactfrequency than those who come from different communitiesAnd the changing frequency of familiar link is lower than thatof temporary link That is to say people who come from thesame community always keep the relatively stable contactwhile contacts among people who come from differentcommunities seem to be uncertain

Based on the discussion above we use ldquocommunity corerdquoto solve our problem Unlike our definition the conceptdenoted by [5] is based on the nondeterministic communitydetection algorithm Generally speaking a community corein mobile social networks is the subset of a community Forlinks in the same community core few changes will occurbetween consecutive timestamps

The study in [9] extracts the core structure of social net-works using (120572 120573) community The authors have discoveredthe core structure by lots of overlapping (120572 120573) communitiesHowever the proposed static heuristic algorithm did notconsider the dynamic features of social networks The studyin [5] analyzed the community core in evolving networks Byrecursive computation stable community cores are detectedand well tracked However the instability of communitiesin [5] is caused by the instability of the nondeterministicdetecting algorithm which does not suit the community coredetection under the instability of links at each timestamp

In this paper we propose a novel approach for communitycore detection in mobile social networks The main idea ofour approach is to find a partition based on stable links in agiving network To the best of our knowledge we are the firstto find the relatively stable community core using the historycumulative contact in mobile social networks and the first tofind the power-law distribution of these contactsrsquo changingbetween consecutive timestamps In order to recognize thechange of community core at each timestamp the trackingmechanism is also concerned

The rest of this paper is organized as follows We intro-duce the preliminaries used in this paper in Section 2 InSection 3 we discuss the characteristics of cumulative stablecontact Then we present our community core detection andtracking algorithm separately in Section 4 We evaluate ouralgorithms in Section 5 and we finally conclude the workwith Section 6

2 Preliminary

In this section we present the notion and themobile networkmodel that we will use throughout the paper

Definition 1 (mobile social network) A mobile social net-work is denoted as 119866 = ⟨119864 119881⟩ where 119881 is the vertex setand 119864 sube 119881 times 119881 is the link set

The nodes and links of a mobile social network willchange according to different timestamps A mobile socialnetwork at 119879 = 119905 is denoted as 119866(119905) = ⟨119864(119905) 119881(119905)⟩ Here119881(119905) is the set of nodes that appears at 119879 = 119905 and 119864(119905) is theset of links that appears at the same timestamp 119881(119905) sube 119881and 119864(119905) sube 119864

Topologies of a certain mobile social network are alwayschanging due to the time variation which is the most dif-ference compared with static networks Like previous workswe treat the continuous time as a sequence of timestampsFurthermore nodes and links may be different from theconsecutive timestamps Hence we use the following fourevents to describe the evolution of network node add noderemove link add and link remove

Definition 2 (cumulative stable contact (CSC)) There is aCSC between two nodes V

119894and V

119895if and only if their history

contact duration is higher than a threshold (we will discussthis threshold in the following section)

As mentioned before the temporary link cannot depictthe relationship between two nodes in the mobile socialnetwork Inversely two nodes that disconnect at119879 = 119905 cannotdemonstrate that they are irrelevant Considering the historyconnection among nodes we use cumulative contact to judgethe stability of links

Definition 3 (community core set) It is denoted as CRS =

CO0CO1 CO

119898 where CO

119894is a community core and

is a subset of 119866 ⋃119894=0

CO119894(119905) sube 119866(119905)

The community core at119879 = 119905 is a partition about the givennetwork the same as the concept ldquocommunityrdquo in previousworks A community set is defined as a partition of a givennetwork CS = 119862

0 1198621 119862

119898 However we only focus on

detecting the ldquouseful linksrdquo rather than carry on a clusterprocess Hence some of the nodes and links may not beincluded in CRS(119905) even if they appear at 119879 = 119905 Thereforethe biggest difference between community and communitycore is ⋃

119894=0CO119894(119905) sube ⋃

119895=0119862119895(119905)

3 Cumulative Stable Link

In this section we study the characteristics of CSC Firsta well-known mobile social network is introduced Then astable link extraction method is proposed to find the CSCFinally we discuss the distribution of the change of CSC(CCSC)

31 Dataset Due to the increasing concern about mobilesocial networks various datasets about peoplersquos behavior arecollected by researchers The studies in [8 10] have collectedthe traces information about attendants of INFOCOM06and SIGCOMM09 separately The features of these datasetsare as follows (1) These datasets include not only thecontact information but also the attributes of attendants TheSIGCOMM09 has 76 attendants and the INFOCOM06 has78 attendants (2) Both of these datasets contain several daystraces information more than 300000 timestamps can beused to describe the evolution of networks

The Scientific World Journal 3

SIGCOMM09 collects the traces information amongattendants in SIGCOMM 2009 The dataset not only recordsthe contact time of each device pair but also includes theprofile of each attendant such as country city instituteand interest The contact information is recorded in theform of lttimestamp user id seen user id gt which meansthe scanning time the scanning device and the discovereddevice Hence the cumulative contact pair at each timestampis easier to obtain

Similar to SIGCOMM09 INFOCOM06 collects the con-tact traces among attendants in INFOCOM 2006 Eachparticipant was asked to fill in a questionnaire that includedname nationality affiliation country and other items Thecontact information is also well refined by the author so thatin the form of ltuser id seen (user id) start time end time gt which means scanning device discovered device andtheir duration contact time

32 Stable Link Extraction Both the SIGCOMM09 andINFOCOM06 contain connection duration between eachpair of nodesWe use a contactmatrixM denoting the contactamong nodes 119898

119894119895(119905) is the cumulative contact duration

between V119894and V

119895from 119879 = 0 to 119879 = 119905

The study in [8] has studied the correlation betweenregularity and familiarity on Cambridge students and it isobserved that most of contacts among nodes reveal a shortduration while few of them have long duration which isdenoted as ldquocommunityrdquo In this paper we use M1015840 = [119898

1015840

119894119895]

to denote whether V119894and V

119895have a contact duration higher

than a threshold 120575sdotmax(119898119894119895)Then we cluster the attendants

into several groups by their friendship graphs which areextracted from the two datasets as follows

1198981015840

119894119895=

119898119894119895 119898119894119895ge 120575 sdotmax (119898)

0 119898119894119895lt 120575 sdotmax (119898)

(1)

33 Distribution of CCSC We first construct a contactduration matrix M = [119898

119894119895] where 119898

119894119895presents the his-

tory contact duration between V119894and V

119895during the whole

lifetime of the network (INFOCOM06 119879 = [6207 340927]SIGCOMM09 119879 = [21 349811]) Without loss of generalitywe denote X and Y as two consecutive M and then computethe change of history contact (CHC) which is depicted as thedistance of X and Y using

Distance (XY) = sum

119895

sum

119894

119904 (119894 119895)

119904 (119894 119895) = 0 119883

119894119895= 119884119894119895

1 119883119894119895

= 119884119894119895

(2)

The distribution of the distance is plotted in the log-logscale (Figure 1) The power-law distribution of intercontacttime in the mobile social network is fully discussed inthe existing literatures However the change of contactsin consecutive timestamps does not follow the power-lawdistribution Then we use M1015840 where 119898

1015840

119894119895denotes whether a

link between V119894and V

119895is a CSC or not and we compute the

distance of two consecutive M1015840 then the CCSC at different

timestamps is obtained (Figure 1(a)) The distribution ofCCSC is plotted in Figure 1(b) using log-log scale It is clearthat the CCSC extracted from SIGCOMM09 follows thepower-law distribution ranging from 2 changes to 6 changesIn INFOCOM06 when the changes range from 2 to 7 theCCSC also follows the power-law distribution The diversityof distribution between history contact duration and changeof cumulative stable contact might be caused by removal ofthe temporary contact Considering two people in the realworld the more familiar the more stable their relationship isMoreover people denoted as ldquofamiliarrdquo have longer contactduration and contact time which has been proven in previousworks According to the discussion above the CSC removesthe temporary links among nodes

4 Community Core Evolution

In this section we first introduce the community coredetection algorithm and then discuss the community coretracking mechanism in mobile social networks

41 Community Core Detection Let us first discuss the topol-ogy change of the network which is constantly updated bynodes and links changing through different timestamps Theincreasing nodes or links can be decomposed as a sequenceof node or link insertions while the decreasing nodes orlinks can be decomposed as a sequence of node or linkremovals We define four events that may cause the evolutionof network node add node remove link add and linkremove However in the definition above the communitycore is based on links between two nodes Hence a singlenode without associated CSC cannot exist in the communitycore Then we refine the events as shown in Figure 2

Link Add The cumulative contact between V119894and V119895is higher

than the current threshold then link 119890119894119895

that is associatedwith two nodes V

119894and V119895is added to a community core CO

Both V119894and V119895will be added to CO even if any of them did

not belong to CO

Link Remove The cumulative contact between V119894and V

119895

is lower than the current threshold then link 119890119894119895

that isassociated with two nodes V

119894and V

119895is removed from a

community core CO If V119894or V119895has no link associated with

other nodes in CO then the corresponding node will beremoved from CO

The basic operation procedure of community core detec-tion at one timestamp is described in Figures 3 and 4 Weextract the decreasing and increasing links at each timestampand then use LRS and LAS to denote the decreasing link setand the increasing link set separately

Giving a certain community core partition the LinkRemove will result in deleting existing community cores(Figure 3) while Link Add will result in expanding the com-munity core or inserting a new community core (Figure 4)The procedure can be divided into two phases Firstly wetreat the Link Remove set and refine the existing communitycore through the existence of connected path If there is nopath connected the link will be removed from CO Secondly

4 The Scientific World Journal

Contact variation

Prob

abili

ty

INFOCOM06SIGCOMM09

100

10minus2

10minus4

10minus6

100 101 102

(a)

Contact variation

Prob

abili

ty

INFOCOM06SIGCOMM09

6 changes

7 changes

100

10minus2

10minus4

10minus6

100 101 102

(b)

Figure 1 (a) Distribution of history contact duration (b) Distribution of CCSC

a b

c

d

e

f

g

med lt 120575 lowast max(m998400)

(a)

ab

cd

e

f

g

mbc gt 120575 lowast max(m998400)

(b)

Figure 2 Two events cause the structure variation of network (a) A link between nodes 119887 and 119888 is added into the network because 1198981015840

119887119888gt

120575 lowastmax(1198981015840) (b) A link between nodes 119889 and 119890 is removed from the network because 1198981015840

119889119890lt 120575 lowastmax(1198981015840)

Select linkij in LRS Yes LRS ne 120601 No End

i and j in the same core No Remove linkij in LRS

Split the core intotwo parts

No

There is a pathfrom i to j

Yes

No

No

YesYes

Yes

i or j not in CRS998400(t + 1)NoRemove linkij inCRS998400(t + 1)

Remove node not inCRS998400(t + 1)

CRS998400(t + 1) larrminus CRS(t)

m998400ij lt 120575 times max(m998400)

i and j in CRS998400(t + 1)

Figure 3 Basic operation procedure of link remove

The Scientific World Journal 5

Select linkij in LAS Yes

Yes

LAS ne 120601 No End

No Remove linkij in LASThe one neighbor nodejoins into the other core No

i and j have morethan one neighbor

NoNo

i and j in the same core Yes

Yes

Add linkij in CRS(t + 1)Create a new

community core

i in COp and j in COq Yes

Yes

i and j have onlyone neighbor

Merge the twocommunity cores

CRS(t + 1) larrminus CRS998400(t + 1)

Update M998400(t + 1)

m998400ij gt 120575 times max(m998400)

(COpq in CRS998400(t + 1))

Figure 4 Basic operation procedure of link add

a

c b

a

c

b

e

d

a c

b

a

cb

e

da

c

be

d

a

cb e

d

a

c

b e

d

a

c

b

e

d

a

c

b

d

a

c b

d

a c

bd

a

c

b

DeathBirth Merging

Splitting Growth Contraction

e

d

Figure 5 Six events provide variation of community cores The red soliddash line denotes the link addremove separately and differentcolors mean different community cores

by updating the max contact duration addition link and itsnodes will be added into the existing community core or anew community core

42 Evolution of Community Core In order to study theevolution process of community core we should track thecommunity core at each timestamp How to distinguish twocommunity cores in the consecutive timestamps is the biggestproblem about tracking

421 Model According to Definition 3 the evolution ofcommunity core can be presented by CRS

0CRS1 CRS

119896

For a community core at 119879 = 119905 CRS(119905) = CO0CO1

CO119898 and 119871(119905) = 119897

0 1198971 119897119899 is denoted as the label set

of community cores identifying a community core at 119879 = 119905

uniquely In the previous literatures there can be seen a broadconsensus on the basic events that can be used to describe theevolution of dynamic communities [3 7 11] We extend andspecify these events as shown in Figure 5

6 The Scientific World Journal

0 05 1 15 2 25 30

10

20

35times105

(a)

0 05 1 15 2 25 30

150

300

450

35times105

(b)

0 05 1 15 2 25 30

10

20

35times105

(c)

0 05 1 15 2 25 30

50

100

35times105

(d)

0 05 1 15 2 25 30

10

20

35times105

(e)

0 05 1 15 2 25 30

80

160

35times105

(f)

0 05 1 15 2 25 30

20

40

60

Link numberNode numberNode degree

35times105

(g)

0 05 1 15 2 25 3 350

100

200

300

Link numberNode numberNode degree

times105

(h)Figure 6 Description of two datasets SIGCOMM09 records the discrete contact time among nodes while INFOCOM06 records thecontinuous contact time (a) Total link number (SIGCOMM09) (b) total link number (INFOCOM06) (c) link add (SIGCOMM09)(d) link add (INFOCOM06) (e) link remove (SIGCOMM09) (f) link remove (INFOCOM06) (g) network feature (SIGCOMM09) and(h) network feature (INFOCOM06)

Birth It is the emergence of a new community core at 119879 =

119905 + 1 and there is no corresponding community core inCRS(119905) A community core COnew which is labeled as 119897new iscreated in CRS(119905 + 1) at 119879 = 119905 + 1 while COnew notin CRS(119905)That is for any 119897new isin 119871(119905 + 1) there is 119897new notin 119871(119905)

Death It is the disappearance of a community core COold at119879 = 119905 + 1 as COold exists in CRS(119905) at 119879 = 119905 There is nocorresponding community core in CRS(119905 + 1) If we use 119897olddenoting core label of COold then the death of a communitycoreCOold means 119897old isin 119871(119905) and for any 119897

119904isin 119871(119905+1) 119897

119904= 119897old

Merging It means that two community cores merge into onecommunity core at 119879 = 119905 + 1

Splitting A community core is divided into two separatecommunity cores

Growth A node joins into a community core And this willresult in no changes between 119871(119905) and 119871(119905 + 1)

Contraction A nodemoves out of a community coreThis alsowill result in no changes between 119871(119905) and 119871(119905 + 1)

422 Tracking Community Cores In the context of themodeldescribed above the most important thing is to identifythe changes of community cores Three questions should beanswered (1) Where does the core come from (2) Whathappened to community core after updating (3)Where is thecore going

The study in [12] is concerned with tracking communitiesbased on detected communities using the Jaccard coeffi-cient Differing from this method we track these cores byenhancing our algorithm When the algorithm runs thetracking process works simultaneously which is triggered bythe variation of links

According to our algorithm there exist only contactfrequency between two core nodes lower than 120575 sdot max(119898

119894119895)

and there is no other path connecting these two nodes inthe original core the splitting process will be triggered

The Scientific World Journal 7

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(a)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(b)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(c)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(d)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(e)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(f)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(g)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(h)

Figure 7 Number of changes in 0-1 matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 = 0204 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

Hence V119894and V

119895have their isolate core sets separately We

use node ID as the core label when creating a new communitycore Initially each node will be tagged a community label asitself The benefits are as follows Firstly the finite namespaceof core label will not cause muddle of naming Secondlycommunity core can be tracked easier when a communitycore is created again The tracking algorithm uses nodeID as initial community core ID if the node is extractedto a community core and when a node is removed fromcommunity core set the node ID is restored

Basic operation procedure of tracking is given as follows

Switch (events)Case Birth (V

119894and V119895form a new core)

119871 (119905 + 1) larr997888 119897new 119897new = 119894 or 119895 (3)

Case Death (CO119894isin CRS(119905) but CO

119894notin CRS(119905 + 1))

Every node in CO119894labeled as its node ID

CaseMerging (CO119894and CO

119895merge into on core)

Nodes with fewer core members change theirIDs to the other core

Case Splitting (V119894and V119895have no path to connect)

Delete 119897old in 119871(119905 + 1) and 119871(119905 + 1) larr 119894 119895 V119894and

V119895change label as their node ID

Case Growth (V119894join into an existing community

core)

V119894changes label as the community core

Case Contraction (V119894moves out of community core)

V119894changes label as its node ID

5 Evaluation

In this section we discuss the evolution of community corewhich is detected and tracked by our algorithm COPRA[13] is a well-known efficient algorithm of fast communitydetection and we choose COPRA to compare with our algo-rithm

51 Contact Variation In order to reveal the communitycore evolution briefly we first show the contact variationof both SIGCOMM09 and INFOCOM06 under the wholelifetime (Figure 6) Due to the difference of data preprocess-ing SIGCOMM09 records the discrete contact time amongnodes while INFOCOM06 records the continuous contacttime According to the description of SIGCOMM09 eachdevice performs a periodic Bluetooth device discovery every120 plusmn 1024 seconds (randomized) for 1024 seconds Whena device finds others it records the current time Howeverthis mechanism can only record one contact time in a periodfor the same pair which will miss some contact informationHence the total link change link add and link remove ofSIGCOMM09 almost have no differences

52 Change of 0-1 Contact Matrix According to M1015840 the 0-1 contact matrix B = [119887

119894119895] can be obtained M1015840 changes

when the time increases and the maximum contact durationamong nodes may be changed which results in the variationof B The change of B during the whole collection procedureis plotted in Figure 7 Consider the following

119887119894119895=

1 1198981015840

119894119895ge 120575 sdotmax (119898

1015840

)

0 1198981015840

119894119895lt 120575 sdotmax (119898

1015840

)

(4)

8 The Scientific World Journal

Nod

e cou

nt70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

Nod

e cou

nt

80

60

40

20

0

(b)

Aver

age n

ode c

ount

80

70

60

50

40

30

20

10

0

120575

01 03 05 07 09

SIGCOMM09INFOCOM06

(c)

Figure 8 Number of nodes in community core (a) Variation in number of nodes in community core in SIGCOMM09 (b) Variation innumber of nodes in community core in INFOCOM06 (c) Average number of nodes in community core in the two datasets under different120575

As shown in Figure 7 most of the changes occur atthe beginning of collection and they are then graduallydiminished over timeThe number of changes reduces by theincreasing of 120575 Meanwhile in SIGCOMM09 the maximumchanges of 0-1 contact matrix under 120575 = 02 04 06 08 are8 6 4 and 2 respectively while in INFOCOM06 the valuesare 12 4 2 and 2 With 120575 increasing the maximum changesof 0-1 contact matrix are reduced

53 Selected Node Count One of the biggest differencesbetween community and community core is the number ofclustered nodes In other words the community core of amobile social network is the subset of the whole communityAccording to previous works in [8] few of nodes can be

classified as ldquofamiliar strangersrdquo and ldquofriendsrdquo and nodes inthe ldquocommunityrdquo are even less Hence only the node pairswhich have high contact frequency can be selected to thecommunity core

Intuitively improving 120575 will result in fewer selectedcontacts and nodes which is illustrated in Figures 8(a) and8(b) Let us first consider SIGCOMM09 when time goesby the selected nodes become more and more stable Inthe first day of data collection the selected node changesdramatically especially under the low 120575 Then the stableduration of selected nodes prolonged and the selected nodesno longer have change after about 2lowast10

5 seconds Comparedwith different 120575 the lower 120575 will result in a higher selectednodes which is meaningless for community cores (when

The Scientific World Journal 9

Nod

e cou

nt35

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Nod

e cou

nt

80

60

40

20

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 9 Number of nodes in community detected by COPRA (a) Node number at each timestamp in communities (SIGCOMM09)(b) Number of node changes in community between two consecutive timestamps (SIGCOMM09) (c) Node number at each timestamp incommunities (SIGCOMM09) (d) Number of node changes in community between two consecutive timestamps (INFOCOM06)

120575 = 02 during the data collection the maximum selectednodes in SIGCOMM09 are 61 and the maximum selectednodes in INFOCOM06 are 71) while the higher the 120575 is thefewer nodes are selected to construct the community core Abrief comparison of average selected nodes under different120575 is depicted in Figure 8(c) The same as discussed abovethe average of selected nodes decreases when 120575 increasesAlthough the selected nodes increase with the decreasingof 120575 the variation of selected nodes in INFOCOM06 stillhas little difference Unlike SIGCOMM09 the selected nodesin INFOCOM06 change frequently even at the end of thedata collection This phenomenon can reflect the fact thatin SIGCOMM09 the ldquofriendsrdquo of attendants are relativelystable and participants usually contact with familiar people

while in INFOCOM06 contacts among strangers are morefrequent than in SIGCOMM09 hence the selected nodechanges more often Nevertheless the variation of selectednodes in SIGCOMM09 and INFOCOM06 is relatively stableafter 2lowast10

5 seconds which is important according to featuresof community core

The number of nodes in communities detected byCOPRA is depicted in Figure 9Different fromour algorithmthe number of nodes detected by COPRA is highly unstableAnd the number of changes between two consecutive times-tamps is also very large compared with our algorithm

54 Number of Community Cores A community core set iscomposed of selected nodes which divides a mobile social

10 The Scientific World Journal

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(a)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(b)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(c)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(d)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(e)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(f)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(g)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(h)

Figure 10 Number of changes in core matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 =02 04 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

network into pieces of closely connected fragments Hencethe community core number in a community set determinesthe fragmentation degree of this mobile social network Afterrunning our algorithm the community core count in eachtimestamp under different 120575 is obtained (Figure 10)

As depicted in Figure 11(a) after about 2 lowast 105 seconds

the community count under each different 120575 becomes fixedwhich is the same as the selected nodes However the corenumber in INFOCOM06 is more complicated Like thevariation of selected node the number of community corefluctuates with time changes see Figure 11(b)

As mentioned before this is due to the frequent contactamong strangers In other words attendants in SGICOMM09have a more fixed social circle than in INFOCOM06 Thelast but the most important feature of community core countis that the core number is not monotonically changing withchangesWe can see in Figure 11(c) that themaximumaveragecore count (asymp771) in SIGCOMM09 appears when 120575 = 02After a reduction to 120575 = 04 the average core count risesapproximate to 487 when 120575 = 5 The same phenomenonappears in INFOCOM06 the core count reaches the maxvalue at 120575 = 4 (asymp931) Then it declines approximate to 84at 120575 = 5 and grows approximate to 905 at 120575 = 6 Thereason for this situation can be explained by two sides Onone hand the higher will filter out more contacts whichmakes more fragments of the mobile network and increasesthe number of community cores On the other hand somenetwork fragments with low contact frequency will be movedout of the community core set entirely which results in thereduction of cores Hence the number of cores fluctuatesunder the different

Stability is very important to the community core Com-pared with Figures 11 and 12 the community core is much

more stable than traditional community detection algorithmThe number of communities detected by COPRA is depictedin Figure 12 which reveals high instability And the numberof changes between two consecutive timestamps is also verylarge compared with our algorithm

55 Change of Core Matrix In order to evaluate the changeof community core set between consecutive timestamps weuse the distance function discussed in Section 33 to illustratethe difference between consecutive community core sets

The distance function requires that the twomatrices havethe same line and row number However the nodes and linksin the community core set between consecutive timestampsare different According to the tracking algorithm nodes inthemobile social networks are assigned an initial communitycore label which is equal to their node ID And the finitenamespace of core label ensures that the label will notbe beyond the scope of node ID Then we construct acommunity core matrix CM = [119888119898

119894119895] where

119888119898119894119895=

node ID 119894 119895 notin CRS

core ID 119894 119895 isin CRS(5)

Note that although node ID is considered in computationit will not influence the final result of distance

The distance under different 120575 is depicted in Figure 10It can be observed that the majority of consecutive corematrices have no changes Only few of them have 1 dif-ferent element and the variation of 2 elements is evenless (Figure 13) With 120575 increasing the variation of core

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

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The Scientific World Journal 3

SIGCOMM09 collects the traces information amongattendants in SIGCOMM 2009 The dataset not only recordsthe contact time of each device pair but also includes theprofile of each attendant such as country city instituteand interest The contact information is recorded in theform of lttimestamp user id seen user id gt which meansthe scanning time the scanning device and the discovereddevice Hence the cumulative contact pair at each timestampis easier to obtain

Similar to SIGCOMM09 INFOCOM06 collects the con-tact traces among attendants in INFOCOM 2006 Eachparticipant was asked to fill in a questionnaire that includedname nationality affiliation country and other items Thecontact information is also well refined by the author so thatin the form of ltuser id seen (user id) start time end time gt which means scanning device discovered device andtheir duration contact time

32 Stable Link Extraction Both the SIGCOMM09 andINFOCOM06 contain connection duration between eachpair of nodesWe use a contactmatrixM denoting the contactamong nodes 119898

119894119895(119905) is the cumulative contact duration

between V119894and V

119895from 119879 = 0 to 119879 = 119905

The study in [8] has studied the correlation betweenregularity and familiarity on Cambridge students and it isobserved that most of contacts among nodes reveal a shortduration while few of them have long duration which isdenoted as ldquocommunityrdquo In this paper we use M1015840 = [119898

1015840

119894119895]

to denote whether V119894and V

119895have a contact duration higher

than a threshold 120575sdotmax(119898119894119895)Then we cluster the attendants

into several groups by their friendship graphs which areextracted from the two datasets as follows

1198981015840

119894119895=

119898119894119895 119898119894119895ge 120575 sdotmax (119898)

0 119898119894119895lt 120575 sdotmax (119898)

(1)

33 Distribution of CCSC We first construct a contactduration matrix M = [119898

119894119895] where 119898

119894119895presents the his-

tory contact duration between V119894and V

119895during the whole

lifetime of the network (INFOCOM06 119879 = [6207 340927]SIGCOMM09 119879 = [21 349811]) Without loss of generalitywe denote X and Y as two consecutive M and then computethe change of history contact (CHC) which is depicted as thedistance of X and Y using

Distance (XY) = sum

119895

sum

119894

119904 (119894 119895)

119904 (119894 119895) = 0 119883

119894119895= 119884119894119895

1 119883119894119895

= 119884119894119895

(2)

The distribution of the distance is plotted in the log-logscale (Figure 1) The power-law distribution of intercontacttime in the mobile social network is fully discussed inthe existing literatures However the change of contactsin consecutive timestamps does not follow the power-lawdistribution Then we use M1015840 where 119898

1015840

119894119895denotes whether a

link between V119894and V

119895is a CSC or not and we compute the

distance of two consecutive M1015840 then the CCSC at different

timestamps is obtained (Figure 1(a)) The distribution ofCCSC is plotted in Figure 1(b) using log-log scale It is clearthat the CCSC extracted from SIGCOMM09 follows thepower-law distribution ranging from 2 changes to 6 changesIn INFOCOM06 when the changes range from 2 to 7 theCCSC also follows the power-law distribution The diversityof distribution between history contact duration and changeof cumulative stable contact might be caused by removal ofthe temporary contact Considering two people in the realworld the more familiar the more stable their relationship isMoreover people denoted as ldquofamiliarrdquo have longer contactduration and contact time which has been proven in previousworks According to the discussion above the CSC removesthe temporary links among nodes

4 Community Core Evolution

In this section we first introduce the community coredetection algorithm and then discuss the community coretracking mechanism in mobile social networks

41 Community Core Detection Let us first discuss the topol-ogy change of the network which is constantly updated bynodes and links changing through different timestamps Theincreasing nodes or links can be decomposed as a sequenceof node or link insertions while the decreasing nodes orlinks can be decomposed as a sequence of node or linkremovals We define four events that may cause the evolutionof network node add node remove link add and linkremove However in the definition above the communitycore is based on links between two nodes Hence a singlenode without associated CSC cannot exist in the communitycore Then we refine the events as shown in Figure 2

Link Add The cumulative contact between V119894and V119895is higher

than the current threshold then link 119890119894119895

that is associatedwith two nodes V

119894and V119895is added to a community core CO

Both V119894and V119895will be added to CO even if any of them did

not belong to CO

Link Remove The cumulative contact between V119894and V

119895

is lower than the current threshold then link 119890119894119895

that isassociated with two nodes V

119894and V

119895is removed from a

community core CO If V119894or V119895has no link associated with

other nodes in CO then the corresponding node will beremoved from CO

The basic operation procedure of community core detec-tion at one timestamp is described in Figures 3 and 4 Weextract the decreasing and increasing links at each timestampand then use LRS and LAS to denote the decreasing link setand the increasing link set separately

Giving a certain community core partition the LinkRemove will result in deleting existing community cores(Figure 3) while Link Add will result in expanding the com-munity core or inserting a new community core (Figure 4)The procedure can be divided into two phases Firstly wetreat the Link Remove set and refine the existing communitycore through the existence of connected path If there is nopath connected the link will be removed from CO Secondly

4 The Scientific World Journal

Contact variation

Prob

abili

ty

INFOCOM06SIGCOMM09

100

10minus2

10minus4

10minus6

100 101 102

(a)

Contact variation

Prob

abili

ty

INFOCOM06SIGCOMM09

6 changes

7 changes

100

10minus2

10minus4

10minus6

100 101 102

(b)

Figure 1 (a) Distribution of history contact duration (b) Distribution of CCSC

a b

c

d

e

f

g

med lt 120575 lowast max(m998400)

(a)

ab

cd

e

f

g

mbc gt 120575 lowast max(m998400)

(b)

Figure 2 Two events cause the structure variation of network (a) A link between nodes 119887 and 119888 is added into the network because 1198981015840

119887119888gt

120575 lowastmax(1198981015840) (b) A link between nodes 119889 and 119890 is removed from the network because 1198981015840

119889119890lt 120575 lowastmax(1198981015840)

Select linkij in LRS Yes LRS ne 120601 No End

i and j in the same core No Remove linkij in LRS

Split the core intotwo parts

No

There is a pathfrom i to j

Yes

No

No

YesYes

Yes

i or j not in CRS998400(t + 1)NoRemove linkij inCRS998400(t + 1)

Remove node not inCRS998400(t + 1)

CRS998400(t + 1) larrminus CRS(t)

m998400ij lt 120575 times max(m998400)

i and j in CRS998400(t + 1)

Figure 3 Basic operation procedure of link remove

The Scientific World Journal 5

Select linkij in LAS Yes

Yes

LAS ne 120601 No End

No Remove linkij in LASThe one neighbor nodejoins into the other core No

i and j have morethan one neighbor

NoNo

i and j in the same core Yes

Yes

Add linkij in CRS(t + 1)Create a new

community core

i in COp and j in COq Yes

Yes

i and j have onlyone neighbor

Merge the twocommunity cores

CRS(t + 1) larrminus CRS998400(t + 1)

Update M998400(t + 1)

m998400ij gt 120575 times max(m998400)

(COpq in CRS998400(t + 1))

Figure 4 Basic operation procedure of link add

a

c b

a

c

b

e

d

a c

b

a

cb

e

da

c

be

d

a

cb e

d

a

c

b e

d

a

c

b

e

d

a

c

b

d

a

c b

d

a c

bd

a

c

b

DeathBirth Merging

Splitting Growth Contraction

e

d

Figure 5 Six events provide variation of community cores The red soliddash line denotes the link addremove separately and differentcolors mean different community cores

by updating the max contact duration addition link and itsnodes will be added into the existing community core or anew community core

42 Evolution of Community Core In order to study theevolution process of community core we should track thecommunity core at each timestamp How to distinguish twocommunity cores in the consecutive timestamps is the biggestproblem about tracking

421 Model According to Definition 3 the evolution ofcommunity core can be presented by CRS

0CRS1 CRS

119896

For a community core at 119879 = 119905 CRS(119905) = CO0CO1

CO119898 and 119871(119905) = 119897

0 1198971 119897119899 is denoted as the label set

of community cores identifying a community core at 119879 = 119905

uniquely In the previous literatures there can be seen a broadconsensus on the basic events that can be used to describe theevolution of dynamic communities [3 7 11] We extend andspecify these events as shown in Figure 5

6 The Scientific World Journal

0 05 1 15 2 25 30

10

20

35times105

(a)

0 05 1 15 2 25 30

150

300

450

35times105

(b)

0 05 1 15 2 25 30

10

20

35times105

(c)

0 05 1 15 2 25 30

50

100

35times105

(d)

0 05 1 15 2 25 30

10

20

35times105

(e)

0 05 1 15 2 25 30

80

160

35times105

(f)

0 05 1 15 2 25 30

20

40

60

Link numberNode numberNode degree

35times105

(g)

0 05 1 15 2 25 3 350

100

200

300

Link numberNode numberNode degree

times105

(h)Figure 6 Description of two datasets SIGCOMM09 records the discrete contact time among nodes while INFOCOM06 records thecontinuous contact time (a) Total link number (SIGCOMM09) (b) total link number (INFOCOM06) (c) link add (SIGCOMM09)(d) link add (INFOCOM06) (e) link remove (SIGCOMM09) (f) link remove (INFOCOM06) (g) network feature (SIGCOMM09) and(h) network feature (INFOCOM06)

Birth It is the emergence of a new community core at 119879 =

119905 + 1 and there is no corresponding community core inCRS(119905) A community core COnew which is labeled as 119897new iscreated in CRS(119905 + 1) at 119879 = 119905 + 1 while COnew notin CRS(119905)That is for any 119897new isin 119871(119905 + 1) there is 119897new notin 119871(119905)

Death It is the disappearance of a community core COold at119879 = 119905 + 1 as COold exists in CRS(119905) at 119879 = 119905 There is nocorresponding community core in CRS(119905 + 1) If we use 119897olddenoting core label of COold then the death of a communitycoreCOold means 119897old isin 119871(119905) and for any 119897

119904isin 119871(119905+1) 119897

119904= 119897old

Merging It means that two community cores merge into onecommunity core at 119879 = 119905 + 1

Splitting A community core is divided into two separatecommunity cores

Growth A node joins into a community core And this willresult in no changes between 119871(119905) and 119871(119905 + 1)

Contraction A nodemoves out of a community coreThis alsowill result in no changes between 119871(119905) and 119871(119905 + 1)

422 Tracking Community Cores In the context of themodeldescribed above the most important thing is to identifythe changes of community cores Three questions should beanswered (1) Where does the core come from (2) Whathappened to community core after updating (3)Where is thecore going

The study in [12] is concerned with tracking communitiesbased on detected communities using the Jaccard coeffi-cient Differing from this method we track these cores byenhancing our algorithm When the algorithm runs thetracking process works simultaneously which is triggered bythe variation of links

According to our algorithm there exist only contactfrequency between two core nodes lower than 120575 sdot max(119898

119894119895)

and there is no other path connecting these two nodes inthe original core the splitting process will be triggered

The Scientific World Journal 7

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(a)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(b)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(c)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(d)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(e)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(f)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(g)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(h)

Figure 7 Number of changes in 0-1 matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 = 0204 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

Hence V119894and V

119895have their isolate core sets separately We

use node ID as the core label when creating a new communitycore Initially each node will be tagged a community label asitself The benefits are as follows Firstly the finite namespaceof core label will not cause muddle of naming Secondlycommunity core can be tracked easier when a communitycore is created again The tracking algorithm uses nodeID as initial community core ID if the node is extractedto a community core and when a node is removed fromcommunity core set the node ID is restored

Basic operation procedure of tracking is given as follows

Switch (events)Case Birth (V

119894and V119895form a new core)

119871 (119905 + 1) larr997888 119897new 119897new = 119894 or 119895 (3)

Case Death (CO119894isin CRS(119905) but CO

119894notin CRS(119905 + 1))

Every node in CO119894labeled as its node ID

CaseMerging (CO119894and CO

119895merge into on core)

Nodes with fewer core members change theirIDs to the other core

Case Splitting (V119894and V119895have no path to connect)

Delete 119897old in 119871(119905 + 1) and 119871(119905 + 1) larr 119894 119895 V119894and

V119895change label as their node ID

Case Growth (V119894join into an existing community

core)

V119894changes label as the community core

Case Contraction (V119894moves out of community core)

V119894changes label as its node ID

5 Evaluation

In this section we discuss the evolution of community corewhich is detected and tracked by our algorithm COPRA[13] is a well-known efficient algorithm of fast communitydetection and we choose COPRA to compare with our algo-rithm

51 Contact Variation In order to reveal the communitycore evolution briefly we first show the contact variationof both SIGCOMM09 and INFOCOM06 under the wholelifetime (Figure 6) Due to the difference of data preprocess-ing SIGCOMM09 records the discrete contact time amongnodes while INFOCOM06 records the continuous contacttime According to the description of SIGCOMM09 eachdevice performs a periodic Bluetooth device discovery every120 plusmn 1024 seconds (randomized) for 1024 seconds Whena device finds others it records the current time Howeverthis mechanism can only record one contact time in a periodfor the same pair which will miss some contact informationHence the total link change link add and link remove ofSIGCOMM09 almost have no differences

52 Change of 0-1 Contact Matrix According to M1015840 the 0-1 contact matrix B = [119887

119894119895] can be obtained M1015840 changes

when the time increases and the maximum contact durationamong nodes may be changed which results in the variationof B The change of B during the whole collection procedureis plotted in Figure 7 Consider the following

119887119894119895=

1 1198981015840

119894119895ge 120575 sdotmax (119898

1015840

)

0 1198981015840

119894119895lt 120575 sdotmax (119898

1015840

)

(4)

8 The Scientific World Journal

Nod

e cou

nt70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

Nod

e cou

nt

80

60

40

20

0

(b)

Aver

age n

ode c

ount

80

70

60

50

40

30

20

10

0

120575

01 03 05 07 09

SIGCOMM09INFOCOM06

(c)

Figure 8 Number of nodes in community core (a) Variation in number of nodes in community core in SIGCOMM09 (b) Variation innumber of nodes in community core in INFOCOM06 (c) Average number of nodes in community core in the two datasets under different120575

As shown in Figure 7 most of the changes occur atthe beginning of collection and they are then graduallydiminished over timeThe number of changes reduces by theincreasing of 120575 Meanwhile in SIGCOMM09 the maximumchanges of 0-1 contact matrix under 120575 = 02 04 06 08 are8 6 4 and 2 respectively while in INFOCOM06 the valuesare 12 4 2 and 2 With 120575 increasing the maximum changesof 0-1 contact matrix are reduced

53 Selected Node Count One of the biggest differencesbetween community and community core is the number ofclustered nodes In other words the community core of amobile social network is the subset of the whole communityAccording to previous works in [8] few of nodes can be

classified as ldquofamiliar strangersrdquo and ldquofriendsrdquo and nodes inthe ldquocommunityrdquo are even less Hence only the node pairswhich have high contact frequency can be selected to thecommunity core

Intuitively improving 120575 will result in fewer selectedcontacts and nodes which is illustrated in Figures 8(a) and8(b) Let us first consider SIGCOMM09 when time goesby the selected nodes become more and more stable Inthe first day of data collection the selected node changesdramatically especially under the low 120575 Then the stableduration of selected nodes prolonged and the selected nodesno longer have change after about 2lowast10

5 seconds Comparedwith different 120575 the lower 120575 will result in a higher selectednodes which is meaningless for community cores (when

The Scientific World Journal 9

Nod

e cou

nt35

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Nod

e cou

nt

80

60

40

20

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 9 Number of nodes in community detected by COPRA (a) Node number at each timestamp in communities (SIGCOMM09)(b) Number of node changes in community between two consecutive timestamps (SIGCOMM09) (c) Node number at each timestamp incommunities (SIGCOMM09) (d) Number of node changes in community between two consecutive timestamps (INFOCOM06)

120575 = 02 during the data collection the maximum selectednodes in SIGCOMM09 are 61 and the maximum selectednodes in INFOCOM06 are 71) while the higher the 120575 is thefewer nodes are selected to construct the community core Abrief comparison of average selected nodes under different120575 is depicted in Figure 8(c) The same as discussed abovethe average of selected nodes decreases when 120575 increasesAlthough the selected nodes increase with the decreasingof 120575 the variation of selected nodes in INFOCOM06 stillhas little difference Unlike SIGCOMM09 the selected nodesin INFOCOM06 change frequently even at the end of thedata collection This phenomenon can reflect the fact thatin SIGCOMM09 the ldquofriendsrdquo of attendants are relativelystable and participants usually contact with familiar people

while in INFOCOM06 contacts among strangers are morefrequent than in SIGCOMM09 hence the selected nodechanges more often Nevertheless the variation of selectednodes in SIGCOMM09 and INFOCOM06 is relatively stableafter 2lowast10

5 seconds which is important according to featuresof community core

The number of nodes in communities detected byCOPRA is depicted in Figure 9Different fromour algorithmthe number of nodes detected by COPRA is highly unstableAnd the number of changes between two consecutive times-tamps is also very large compared with our algorithm

54 Number of Community Cores A community core set iscomposed of selected nodes which divides a mobile social

10 The Scientific World Journal

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(a)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(b)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(c)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(d)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(e)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(f)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(g)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(h)

Figure 10 Number of changes in core matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 =02 04 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

network into pieces of closely connected fragments Hencethe community core number in a community set determinesthe fragmentation degree of this mobile social network Afterrunning our algorithm the community core count in eachtimestamp under different 120575 is obtained (Figure 10)

As depicted in Figure 11(a) after about 2 lowast 105 seconds

the community count under each different 120575 becomes fixedwhich is the same as the selected nodes However the corenumber in INFOCOM06 is more complicated Like thevariation of selected node the number of community corefluctuates with time changes see Figure 11(b)

As mentioned before this is due to the frequent contactamong strangers In other words attendants in SGICOMM09have a more fixed social circle than in INFOCOM06 Thelast but the most important feature of community core countis that the core number is not monotonically changing withchangesWe can see in Figure 11(c) that themaximumaveragecore count (asymp771) in SIGCOMM09 appears when 120575 = 02After a reduction to 120575 = 04 the average core count risesapproximate to 487 when 120575 = 5 The same phenomenonappears in INFOCOM06 the core count reaches the maxvalue at 120575 = 4 (asymp931) Then it declines approximate to 84at 120575 = 5 and grows approximate to 905 at 120575 = 6 Thereason for this situation can be explained by two sides Onone hand the higher will filter out more contacts whichmakes more fragments of the mobile network and increasesthe number of community cores On the other hand somenetwork fragments with low contact frequency will be movedout of the community core set entirely which results in thereduction of cores Hence the number of cores fluctuatesunder the different

Stability is very important to the community core Com-pared with Figures 11 and 12 the community core is much

more stable than traditional community detection algorithmThe number of communities detected by COPRA is depictedin Figure 12 which reveals high instability And the numberof changes between two consecutive timestamps is also verylarge compared with our algorithm

55 Change of Core Matrix In order to evaluate the changeof community core set between consecutive timestamps weuse the distance function discussed in Section 33 to illustratethe difference between consecutive community core sets

The distance function requires that the twomatrices havethe same line and row number However the nodes and linksin the community core set between consecutive timestampsare different According to the tracking algorithm nodes inthemobile social networks are assigned an initial communitycore label which is equal to their node ID And the finitenamespace of core label ensures that the label will notbe beyond the scope of node ID Then we construct acommunity core matrix CM = [119888119898

119894119895] where

119888119898119894119895=

node ID 119894 119895 notin CRS

core ID 119894 119895 isin CRS(5)

Note that although node ID is considered in computationit will not influence the final result of distance

The distance under different 120575 is depicted in Figure 10It can be observed that the majority of consecutive corematrices have no changes Only few of them have 1 dif-ferent element and the variation of 2 elements is evenless (Figure 13) With 120575 increasing the variation of core

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

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4 The Scientific World Journal

Contact variation

Prob

abili

ty

INFOCOM06SIGCOMM09

100

10minus2

10minus4

10minus6

100 101 102

(a)

Contact variation

Prob

abili

ty

INFOCOM06SIGCOMM09

6 changes

7 changes

100

10minus2

10minus4

10minus6

100 101 102

(b)

Figure 1 (a) Distribution of history contact duration (b) Distribution of CCSC

a b

c

d

e

f

g

med lt 120575 lowast max(m998400)

(a)

ab

cd

e

f

g

mbc gt 120575 lowast max(m998400)

(b)

Figure 2 Two events cause the structure variation of network (a) A link between nodes 119887 and 119888 is added into the network because 1198981015840

119887119888gt

120575 lowastmax(1198981015840) (b) A link between nodes 119889 and 119890 is removed from the network because 1198981015840

119889119890lt 120575 lowastmax(1198981015840)

Select linkij in LRS Yes LRS ne 120601 No End

i and j in the same core No Remove linkij in LRS

Split the core intotwo parts

No

There is a pathfrom i to j

Yes

No

No

YesYes

Yes

i or j not in CRS998400(t + 1)NoRemove linkij inCRS998400(t + 1)

Remove node not inCRS998400(t + 1)

CRS998400(t + 1) larrminus CRS(t)

m998400ij lt 120575 times max(m998400)

i and j in CRS998400(t + 1)

Figure 3 Basic operation procedure of link remove

The Scientific World Journal 5

Select linkij in LAS Yes

Yes

LAS ne 120601 No End

No Remove linkij in LASThe one neighbor nodejoins into the other core No

i and j have morethan one neighbor

NoNo

i and j in the same core Yes

Yes

Add linkij in CRS(t + 1)Create a new

community core

i in COp and j in COq Yes

Yes

i and j have onlyone neighbor

Merge the twocommunity cores

CRS(t + 1) larrminus CRS998400(t + 1)

Update M998400(t + 1)

m998400ij gt 120575 times max(m998400)

(COpq in CRS998400(t + 1))

Figure 4 Basic operation procedure of link add

a

c b

a

c

b

e

d

a c

b

a

cb

e

da

c

be

d

a

cb e

d

a

c

b e

d

a

c

b

e

d

a

c

b

d

a

c b

d

a c

bd

a

c

b

DeathBirth Merging

Splitting Growth Contraction

e

d

Figure 5 Six events provide variation of community cores The red soliddash line denotes the link addremove separately and differentcolors mean different community cores

by updating the max contact duration addition link and itsnodes will be added into the existing community core or anew community core

42 Evolution of Community Core In order to study theevolution process of community core we should track thecommunity core at each timestamp How to distinguish twocommunity cores in the consecutive timestamps is the biggestproblem about tracking

421 Model According to Definition 3 the evolution ofcommunity core can be presented by CRS

0CRS1 CRS

119896

For a community core at 119879 = 119905 CRS(119905) = CO0CO1

CO119898 and 119871(119905) = 119897

0 1198971 119897119899 is denoted as the label set

of community cores identifying a community core at 119879 = 119905

uniquely In the previous literatures there can be seen a broadconsensus on the basic events that can be used to describe theevolution of dynamic communities [3 7 11] We extend andspecify these events as shown in Figure 5

6 The Scientific World Journal

0 05 1 15 2 25 30

10

20

35times105

(a)

0 05 1 15 2 25 30

150

300

450

35times105

(b)

0 05 1 15 2 25 30

10

20

35times105

(c)

0 05 1 15 2 25 30

50

100

35times105

(d)

0 05 1 15 2 25 30

10

20

35times105

(e)

0 05 1 15 2 25 30

80

160

35times105

(f)

0 05 1 15 2 25 30

20

40

60

Link numberNode numberNode degree

35times105

(g)

0 05 1 15 2 25 3 350

100

200

300

Link numberNode numberNode degree

times105

(h)Figure 6 Description of two datasets SIGCOMM09 records the discrete contact time among nodes while INFOCOM06 records thecontinuous contact time (a) Total link number (SIGCOMM09) (b) total link number (INFOCOM06) (c) link add (SIGCOMM09)(d) link add (INFOCOM06) (e) link remove (SIGCOMM09) (f) link remove (INFOCOM06) (g) network feature (SIGCOMM09) and(h) network feature (INFOCOM06)

Birth It is the emergence of a new community core at 119879 =

119905 + 1 and there is no corresponding community core inCRS(119905) A community core COnew which is labeled as 119897new iscreated in CRS(119905 + 1) at 119879 = 119905 + 1 while COnew notin CRS(119905)That is for any 119897new isin 119871(119905 + 1) there is 119897new notin 119871(119905)

Death It is the disappearance of a community core COold at119879 = 119905 + 1 as COold exists in CRS(119905) at 119879 = 119905 There is nocorresponding community core in CRS(119905 + 1) If we use 119897olddenoting core label of COold then the death of a communitycoreCOold means 119897old isin 119871(119905) and for any 119897

119904isin 119871(119905+1) 119897

119904= 119897old

Merging It means that two community cores merge into onecommunity core at 119879 = 119905 + 1

Splitting A community core is divided into two separatecommunity cores

Growth A node joins into a community core And this willresult in no changes between 119871(119905) and 119871(119905 + 1)

Contraction A nodemoves out of a community coreThis alsowill result in no changes between 119871(119905) and 119871(119905 + 1)

422 Tracking Community Cores In the context of themodeldescribed above the most important thing is to identifythe changes of community cores Three questions should beanswered (1) Where does the core come from (2) Whathappened to community core after updating (3)Where is thecore going

The study in [12] is concerned with tracking communitiesbased on detected communities using the Jaccard coeffi-cient Differing from this method we track these cores byenhancing our algorithm When the algorithm runs thetracking process works simultaneously which is triggered bythe variation of links

According to our algorithm there exist only contactfrequency between two core nodes lower than 120575 sdot max(119898

119894119895)

and there is no other path connecting these two nodes inthe original core the splitting process will be triggered

The Scientific World Journal 7

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(a)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(b)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(c)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(d)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(e)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(f)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(g)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(h)

Figure 7 Number of changes in 0-1 matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 = 0204 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

Hence V119894and V

119895have their isolate core sets separately We

use node ID as the core label when creating a new communitycore Initially each node will be tagged a community label asitself The benefits are as follows Firstly the finite namespaceof core label will not cause muddle of naming Secondlycommunity core can be tracked easier when a communitycore is created again The tracking algorithm uses nodeID as initial community core ID if the node is extractedto a community core and when a node is removed fromcommunity core set the node ID is restored

Basic operation procedure of tracking is given as follows

Switch (events)Case Birth (V

119894and V119895form a new core)

119871 (119905 + 1) larr997888 119897new 119897new = 119894 or 119895 (3)

Case Death (CO119894isin CRS(119905) but CO

119894notin CRS(119905 + 1))

Every node in CO119894labeled as its node ID

CaseMerging (CO119894and CO

119895merge into on core)

Nodes with fewer core members change theirIDs to the other core

Case Splitting (V119894and V119895have no path to connect)

Delete 119897old in 119871(119905 + 1) and 119871(119905 + 1) larr 119894 119895 V119894and

V119895change label as their node ID

Case Growth (V119894join into an existing community

core)

V119894changes label as the community core

Case Contraction (V119894moves out of community core)

V119894changes label as its node ID

5 Evaluation

In this section we discuss the evolution of community corewhich is detected and tracked by our algorithm COPRA[13] is a well-known efficient algorithm of fast communitydetection and we choose COPRA to compare with our algo-rithm

51 Contact Variation In order to reveal the communitycore evolution briefly we first show the contact variationof both SIGCOMM09 and INFOCOM06 under the wholelifetime (Figure 6) Due to the difference of data preprocess-ing SIGCOMM09 records the discrete contact time amongnodes while INFOCOM06 records the continuous contacttime According to the description of SIGCOMM09 eachdevice performs a periodic Bluetooth device discovery every120 plusmn 1024 seconds (randomized) for 1024 seconds Whena device finds others it records the current time Howeverthis mechanism can only record one contact time in a periodfor the same pair which will miss some contact informationHence the total link change link add and link remove ofSIGCOMM09 almost have no differences

52 Change of 0-1 Contact Matrix According to M1015840 the 0-1 contact matrix B = [119887

119894119895] can be obtained M1015840 changes

when the time increases and the maximum contact durationamong nodes may be changed which results in the variationof B The change of B during the whole collection procedureis plotted in Figure 7 Consider the following

119887119894119895=

1 1198981015840

119894119895ge 120575 sdotmax (119898

1015840

)

0 1198981015840

119894119895lt 120575 sdotmax (119898

1015840

)

(4)

8 The Scientific World Journal

Nod

e cou

nt70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

Nod

e cou

nt

80

60

40

20

0

(b)

Aver

age n

ode c

ount

80

70

60

50

40

30

20

10

0

120575

01 03 05 07 09

SIGCOMM09INFOCOM06

(c)

Figure 8 Number of nodes in community core (a) Variation in number of nodes in community core in SIGCOMM09 (b) Variation innumber of nodes in community core in INFOCOM06 (c) Average number of nodes in community core in the two datasets under different120575

As shown in Figure 7 most of the changes occur atthe beginning of collection and they are then graduallydiminished over timeThe number of changes reduces by theincreasing of 120575 Meanwhile in SIGCOMM09 the maximumchanges of 0-1 contact matrix under 120575 = 02 04 06 08 are8 6 4 and 2 respectively while in INFOCOM06 the valuesare 12 4 2 and 2 With 120575 increasing the maximum changesof 0-1 contact matrix are reduced

53 Selected Node Count One of the biggest differencesbetween community and community core is the number ofclustered nodes In other words the community core of amobile social network is the subset of the whole communityAccording to previous works in [8] few of nodes can be

classified as ldquofamiliar strangersrdquo and ldquofriendsrdquo and nodes inthe ldquocommunityrdquo are even less Hence only the node pairswhich have high contact frequency can be selected to thecommunity core

Intuitively improving 120575 will result in fewer selectedcontacts and nodes which is illustrated in Figures 8(a) and8(b) Let us first consider SIGCOMM09 when time goesby the selected nodes become more and more stable Inthe first day of data collection the selected node changesdramatically especially under the low 120575 Then the stableduration of selected nodes prolonged and the selected nodesno longer have change after about 2lowast10

5 seconds Comparedwith different 120575 the lower 120575 will result in a higher selectednodes which is meaningless for community cores (when

The Scientific World Journal 9

Nod

e cou

nt35

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Nod

e cou

nt

80

60

40

20

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 9 Number of nodes in community detected by COPRA (a) Node number at each timestamp in communities (SIGCOMM09)(b) Number of node changes in community between two consecutive timestamps (SIGCOMM09) (c) Node number at each timestamp incommunities (SIGCOMM09) (d) Number of node changes in community between two consecutive timestamps (INFOCOM06)

120575 = 02 during the data collection the maximum selectednodes in SIGCOMM09 are 61 and the maximum selectednodes in INFOCOM06 are 71) while the higher the 120575 is thefewer nodes are selected to construct the community core Abrief comparison of average selected nodes under different120575 is depicted in Figure 8(c) The same as discussed abovethe average of selected nodes decreases when 120575 increasesAlthough the selected nodes increase with the decreasingof 120575 the variation of selected nodes in INFOCOM06 stillhas little difference Unlike SIGCOMM09 the selected nodesin INFOCOM06 change frequently even at the end of thedata collection This phenomenon can reflect the fact thatin SIGCOMM09 the ldquofriendsrdquo of attendants are relativelystable and participants usually contact with familiar people

while in INFOCOM06 contacts among strangers are morefrequent than in SIGCOMM09 hence the selected nodechanges more often Nevertheless the variation of selectednodes in SIGCOMM09 and INFOCOM06 is relatively stableafter 2lowast10

5 seconds which is important according to featuresof community core

The number of nodes in communities detected byCOPRA is depicted in Figure 9Different fromour algorithmthe number of nodes detected by COPRA is highly unstableAnd the number of changes between two consecutive times-tamps is also very large compared with our algorithm

54 Number of Community Cores A community core set iscomposed of selected nodes which divides a mobile social

10 The Scientific World Journal

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(a)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(b)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(c)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(d)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(e)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(f)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(g)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(h)

Figure 10 Number of changes in core matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 =02 04 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

network into pieces of closely connected fragments Hencethe community core number in a community set determinesthe fragmentation degree of this mobile social network Afterrunning our algorithm the community core count in eachtimestamp under different 120575 is obtained (Figure 10)

As depicted in Figure 11(a) after about 2 lowast 105 seconds

the community count under each different 120575 becomes fixedwhich is the same as the selected nodes However the corenumber in INFOCOM06 is more complicated Like thevariation of selected node the number of community corefluctuates with time changes see Figure 11(b)

As mentioned before this is due to the frequent contactamong strangers In other words attendants in SGICOMM09have a more fixed social circle than in INFOCOM06 Thelast but the most important feature of community core countis that the core number is not monotonically changing withchangesWe can see in Figure 11(c) that themaximumaveragecore count (asymp771) in SIGCOMM09 appears when 120575 = 02After a reduction to 120575 = 04 the average core count risesapproximate to 487 when 120575 = 5 The same phenomenonappears in INFOCOM06 the core count reaches the maxvalue at 120575 = 4 (asymp931) Then it declines approximate to 84at 120575 = 5 and grows approximate to 905 at 120575 = 6 Thereason for this situation can be explained by two sides Onone hand the higher will filter out more contacts whichmakes more fragments of the mobile network and increasesthe number of community cores On the other hand somenetwork fragments with low contact frequency will be movedout of the community core set entirely which results in thereduction of cores Hence the number of cores fluctuatesunder the different

Stability is very important to the community core Com-pared with Figures 11 and 12 the community core is much

more stable than traditional community detection algorithmThe number of communities detected by COPRA is depictedin Figure 12 which reveals high instability And the numberof changes between two consecutive timestamps is also verylarge compared with our algorithm

55 Change of Core Matrix In order to evaluate the changeof community core set between consecutive timestamps weuse the distance function discussed in Section 33 to illustratethe difference between consecutive community core sets

The distance function requires that the twomatrices havethe same line and row number However the nodes and linksin the community core set between consecutive timestampsare different According to the tracking algorithm nodes inthemobile social networks are assigned an initial communitycore label which is equal to their node ID And the finitenamespace of core label ensures that the label will notbe beyond the scope of node ID Then we construct acommunity core matrix CM = [119888119898

119894119895] where

119888119898119894119895=

node ID 119894 119895 notin CRS

core ID 119894 119895 isin CRS(5)

Note that although node ID is considered in computationit will not influence the final result of distance

The distance under different 120575 is depicted in Figure 10It can be observed that the majority of consecutive corematrices have no changes Only few of them have 1 dif-ferent element and the variation of 2 elements is evenless (Figure 13) With 120575 increasing the variation of core

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

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The Scientific World Journal 5

Select linkij in LAS Yes

Yes

LAS ne 120601 No End

No Remove linkij in LASThe one neighbor nodejoins into the other core No

i and j have morethan one neighbor

NoNo

i and j in the same core Yes

Yes

Add linkij in CRS(t + 1)Create a new

community core

i in COp and j in COq Yes

Yes

i and j have onlyone neighbor

Merge the twocommunity cores

CRS(t + 1) larrminus CRS998400(t + 1)

Update M998400(t + 1)

m998400ij gt 120575 times max(m998400)

(COpq in CRS998400(t + 1))

Figure 4 Basic operation procedure of link add

a

c b

a

c

b

e

d

a c

b

a

cb

e

da

c

be

d

a

cb e

d

a

c

b e

d

a

c

b

e

d

a

c

b

d

a

c b

d

a c

bd

a

c

b

DeathBirth Merging

Splitting Growth Contraction

e

d

Figure 5 Six events provide variation of community cores The red soliddash line denotes the link addremove separately and differentcolors mean different community cores

by updating the max contact duration addition link and itsnodes will be added into the existing community core or anew community core

42 Evolution of Community Core In order to study theevolution process of community core we should track thecommunity core at each timestamp How to distinguish twocommunity cores in the consecutive timestamps is the biggestproblem about tracking

421 Model According to Definition 3 the evolution ofcommunity core can be presented by CRS

0CRS1 CRS

119896

For a community core at 119879 = 119905 CRS(119905) = CO0CO1

CO119898 and 119871(119905) = 119897

0 1198971 119897119899 is denoted as the label set

of community cores identifying a community core at 119879 = 119905

uniquely In the previous literatures there can be seen a broadconsensus on the basic events that can be used to describe theevolution of dynamic communities [3 7 11] We extend andspecify these events as shown in Figure 5

6 The Scientific World Journal

0 05 1 15 2 25 30

10

20

35times105

(a)

0 05 1 15 2 25 30

150

300

450

35times105

(b)

0 05 1 15 2 25 30

10

20

35times105

(c)

0 05 1 15 2 25 30

50

100

35times105

(d)

0 05 1 15 2 25 30

10

20

35times105

(e)

0 05 1 15 2 25 30

80

160

35times105

(f)

0 05 1 15 2 25 30

20

40

60

Link numberNode numberNode degree

35times105

(g)

0 05 1 15 2 25 3 350

100

200

300

Link numberNode numberNode degree

times105

(h)Figure 6 Description of two datasets SIGCOMM09 records the discrete contact time among nodes while INFOCOM06 records thecontinuous contact time (a) Total link number (SIGCOMM09) (b) total link number (INFOCOM06) (c) link add (SIGCOMM09)(d) link add (INFOCOM06) (e) link remove (SIGCOMM09) (f) link remove (INFOCOM06) (g) network feature (SIGCOMM09) and(h) network feature (INFOCOM06)

Birth It is the emergence of a new community core at 119879 =

119905 + 1 and there is no corresponding community core inCRS(119905) A community core COnew which is labeled as 119897new iscreated in CRS(119905 + 1) at 119879 = 119905 + 1 while COnew notin CRS(119905)That is for any 119897new isin 119871(119905 + 1) there is 119897new notin 119871(119905)

Death It is the disappearance of a community core COold at119879 = 119905 + 1 as COold exists in CRS(119905) at 119879 = 119905 There is nocorresponding community core in CRS(119905 + 1) If we use 119897olddenoting core label of COold then the death of a communitycoreCOold means 119897old isin 119871(119905) and for any 119897

119904isin 119871(119905+1) 119897

119904= 119897old

Merging It means that two community cores merge into onecommunity core at 119879 = 119905 + 1

Splitting A community core is divided into two separatecommunity cores

Growth A node joins into a community core And this willresult in no changes between 119871(119905) and 119871(119905 + 1)

Contraction A nodemoves out of a community coreThis alsowill result in no changes between 119871(119905) and 119871(119905 + 1)

422 Tracking Community Cores In the context of themodeldescribed above the most important thing is to identifythe changes of community cores Three questions should beanswered (1) Where does the core come from (2) Whathappened to community core after updating (3)Where is thecore going

The study in [12] is concerned with tracking communitiesbased on detected communities using the Jaccard coeffi-cient Differing from this method we track these cores byenhancing our algorithm When the algorithm runs thetracking process works simultaneously which is triggered bythe variation of links

According to our algorithm there exist only contactfrequency between two core nodes lower than 120575 sdot max(119898

119894119895)

and there is no other path connecting these two nodes inthe original core the splitting process will be triggered

The Scientific World Journal 7

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(a)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(b)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(c)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(d)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(e)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(f)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(g)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(h)

Figure 7 Number of changes in 0-1 matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 = 0204 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

Hence V119894and V

119895have their isolate core sets separately We

use node ID as the core label when creating a new communitycore Initially each node will be tagged a community label asitself The benefits are as follows Firstly the finite namespaceof core label will not cause muddle of naming Secondlycommunity core can be tracked easier when a communitycore is created again The tracking algorithm uses nodeID as initial community core ID if the node is extractedto a community core and when a node is removed fromcommunity core set the node ID is restored

Basic operation procedure of tracking is given as follows

Switch (events)Case Birth (V

119894and V119895form a new core)

119871 (119905 + 1) larr997888 119897new 119897new = 119894 or 119895 (3)

Case Death (CO119894isin CRS(119905) but CO

119894notin CRS(119905 + 1))

Every node in CO119894labeled as its node ID

CaseMerging (CO119894and CO

119895merge into on core)

Nodes with fewer core members change theirIDs to the other core

Case Splitting (V119894and V119895have no path to connect)

Delete 119897old in 119871(119905 + 1) and 119871(119905 + 1) larr 119894 119895 V119894and

V119895change label as their node ID

Case Growth (V119894join into an existing community

core)

V119894changes label as the community core

Case Contraction (V119894moves out of community core)

V119894changes label as its node ID

5 Evaluation

In this section we discuss the evolution of community corewhich is detected and tracked by our algorithm COPRA[13] is a well-known efficient algorithm of fast communitydetection and we choose COPRA to compare with our algo-rithm

51 Contact Variation In order to reveal the communitycore evolution briefly we first show the contact variationof both SIGCOMM09 and INFOCOM06 under the wholelifetime (Figure 6) Due to the difference of data preprocess-ing SIGCOMM09 records the discrete contact time amongnodes while INFOCOM06 records the continuous contacttime According to the description of SIGCOMM09 eachdevice performs a periodic Bluetooth device discovery every120 plusmn 1024 seconds (randomized) for 1024 seconds Whena device finds others it records the current time Howeverthis mechanism can only record one contact time in a periodfor the same pair which will miss some contact informationHence the total link change link add and link remove ofSIGCOMM09 almost have no differences

52 Change of 0-1 Contact Matrix According to M1015840 the 0-1 contact matrix B = [119887

119894119895] can be obtained M1015840 changes

when the time increases and the maximum contact durationamong nodes may be changed which results in the variationof B The change of B during the whole collection procedureis plotted in Figure 7 Consider the following

119887119894119895=

1 1198981015840

119894119895ge 120575 sdotmax (119898

1015840

)

0 1198981015840

119894119895lt 120575 sdotmax (119898

1015840

)

(4)

8 The Scientific World Journal

Nod

e cou

nt70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

Nod

e cou

nt

80

60

40

20

0

(b)

Aver

age n

ode c

ount

80

70

60

50

40

30

20

10

0

120575

01 03 05 07 09

SIGCOMM09INFOCOM06

(c)

Figure 8 Number of nodes in community core (a) Variation in number of nodes in community core in SIGCOMM09 (b) Variation innumber of nodes in community core in INFOCOM06 (c) Average number of nodes in community core in the two datasets under different120575

As shown in Figure 7 most of the changes occur atthe beginning of collection and they are then graduallydiminished over timeThe number of changes reduces by theincreasing of 120575 Meanwhile in SIGCOMM09 the maximumchanges of 0-1 contact matrix under 120575 = 02 04 06 08 are8 6 4 and 2 respectively while in INFOCOM06 the valuesare 12 4 2 and 2 With 120575 increasing the maximum changesof 0-1 contact matrix are reduced

53 Selected Node Count One of the biggest differencesbetween community and community core is the number ofclustered nodes In other words the community core of amobile social network is the subset of the whole communityAccording to previous works in [8] few of nodes can be

classified as ldquofamiliar strangersrdquo and ldquofriendsrdquo and nodes inthe ldquocommunityrdquo are even less Hence only the node pairswhich have high contact frequency can be selected to thecommunity core

Intuitively improving 120575 will result in fewer selectedcontacts and nodes which is illustrated in Figures 8(a) and8(b) Let us first consider SIGCOMM09 when time goesby the selected nodes become more and more stable Inthe first day of data collection the selected node changesdramatically especially under the low 120575 Then the stableduration of selected nodes prolonged and the selected nodesno longer have change after about 2lowast10

5 seconds Comparedwith different 120575 the lower 120575 will result in a higher selectednodes which is meaningless for community cores (when

The Scientific World Journal 9

Nod

e cou

nt35

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Nod

e cou

nt

80

60

40

20

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 9 Number of nodes in community detected by COPRA (a) Node number at each timestamp in communities (SIGCOMM09)(b) Number of node changes in community between two consecutive timestamps (SIGCOMM09) (c) Node number at each timestamp incommunities (SIGCOMM09) (d) Number of node changes in community between two consecutive timestamps (INFOCOM06)

120575 = 02 during the data collection the maximum selectednodes in SIGCOMM09 are 61 and the maximum selectednodes in INFOCOM06 are 71) while the higher the 120575 is thefewer nodes are selected to construct the community core Abrief comparison of average selected nodes under different120575 is depicted in Figure 8(c) The same as discussed abovethe average of selected nodes decreases when 120575 increasesAlthough the selected nodes increase with the decreasingof 120575 the variation of selected nodes in INFOCOM06 stillhas little difference Unlike SIGCOMM09 the selected nodesin INFOCOM06 change frequently even at the end of thedata collection This phenomenon can reflect the fact thatin SIGCOMM09 the ldquofriendsrdquo of attendants are relativelystable and participants usually contact with familiar people

while in INFOCOM06 contacts among strangers are morefrequent than in SIGCOMM09 hence the selected nodechanges more often Nevertheless the variation of selectednodes in SIGCOMM09 and INFOCOM06 is relatively stableafter 2lowast10

5 seconds which is important according to featuresof community core

The number of nodes in communities detected byCOPRA is depicted in Figure 9Different fromour algorithmthe number of nodes detected by COPRA is highly unstableAnd the number of changes between two consecutive times-tamps is also very large compared with our algorithm

54 Number of Community Cores A community core set iscomposed of selected nodes which divides a mobile social

10 The Scientific World Journal

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(a)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(b)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(c)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(d)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(e)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(f)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(g)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(h)

Figure 10 Number of changes in core matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 =02 04 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

network into pieces of closely connected fragments Hencethe community core number in a community set determinesthe fragmentation degree of this mobile social network Afterrunning our algorithm the community core count in eachtimestamp under different 120575 is obtained (Figure 10)

As depicted in Figure 11(a) after about 2 lowast 105 seconds

the community count under each different 120575 becomes fixedwhich is the same as the selected nodes However the corenumber in INFOCOM06 is more complicated Like thevariation of selected node the number of community corefluctuates with time changes see Figure 11(b)

As mentioned before this is due to the frequent contactamong strangers In other words attendants in SGICOMM09have a more fixed social circle than in INFOCOM06 Thelast but the most important feature of community core countis that the core number is not monotonically changing withchangesWe can see in Figure 11(c) that themaximumaveragecore count (asymp771) in SIGCOMM09 appears when 120575 = 02After a reduction to 120575 = 04 the average core count risesapproximate to 487 when 120575 = 5 The same phenomenonappears in INFOCOM06 the core count reaches the maxvalue at 120575 = 4 (asymp931) Then it declines approximate to 84at 120575 = 5 and grows approximate to 905 at 120575 = 6 Thereason for this situation can be explained by two sides Onone hand the higher will filter out more contacts whichmakes more fragments of the mobile network and increasesthe number of community cores On the other hand somenetwork fragments with low contact frequency will be movedout of the community core set entirely which results in thereduction of cores Hence the number of cores fluctuatesunder the different

Stability is very important to the community core Com-pared with Figures 11 and 12 the community core is much

more stable than traditional community detection algorithmThe number of communities detected by COPRA is depictedin Figure 12 which reveals high instability And the numberof changes between two consecutive timestamps is also verylarge compared with our algorithm

55 Change of Core Matrix In order to evaluate the changeof community core set between consecutive timestamps weuse the distance function discussed in Section 33 to illustratethe difference between consecutive community core sets

The distance function requires that the twomatrices havethe same line and row number However the nodes and linksin the community core set between consecutive timestampsare different According to the tracking algorithm nodes inthemobile social networks are assigned an initial communitycore label which is equal to their node ID And the finitenamespace of core label ensures that the label will notbe beyond the scope of node ID Then we construct acommunity core matrix CM = [119888119898

119894119895] where

119888119898119894119895=

node ID 119894 119895 notin CRS

core ID 119894 119895 isin CRS(5)

Note that although node ID is considered in computationit will not influence the final result of distance

The distance under different 120575 is depicted in Figure 10It can be observed that the majority of consecutive corematrices have no changes Only few of them have 1 dif-ferent element and the variation of 2 elements is evenless (Figure 13) With 120575 increasing the variation of core

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

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6 The Scientific World Journal

0 05 1 15 2 25 30

10

20

35times105

(a)

0 05 1 15 2 25 30

150

300

450

35times105

(b)

0 05 1 15 2 25 30

10

20

35times105

(c)

0 05 1 15 2 25 30

50

100

35times105

(d)

0 05 1 15 2 25 30

10

20

35times105

(e)

0 05 1 15 2 25 30

80

160

35times105

(f)

0 05 1 15 2 25 30

20

40

60

Link numberNode numberNode degree

35times105

(g)

0 05 1 15 2 25 3 350

100

200

300

Link numberNode numberNode degree

times105

(h)Figure 6 Description of two datasets SIGCOMM09 records the discrete contact time among nodes while INFOCOM06 records thecontinuous contact time (a) Total link number (SIGCOMM09) (b) total link number (INFOCOM06) (c) link add (SIGCOMM09)(d) link add (INFOCOM06) (e) link remove (SIGCOMM09) (f) link remove (INFOCOM06) (g) network feature (SIGCOMM09) and(h) network feature (INFOCOM06)

Birth It is the emergence of a new community core at 119879 =

119905 + 1 and there is no corresponding community core inCRS(119905) A community core COnew which is labeled as 119897new iscreated in CRS(119905 + 1) at 119879 = 119905 + 1 while COnew notin CRS(119905)That is for any 119897new isin 119871(119905 + 1) there is 119897new notin 119871(119905)

Death It is the disappearance of a community core COold at119879 = 119905 + 1 as COold exists in CRS(119905) at 119879 = 119905 There is nocorresponding community core in CRS(119905 + 1) If we use 119897olddenoting core label of COold then the death of a communitycoreCOold means 119897old isin 119871(119905) and for any 119897

119904isin 119871(119905+1) 119897

119904= 119897old

Merging It means that two community cores merge into onecommunity core at 119879 = 119905 + 1

Splitting A community core is divided into two separatecommunity cores

Growth A node joins into a community core And this willresult in no changes between 119871(119905) and 119871(119905 + 1)

Contraction A nodemoves out of a community coreThis alsowill result in no changes between 119871(119905) and 119871(119905 + 1)

422 Tracking Community Cores In the context of themodeldescribed above the most important thing is to identifythe changes of community cores Three questions should beanswered (1) Where does the core come from (2) Whathappened to community core after updating (3)Where is thecore going

The study in [12] is concerned with tracking communitiesbased on detected communities using the Jaccard coeffi-cient Differing from this method we track these cores byenhancing our algorithm When the algorithm runs thetracking process works simultaneously which is triggered bythe variation of links

According to our algorithm there exist only contactfrequency between two core nodes lower than 120575 sdot max(119898

119894119895)

and there is no other path connecting these two nodes inthe original core the splitting process will be triggered

The Scientific World Journal 7

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(a)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(b)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(c)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(d)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(e)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(f)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(g)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(h)

Figure 7 Number of changes in 0-1 matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 = 0204 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

Hence V119894and V

119895have their isolate core sets separately We

use node ID as the core label when creating a new communitycore Initially each node will be tagged a community label asitself The benefits are as follows Firstly the finite namespaceof core label will not cause muddle of naming Secondlycommunity core can be tracked easier when a communitycore is created again The tracking algorithm uses nodeID as initial community core ID if the node is extractedto a community core and when a node is removed fromcommunity core set the node ID is restored

Basic operation procedure of tracking is given as follows

Switch (events)Case Birth (V

119894and V119895form a new core)

119871 (119905 + 1) larr997888 119897new 119897new = 119894 or 119895 (3)

Case Death (CO119894isin CRS(119905) but CO

119894notin CRS(119905 + 1))

Every node in CO119894labeled as its node ID

CaseMerging (CO119894and CO

119895merge into on core)

Nodes with fewer core members change theirIDs to the other core

Case Splitting (V119894and V119895have no path to connect)

Delete 119897old in 119871(119905 + 1) and 119871(119905 + 1) larr 119894 119895 V119894and

V119895change label as their node ID

Case Growth (V119894join into an existing community

core)

V119894changes label as the community core

Case Contraction (V119894moves out of community core)

V119894changes label as its node ID

5 Evaluation

In this section we discuss the evolution of community corewhich is detected and tracked by our algorithm COPRA[13] is a well-known efficient algorithm of fast communitydetection and we choose COPRA to compare with our algo-rithm

51 Contact Variation In order to reveal the communitycore evolution briefly we first show the contact variationof both SIGCOMM09 and INFOCOM06 under the wholelifetime (Figure 6) Due to the difference of data preprocess-ing SIGCOMM09 records the discrete contact time amongnodes while INFOCOM06 records the continuous contacttime According to the description of SIGCOMM09 eachdevice performs a periodic Bluetooth device discovery every120 plusmn 1024 seconds (randomized) for 1024 seconds Whena device finds others it records the current time Howeverthis mechanism can only record one contact time in a periodfor the same pair which will miss some contact informationHence the total link change link add and link remove ofSIGCOMM09 almost have no differences

52 Change of 0-1 Contact Matrix According to M1015840 the 0-1 contact matrix B = [119887

119894119895] can be obtained M1015840 changes

when the time increases and the maximum contact durationamong nodes may be changed which results in the variationof B The change of B during the whole collection procedureis plotted in Figure 7 Consider the following

119887119894119895=

1 1198981015840

119894119895ge 120575 sdotmax (119898

1015840

)

0 1198981015840

119894119895lt 120575 sdotmax (119898

1015840

)

(4)

8 The Scientific World Journal

Nod

e cou

nt70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

Nod

e cou

nt

80

60

40

20

0

(b)

Aver

age n

ode c

ount

80

70

60

50

40

30

20

10

0

120575

01 03 05 07 09

SIGCOMM09INFOCOM06

(c)

Figure 8 Number of nodes in community core (a) Variation in number of nodes in community core in SIGCOMM09 (b) Variation innumber of nodes in community core in INFOCOM06 (c) Average number of nodes in community core in the two datasets under different120575

As shown in Figure 7 most of the changes occur atthe beginning of collection and they are then graduallydiminished over timeThe number of changes reduces by theincreasing of 120575 Meanwhile in SIGCOMM09 the maximumchanges of 0-1 contact matrix under 120575 = 02 04 06 08 are8 6 4 and 2 respectively while in INFOCOM06 the valuesare 12 4 2 and 2 With 120575 increasing the maximum changesof 0-1 contact matrix are reduced

53 Selected Node Count One of the biggest differencesbetween community and community core is the number ofclustered nodes In other words the community core of amobile social network is the subset of the whole communityAccording to previous works in [8] few of nodes can be

classified as ldquofamiliar strangersrdquo and ldquofriendsrdquo and nodes inthe ldquocommunityrdquo are even less Hence only the node pairswhich have high contact frequency can be selected to thecommunity core

Intuitively improving 120575 will result in fewer selectedcontacts and nodes which is illustrated in Figures 8(a) and8(b) Let us first consider SIGCOMM09 when time goesby the selected nodes become more and more stable Inthe first day of data collection the selected node changesdramatically especially under the low 120575 Then the stableduration of selected nodes prolonged and the selected nodesno longer have change after about 2lowast10

5 seconds Comparedwith different 120575 the lower 120575 will result in a higher selectednodes which is meaningless for community cores (when

The Scientific World Journal 9

Nod

e cou

nt35

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Nod

e cou

nt

80

60

40

20

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 9 Number of nodes in community detected by COPRA (a) Node number at each timestamp in communities (SIGCOMM09)(b) Number of node changes in community between two consecutive timestamps (SIGCOMM09) (c) Node number at each timestamp incommunities (SIGCOMM09) (d) Number of node changes in community between two consecutive timestamps (INFOCOM06)

120575 = 02 during the data collection the maximum selectednodes in SIGCOMM09 are 61 and the maximum selectednodes in INFOCOM06 are 71) while the higher the 120575 is thefewer nodes are selected to construct the community core Abrief comparison of average selected nodes under different120575 is depicted in Figure 8(c) The same as discussed abovethe average of selected nodes decreases when 120575 increasesAlthough the selected nodes increase with the decreasingof 120575 the variation of selected nodes in INFOCOM06 stillhas little difference Unlike SIGCOMM09 the selected nodesin INFOCOM06 change frequently even at the end of thedata collection This phenomenon can reflect the fact thatin SIGCOMM09 the ldquofriendsrdquo of attendants are relativelystable and participants usually contact with familiar people

while in INFOCOM06 contacts among strangers are morefrequent than in SIGCOMM09 hence the selected nodechanges more often Nevertheless the variation of selectednodes in SIGCOMM09 and INFOCOM06 is relatively stableafter 2lowast10

5 seconds which is important according to featuresof community core

The number of nodes in communities detected byCOPRA is depicted in Figure 9Different fromour algorithmthe number of nodes detected by COPRA is highly unstableAnd the number of changes between two consecutive times-tamps is also very large compared with our algorithm

54 Number of Community Cores A community core set iscomposed of selected nodes which divides a mobile social

10 The Scientific World Journal

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(a)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(b)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(c)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(d)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(e)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(f)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(g)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(h)

Figure 10 Number of changes in core matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 =02 04 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

network into pieces of closely connected fragments Hencethe community core number in a community set determinesthe fragmentation degree of this mobile social network Afterrunning our algorithm the community core count in eachtimestamp under different 120575 is obtained (Figure 10)

As depicted in Figure 11(a) after about 2 lowast 105 seconds

the community count under each different 120575 becomes fixedwhich is the same as the selected nodes However the corenumber in INFOCOM06 is more complicated Like thevariation of selected node the number of community corefluctuates with time changes see Figure 11(b)

As mentioned before this is due to the frequent contactamong strangers In other words attendants in SGICOMM09have a more fixed social circle than in INFOCOM06 Thelast but the most important feature of community core countis that the core number is not monotonically changing withchangesWe can see in Figure 11(c) that themaximumaveragecore count (asymp771) in SIGCOMM09 appears when 120575 = 02After a reduction to 120575 = 04 the average core count risesapproximate to 487 when 120575 = 5 The same phenomenonappears in INFOCOM06 the core count reaches the maxvalue at 120575 = 4 (asymp931) Then it declines approximate to 84at 120575 = 5 and grows approximate to 905 at 120575 = 6 Thereason for this situation can be explained by two sides Onone hand the higher will filter out more contacts whichmakes more fragments of the mobile network and increasesthe number of community cores On the other hand somenetwork fragments with low contact frequency will be movedout of the community core set entirely which results in thereduction of cores Hence the number of cores fluctuatesunder the different

Stability is very important to the community core Com-pared with Figures 11 and 12 the community core is much

more stable than traditional community detection algorithmThe number of communities detected by COPRA is depictedin Figure 12 which reveals high instability And the numberof changes between two consecutive timestamps is also verylarge compared with our algorithm

55 Change of Core Matrix In order to evaluate the changeof community core set between consecutive timestamps weuse the distance function discussed in Section 33 to illustratethe difference between consecutive community core sets

The distance function requires that the twomatrices havethe same line and row number However the nodes and linksin the community core set between consecutive timestampsare different According to the tracking algorithm nodes inthemobile social networks are assigned an initial communitycore label which is equal to their node ID And the finitenamespace of core label ensures that the label will notbe beyond the scope of node ID Then we construct acommunity core matrix CM = [119888119898

119894119895] where

119888119898119894119895=

node ID 119894 119895 notin CRS

core ID 119894 119895 isin CRS(5)

Note that although node ID is considered in computationit will not influence the final result of distance

The distance under different 120575 is depicted in Figure 10It can be observed that the majority of consecutive corematrices have no changes Only few of them have 1 dif-ferent element and the variation of 2 elements is evenless (Figure 13) With 120575 increasing the variation of core

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

Submit your manuscripts athttpwwwhindawicom

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ReconfigurableComputing

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Applied Computational Intelligence and Soft Computing

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Electrical and Computer Engineering

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 7

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(a)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(b)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(c)

Num

ber o

f cha

nges 8

6

4

2

0

Time0 1 2 3

times105

(d)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(e)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(f)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(g)

Time0 1 2 3

times105

Num

ber o

f cha

nges 15

10

5

0

(h)

Figure 7 Number of changes in 0-1 matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 = 0204 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

Hence V119894and V

119895have their isolate core sets separately We

use node ID as the core label when creating a new communitycore Initially each node will be tagged a community label asitself The benefits are as follows Firstly the finite namespaceof core label will not cause muddle of naming Secondlycommunity core can be tracked easier when a communitycore is created again The tracking algorithm uses nodeID as initial community core ID if the node is extractedto a community core and when a node is removed fromcommunity core set the node ID is restored

Basic operation procedure of tracking is given as follows

Switch (events)Case Birth (V

119894and V119895form a new core)

119871 (119905 + 1) larr997888 119897new 119897new = 119894 or 119895 (3)

Case Death (CO119894isin CRS(119905) but CO

119894notin CRS(119905 + 1))

Every node in CO119894labeled as its node ID

CaseMerging (CO119894and CO

119895merge into on core)

Nodes with fewer core members change theirIDs to the other core

Case Splitting (V119894and V119895have no path to connect)

Delete 119897old in 119871(119905 + 1) and 119871(119905 + 1) larr 119894 119895 V119894and

V119895change label as their node ID

Case Growth (V119894join into an existing community

core)

V119894changes label as the community core

Case Contraction (V119894moves out of community core)

V119894changes label as its node ID

5 Evaluation

In this section we discuss the evolution of community corewhich is detected and tracked by our algorithm COPRA[13] is a well-known efficient algorithm of fast communitydetection and we choose COPRA to compare with our algo-rithm

51 Contact Variation In order to reveal the communitycore evolution briefly we first show the contact variationof both SIGCOMM09 and INFOCOM06 under the wholelifetime (Figure 6) Due to the difference of data preprocess-ing SIGCOMM09 records the discrete contact time amongnodes while INFOCOM06 records the continuous contacttime According to the description of SIGCOMM09 eachdevice performs a periodic Bluetooth device discovery every120 plusmn 1024 seconds (randomized) for 1024 seconds Whena device finds others it records the current time Howeverthis mechanism can only record one contact time in a periodfor the same pair which will miss some contact informationHence the total link change link add and link remove ofSIGCOMM09 almost have no differences

52 Change of 0-1 Contact Matrix According to M1015840 the 0-1 contact matrix B = [119887

119894119895] can be obtained M1015840 changes

when the time increases and the maximum contact durationamong nodes may be changed which results in the variationof B The change of B during the whole collection procedureis plotted in Figure 7 Consider the following

119887119894119895=

1 1198981015840

119894119895ge 120575 sdotmax (119898

1015840

)

0 1198981015840

119894119895lt 120575 sdotmax (119898

1015840

)

(4)

8 The Scientific World Journal

Nod

e cou

nt70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

Nod

e cou

nt

80

60

40

20

0

(b)

Aver

age n

ode c

ount

80

70

60

50

40

30

20

10

0

120575

01 03 05 07 09

SIGCOMM09INFOCOM06

(c)

Figure 8 Number of nodes in community core (a) Variation in number of nodes in community core in SIGCOMM09 (b) Variation innumber of nodes in community core in INFOCOM06 (c) Average number of nodes in community core in the two datasets under different120575

As shown in Figure 7 most of the changes occur atthe beginning of collection and they are then graduallydiminished over timeThe number of changes reduces by theincreasing of 120575 Meanwhile in SIGCOMM09 the maximumchanges of 0-1 contact matrix under 120575 = 02 04 06 08 are8 6 4 and 2 respectively while in INFOCOM06 the valuesare 12 4 2 and 2 With 120575 increasing the maximum changesof 0-1 contact matrix are reduced

53 Selected Node Count One of the biggest differencesbetween community and community core is the number ofclustered nodes In other words the community core of amobile social network is the subset of the whole communityAccording to previous works in [8] few of nodes can be

classified as ldquofamiliar strangersrdquo and ldquofriendsrdquo and nodes inthe ldquocommunityrdquo are even less Hence only the node pairswhich have high contact frequency can be selected to thecommunity core

Intuitively improving 120575 will result in fewer selectedcontacts and nodes which is illustrated in Figures 8(a) and8(b) Let us first consider SIGCOMM09 when time goesby the selected nodes become more and more stable Inthe first day of data collection the selected node changesdramatically especially under the low 120575 Then the stableduration of selected nodes prolonged and the selected nodesno longer have change after about 2lowast10

5 seconds Comparedwith different 120575 the lower 120575 will result in a higher selectednodes which is meaningless for community cores (when

The Scientific World Journal 9

Nod

e cou

nt35

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Nod

e cou

nt

80

60

40

20

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 9 Number of nodes in community detected by COPRA (a) Node number at each timestamp in communities (SIGCOMM09)(b) Number of node changes in community between two consecutive timestamps (SIGCOMM09) (c) Node number at each timestamp incommunities (SIGCOMM09) (d) Number of node changes in community between two consecutive timestamps (INFOCOM06)

120575 = 02 during the data collection the maximum selectednodes in SIGCOMM09 are 61 and the maximum selectednodes in INFOCOM06 are 71) while the higher the 120575 is thefewer nodes are selected to construct the community core Abrief comparison of average selected nodes under different120575 is depicted in Figure 8(c) The same as discussed abovethe average of selected nodes decreases when 120575 increasesAlthough the selected nodes increase with the decreasingof 120575 the variation of selected nodes in INFOCOM06 stillhas little difference Unlike SIGCOMM09 the selected nodesin INFOCOM06 change frequently even at the end of thedata collection This phenomenon can reflect the fact thatin SIGCOMM09 the ldquofriendsrdquo of attendants are relativelystable and participants usually contact with familiar people

while in INFOCOM06 contacts among strangers are morefrequent than in SIGCOMM09 hence the selected nodechanges more often Nevertheless the variation of selectednodes in SIGCOMM09 and INFOCOM06 is relatively stableafter 2lowast10

5 seconds which is important according to featuresof community core

The number of nodes in communities detected byCOPRA is depicted in Figure 9Different fromour algorithmthe number of nodes detected by COPRA is highly unstableAnd the number of changes between two consecutive times-tamps is also very large compared with our algorithm

54 Number of Community Cores A community core set iscomposed of selected nodes which divides a mobile social

10 The Scientific World Journal

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(a)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(b)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(c)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(d)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(e)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(f)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(g)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(h)

Figure 10 Number of changes in core matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 =02 04 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

network into pieces of closely connected fragments Hencethe community core number in a community set determinesthe fragmentation degree of this mobile social network Afterrunning our algorithm the community core count in eachtimestamp under different 120575 is obtained (Figure 10)

As depicted in Figure 11(a) after about 2 lowast 105 seconds

the community count under each different 120575 becomes fixedwhich is the same as the selected nodes However the corenumber in INFOCOM06 is more complicated Like thevariation of selected node the number of community corefluctuates with time changes see Figure 11(b)

As mentioned before this is due to the frequent contactamong strangers In other words attendants in SGICOMM09have a more fixed social circle than in INFOCOM06 Thelast but the most important feature of community core countis that the core number is not monotonically changing withchangesWe can see in Figure 11(c) that themaximumaveragecore count (asymp771) in SIGCOMM09 appears when 120575 = 02After a reduction to 120575 = 04 the average core count risesapproximate to 487 when 120575 = 5 The same phenomenonappears in INFOCOM06 the core count reaches the maxvalue at 120575 = 4 (asymp931) Then it declines approximate to 84at 120575 = 5 and grows approximate to 905 at 120575 = 6 Thereason for this situation can be explained by two sides Onone hand the higher will filter out more contacts whichmakes more fragments of the mobile network and increasesthe number of community cores On the other hand somenetwork fragments with low contact frequency will be movedout of the community core set entirely which results in thereduction of cores Hence the number of cores fluctuatesunder the different

Stability is very important to the community core Com-pared with Figures 11 and 12 the community core is much

more stable than traditional community detection algorithmThe number of communities detected by COPRA is depictedin Figure 12 which reveals high instability And the numberof changes between two consecutive timestamps is also verylarge compared with our algorithm

55 Change of Core Matrix In order to evaluate the changeof community core set between consecutive timestamps weuse the distance function discussed in Section 33 to illustratethe difference between consecutive community core sets

The distance function requires that the twomatrices havethe same line and row number However the nodes and linksin the community core set between consecutive timestampsare different According to the tracking algorithm nodes inthemobile social networks are assigned an initial communitycore label which is equal to their node ID And the finitenamespace of core label ensures that the label will notbe beyond the scope of node ID Then we construct acommunity core matrix CM = [119888119898

119894119895] where

119888119898119894119895=

node ID 119894 119895 notin CRS

core ID 119894 119895 isin CRS(5)

Note that although node ID is considered in computationit will not influence the final result of distance

The distance under different 120575 is depicted in Figure 10It can be observed that the majority of consecutive corematrices have no changes Only few of them have 1 dif-ferent element and the variation of 2 elements is evenless (Figure 13) With 120575 increasing the variation of core

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

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Human-ComputerInteraction

Advances in

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8 The Scientific World Journal

Nod

e cou

nt70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02

120575 = 04

120575 = 06

120575 = 08

Nod

e cou

nt

80

60

40

20

0

(b)

Aver

age n

ode c

ount

80

70

60

50

40

30

20

10

0

120575

01 03 05 07 09

SIGCOMM09INFOCOM06

(c)

Figure 8 Number of nodes in community core (a) Variation in number of nodes in community core in SIGCOMM09 (b) Variation innumber of nodes in community core in INFOCOM06 (c) Average number of nodes in community core in the two datasets under different120575

As shown in Figure 7 most of the changes occur atthe beginning of collection and they are then graduallydiminished over timeThe number of changes reduces by theincreasing of 120575 Meanwhile in SIGCOMM09 the maximumchanges of 0-1 contact matrix under 120575 = 02 04 06 08 are8 6 4 and 2 respectively while in INFOCOM06 the valuesare 12 4 2 and 2 With 120575 increasing the maximum changesof 0-1 contact matrix are reduced

53 Selected Node Count One of the biggest differencesbetween community and community core is the number ofclustered nodes In other words the community core of amobile social network is the subset of the whole communityAccording to previous works in [8] few of nodes can be

classified as ldquofamiliar strangersrdquo and ldquofriendsrdquo and nodes inthe ldquocommunityrdquo are even less Hence only the node pairswhich have high contact frequency can be selected to thecommunity core

Intuitively improving 120575 will result in fewer selectedcontacts and nodes which is illustrated in Figures 8(a) and8(b) Let us first consider SIGCOMM09 when time goesby the selected nodes become more and more stable Inthe first day of data collection the selected node changesdramatically especially under the low 120575 Then the stableduration of selected nodes prolonged and the selected nodesno longer have change after about 2lowast10

5 seconds Comparedwith different 120575 the lower 120575 will result in a higher selectednodes which is meaningless for community cores (when

The Scientific World Journal 9

Nod

e cou

nt35

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Nod

e cou

nt

80

60

40

20

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 9 Number of nodes in community detected by COPRA (a) Node number at each timestamp in communities (SIGCOMM09)(b) Number of node changes in community between two consecutive timestamps (SIGCOMM09) (c) Node number at each timestamp incommunities (SIGCOMM09) (d) Number of node changes in community between two consecutive timestamps (INFOCOM06)

120575 = 02 during the data collection the maximum selectednodes in SIGCOMM09 are 61 and the maximum selectednodes in INFOCOM06 are 71) while the higher the 120575 is thefewer nodes are selected to construct the community core Abrief comparison of average selected nodes under different120575 is depicted in Figure 8(c) The same as discussed abovethe average of selected nodes decreases when 120575 increasesAlthough the selected nodes increase with the decreasingof 120575 the variation of selected nodes in INFOCOM06 stillhas little difference Unlike SIGCOMM09 the selected nodesin INFOCOM06 change frequently even at the end of thedata collection This phenomenon can reflect the fact thatin SIGCOMM09 the ldquofriendsrdquo of attendants are relativelystable and participants usually contact with familiar people

while in INFOCOM06 contacts among strangers are morefrequent than in SIGCOMM09 hence the selected nodechanges more often Nevertheless the variation of selectednodes in SIGCOMM09 and INFOCOM06 is relatively stableafter 2lowast10

5 seconds which is important according to featuresof community core

The number of nodes in communities detected byCOPRA is depicted in Figure 9Different fromour algorithmthe number of nodes detected by COPRA is highly unstableAnd the number of changes between two consecutive times-tamps is also very large compared with our algorithm

54 Number of Community Cores A community core set iscomposed of selected nodes which divides a mobile social

10 The Scientific World Journal

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(a)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(b)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(c)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(d)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(e)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(f)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(g)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(h)

Figure 10 Number of changes in core matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 =02 04 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

network into pieces of closely connected fragments Hencethe community core number in a community set determinesthe fragmentation degree of this mobile social network Afterrunning our algorithm the community core count in eachtimestamp under different 120575 is obtained (Figure 10)

As depicted in Figure 11(a) after about 2 lowast 105 seconds

the community count under each different 120575 becomes fixedwhich is the same as the selected nodes However the corenumber in INFOCOM06 is more complicated Like thevariation of selected node the number of community corefluctuates with time changes see Figure 11(b)

As mentioned before this is due to the frequent contactamong strangers In other words attendants in SGICOMM09have a more fixed social circle than in INFOCOM06 Thelast but the most important feature of community core countis that the core number is not monotonically changing withchangesWe can see in Figure 11(c) that themaximumaveragecore count (asymp771) in SIGCOMM09 appears when 120575 = 02After a reduction to 120575 = 04 the average core count risesapproximate to 487 when 120575 = 5 The same phenomenonappears in INFOCOM06 the core count reaches the maxvalue at 120575 = 4 (asymp931) Then it declines approximate to 84at 120575 = 5 and grows approximate to 905 at 120575 = 6 Thereason for this situation can be explained by two sides Onone hand the higher will filter out more contacts whichmakes more fragments of the mobile network and increasesthe number of community cores On the other hand somenetwork fragments with low contact frequency will be movedout of the community core set entirely which results in thereduction of cores Hence the number of cores fluctuatesunder the different

Stability is very important to the community core Com-pared with Figures 11 and 12 the community core is much

more stable than traditional community detection algorithmThe number of communities detected by COPRA is depictedin Figure 12 which reveals high instability And the numberof changes between two consecutive timestamps is also verylarge compared with our algorithm

55 Change of Core Matrix In order to evaluate the changeof community core set between consecutive timestamps weuse the distance function discussed in Section 33 to illustratethe difference between consecutive community core sets

The distance function requires that the twomatrices havethe same line and row number However the nodes and linksin the community core set between consecutive timestampsare different According to the tracking algorithm nodes inthemobile social networks are assigned an initial communitycore label which is equal to their node ID And the finitenamespace of core label ensures that the label will notbe beyond the scope of node ID Then we construct acommunity core matrix CM = [119888119898

119894119895] where

119888119898119894119895=

node ID 119894 119895 notin CRS

core ID 119894 119895 isin CRS(5)

Note that although node ID is considered in computationit will not influence the final result of distance

The distance under different 120575 is depicted in Figure 10It can be observed that the majority of consecutive corematrices have no changes Only few of them have 1 dif-ferent element and the variation of 2 elements is evenless (Figure 13) With 120575 increasing the variation of core

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 9

Nod

e cou

nt35

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Nod

e cou

nt

80

60

40

20

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 9 Number of nodes in community detected by COPRA (a) Node number at each timestamp in communities (SIGCOMM09)(b) Number of node changes in community between two consecutive timestamps (SIGCOMM09) (c) Node number at each timestamp incommunities (SIGCOMM09) (d) Number of node changes in community between two consecutive timestamps (INFOCOM06)

120575 = 02 during the data collection the maximum selectednodes in SIGCOMM09 are 61 and the maximum selectednodes in INFOCOM06 are 71) while the higher the 120575 is thefewer nodes are selected to construct the community core Abrief comparison of average selected nodes under different120575 is depicted in Figure 8(c) The same as discussed abovethe average of selected nodes decreases when 120575 increasesAlthough the selected nodes increase with the decreasingof 120575 the variation of selected nodes in INFOCOM06 stillhas little difference Unlike SIGCOMM09 the selected nodesin INFOCOM06 change frequently even at the end of thedata collection This phenomenon can reflect the fact thatin SIGCOMM09 the ldquofriendsrdquo of attendants are relativelystable and participants usually contact with familiar people

while in INFOCOM06 contacts among strangers are morefrequent than in SIGCOMM09 hence the selected nodechanges more often Nevertheless the variation of selectednodes in SIGCOMM09 and INFOCOM06 is relatively stableafter 2lowast10

5 seconds which is important according to featuresof community core

The number of nodes in communities detected byCOPRA is depicted in Figure 9Different fromour algorithmthe number of nodes detected by COPRA is highly unstableAnd the number of changes between two consecutive times-tamps is also very large compared with our algorithm

54 Number of Community Cores A community core set iscomposed of selected nodes which divides a mobile social

10 The Scientific World Journal

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(a)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(b)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(c)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(d)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(e)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(f)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(g)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(h)

Figure 10 Number of changes in core matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 =02 04 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

network into pieces of closely connected fragments Hencethe community core number in a community set determinesthe fragmentation degree of this mobile social network Afterrunning our algorithm the community core count in eachtimestamp under different 120575 is obtained (Figure 10)

As depicted in Figure 11(a) after about 2 lowast 105 seconds

the community count under each different 120575 becomes fixedwhich is the same as the selected nodes However the corenumber in INFOCOM06 is more complicated Like thevariation of selected node the number of community corefluctuates with time changes see Figure 11(b)

As mentioned before this is due to the frequent contactamong strangers In other words attendants in SGICOMM09have a more fixed social circle than in INFOCOM06 Thelast but the most important feature of community core countis that the core number is not monotonically changing withchangesWe can see in Figure 11(c) that themaximumaveragecore count (asymp771) in SIGCOMM09 appears when 120575 = 02After a reduction to 120575 = 04 the average core count risesapproximate to 487 when 120575 = 5 The same phenomenonappears in INFOCOM06 the core count reaches the maxvalue at 120575 = 4 (asymp931) Then it declines approximate to 84at 120575 = 5 and grows approximate to 905 at 120575 = 6 Thereason for this situation can be explained by two sides Onone hand the higher will filter out more contacts whichmakes more fragments of the mobile network and increasesthe number of community cores On the other hand somenetwork fragments with low contact frequency will be movedout of the community core set entirely which results in thereduction of cores Hence the number of cores fluctuatesunder the different

Stability is very important to the community core Com-pared with Figures 11 and 12 the community core is much

more stable than traditional community detection algorithmThe number of communities detected by COPRA is depictedin Figure 12 which reveals high instability And the numberof changes between two consecutive timestamps is also verylarge compared with our algorithm

55 Change of Core Matrix In order to evaluate the changeof community core set between consecutive timestamps weuse the distance function discussed in Section 33 to illustratethe difference between consecutive community core sets

The distance function requires that the twomatrices havethe same line and row number However the nodes and linksin the community core set between consecutive timestampsare different According to the tracking algorithm nodes inthemobile social networks are assigned an initial communitycore label which is equal to their node ID And the finitenamespace of core label ensures that the label will notbe beyond the scope of node ID Then we construct acommunity core matrix CM = [119888119898

119894119895] where

119888119898119894119895=

node ID 119894 119895 notin CRS

core ID 119894 119895 isin CRS(5)

Note that although node ID is considered in computationit will not influence the final result of distance

The distance under different 120575 is depicted in Figure 10It can be observed that the majority of consecutive corematrices have no changes Only few of them have 1 dif-ferent element and the variation of 2 elements is evenless (Figure 13) With 120575 increasing the variation of core

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 The Scientific World Journal

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(a)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(b)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(c)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(d)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(e)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(f)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(g)

Chan

ge o

f cor

e mat

rix 3

2

1

0

Time0 1 2 3

times105

(h)

Figure 10 Number of changes in core matrix between two consecutive timestamps Above number of changes in SIGCOMM09 under 120575 =02 04 06 and 08 (from (a) to (d)) Below number of changes in INFOCOM06 under 120575 = 02 04 06 and 08 (from (e) to (h))

network into pieces of closely connected fragments Hencethe community core number in a community set determinesthe fragmentation degree of this mobile social network Afterrunning our algorithm the community core count in eachtimestamp under different 120575 is obtained (Figure 10)

As depicted in Figure 11(a) after about 2 lowast 105 seconds

the community count under each different 120575 becomes fixedwhich is the same as the selected nodes However the corenumber in INFOCOM06 is more complicated Like thevariation of selected node the number of community corefluctuates with time changes see Figure 11(b)

As mentioned before this is due to the frequent contactamong strangers In other words attendants in SGICOMM09have a more fixed social circle than in INFOCOM06 Thelast but the most important feature of community core countis that the core number is not monotonically changing withchangesWe can see in Figure 11(c) that themaximumaveragecore count (asymp771) in SIGCOMM09 appears when 120575 = 02After a reduction to 120575 = 04 the average core count risesapproximate to 487 when 120575 = 5 The same phenomenonappears in INFOCOM06 the core count reaches the maxvalue at 120575 = 4 (asymp931) Then it declines approximate to 84at 120575 = 5 and grows approximate to 905 at 120575 = 6 Thereason for this situation can be explained by two sides Onone hand the higher will filter out more contacts whichmakes more fragments of the mobile network and increasesthe number of community cores On the other hand somenetwork fragments with low contact frequency will be movedout of the community core set entirely which results in thereduction of cores Hence the number of cores fluctuatesunder the different

Stability is very important to the community core Com-pared with Figures 11 and 12 the community core is much

more stable than traditional community detection algorithmThe number of communities detected by COPRA is depictedin Figure 12 which reveals high instability And the numberof changes between two consecutive timestamps is also verylarge compared with our algorithm

55 Change of Core Matrix In order to evaluate the changeof community core set between consecutive timestamps weuse the distance function discussed in Section 33 to illustratethe difference between consecutive community core sets

The distance function requires that the twomatrices havethe same line and row number However the nodes and linksin the community core set between consecutive timestampsare different According to the tracking algorithm nodes inthemobile social networks are assigned an initial communitycore label which is equal to their node ID And the finitenamespace of core label ensures that the label will notbe beyond the scope of node ID Then we construct acommunity core matrix CM = [119888119898

119894119895] where

119888119898119894119895=

node ID 119894 119895 notin CRS

core ID 119894 119895 isin CRS(5)

Note that although node ID is considered in computationit will not influence the final result of distance

The distance under different 120575 is depicted in Figure 10It can be observed that the majority of consecutive corematrices have no changes Only few of them have 1 dif-ferent element and the variation of 2 elements is evenless (Figure 13) With 120575 increasing the variation of core

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 11

Num

ber o

f cor

es10

8

6

4

2

0

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

(a)

Time0 05 1 15 2 25 3

times105

120575 = 02120575 = 04

120575 = 05

120575 = 06120575 = 08

Num

ber o

f cor

es

12

10

8

6

4

2

0

(b)

Aver

age c

ore n

umbe

r

10

9

8

7

6

5

4

3

2

1

001 02 03 04 05 06 07 08 09

120575

SIGCOMM09INFOCOM06

(c)

Figure 11 Number of community cores (a) Variation in number of community core in SIGCOMM09 (b) Variation in number of communitycores in INFOCOM06 (c) Average number of community cores in the two datasets under different 120575

matrix reduces accordingly This illustrates that the higherthe 120575 is the more sensitive the community core becomesConsidering the variation time most of the changes occur atthe beginning of data collection which is also consistent withthe change of selected nodes and the core number

56 Community Core Tracking In this part we focus on thevisualization of community core evolution First we extract

the core ID of each node from community core matrix If anode does not belong to any cores it is labeled as its nodeID Then we get the community core at each timestampFinally the ID including only one node is removed Thecommunity core set extracted from the two datasets ispresented in Figure 14 According to the selected node andthe core number under the different 120575 we choose 120575 = 04 inSIGCOMM09 and 120575 = 06 in INFOCOM06 to display theevolution of community cores

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

12 The Scientific World Journal

Num

ber o

f com

mun

ities

30

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(a)

Num

ber o

f cha

nges

25

20

15

10

5

0

Time0 05 1 15 2 25 3

times105

(b)

Num

ber o

f com

mun

ities

70

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(c)

Num

ber o

f cha

nges

60

50

40

30

20

10

0

Time0 05 1 15 2 25 3

times105

(d)

Figure 12 Number of communities detected by COPRA (a) Number of communities at each timestamp (SIGCOMM09) (b) Number ofcommunity changes between two consecutive timestamps (SIGCOMM09) (c) Number of communities at each timestamp (INFOCOM06)(d) Number of community changes between two consecutive timestamps (INFOCOM06)

Figures 14(a) and 14(c) include nodes in the communitycores and the noncore nodes If a node does not belong toany community cores its ID will be a straight line Besides ifa node is selected as the community core the ID will changeto the core ID Figures 14(b) and 14(d) are the refined coretraces including nodes belonging to the community coresonly Compared with Figure 11 the evolution of communitycores appears directly

6 Conclusions

In this paper we mainly analyze the community core inmobile social networks Firstly the change of cumulativestable contact is discussed And then we propose the

community core detection algorithm to extract the commu-nity core from two experimental mobile social networksFinally we introduce a label-based community core trackingalgorithm which can briefly display the evolution of commu-nity core Compared with traditional community detectionalgorithms we show that the community core extracted byour algorithm is stable and that it can be further used innetwork analysis in mobile social networks

Acknowledgment

An earlier version of this paper was presented at the 2013ASEIEEE International Conference on Social ComputingThis paper is being jointly supported in part by the National

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World Journal 13

01 02 03 04 05 06 07 08 090

50

100

150

200

250

300

Num

ber o

f cha

nges

SIGCOMM09 1 changeSIGCOMM09 2 changes

INFOCOM06 1 changeINFOCOM06 2 changes

120575

Figure 13 Number of changes in core matrix with different 120575

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(a)

Nod

e ID

80604020

0

Time0 05 1 15 2 25 3

times105

(b)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(c)

Time0 05 1 15 2 25 3

times105

Nod

e ID

10080604020

(d)

Figure 14 (a) Original community core evolution in SIGCOMM09 with 120575 = 04 (b) Refined community core evolution in SIGCOMM09with 120575 = 04 (c) Original community core evolution in INFOCOM06 with 120575 = 06 (d) Refined community core evolution in INFOCOM06with 120575 = 06

Natural Science Foundation of China under Grant nos61303062 and 60903225

References

[1] Y Zhang and D Y Yeung ldquoOverlapping community detectionvia bounded nonnegative matrix tri-factorizationrdquo in Proceed-ing of the ACM SIGKDD International Conference on KnowledgeDiscovery andDataMining pp 606ndash614 Beijing ChinaAugust2012

[2] W Lin X Kong P S Yu Q Wu Y Jia and C Li ldquoCommunitydetection in incomplete information networksrdquo in Proceedingsof International Conference onWorldWideWeb (WWW rsquo12) pp341ndash350 Lyon France April 2012

[3] P Brodka S Saganowski and P Kazienko ldquoGroup evolutiondiscovery in social networksrdquo in Proceeding of the InternationalConference on Advances in Social Networks Analysis and Mining(ASONAM rsquo11) pp 247ndash253 Kaohsiung Taiwan July 2011

[4] R Cazabet F Amblard and C Hanachi ldquoDetection of overlap-ping communities in dynamical social networksrdquo in Proceedingof the 2nd IEEE International Conference on Social Computing(SocialCom rsquo10) pp 309ndash314 Minneapolis Minn USA August2010

[5] M Seifi and J L Guillaume ldquoCommunity cores in evolvingnetworksrdquo in Proceedings of the International Conference com-panion on World Wide Web (WWW rsquo12) pp 1173ndash1180 LyonFrance April 2012

[6] A Chaintreau P Hui J Crowcroft C Diot R Gass and JScott ldquoImpact of human mobility on opportunistic forwarding

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

14 The Scientific World Journal

algorithmsrdquo IEEE Transactions onMobile Computing vol 6 no6 pp 606ndash620 2007

[7] N P Nguyen T N Dinh Y Xuan and M T Thai ldquoAdaptivealgorithms for detecting community structure in dynamicsocial networksrdquo in Proceedings of the IEEE Conference onComputer Communications (INFOCOM rsquo11) pp 2282ndash2290Shanghai China April 2011

[8] P Hui ldquoPeople are the network experimental design andevaluation of social-based forwarding algorithmsrdquo Tech RepUCAM-CL-TR-713 2008

[9] L Wang J E Hopcroft J He H Liang and S SuwajanakornldquoExtracting the core structure of social networks using (120572 120573)-communitiesrdquo Internet Mathematics vol 9 no 1 pp 58ndash812013

[10] A K Pietilainen E Oliver J LeBrun G Varghese and CDiot ldquoMobiClique middleware for mobile social networkingrdquoin Proceedings of the 2nd ACM Workshop on Online SocialNetworks (WOSN rsquo09) pp 49ndash54 Barcelona Spain August2009

[11] S Asur S Parthasarathy and D Ucar ldquoAn event-based frame-work for characterizing the evolutionary behavior of interactiongraphsrdquo ACM Transactions on Knowledge Discovery from Datavol 3 no 4 article 16 2009

[12] M Shiga I Takigawa andHMamitsuka ldquoA spectral clusteringapproach to optimally combining numericalvectors with amodular networkrdquo in Proceeding of the 13th ACM SIGKDDInternational Conference on Knowledge Discovery and DataMining (KDD rsquo07) pp 647ndash656 San Jose Calif USA August2007

[13] S Gregory ldquoFinding overlapping communities in networks bylabel propagationrdquo New Journal of Physics vol 12 Article ID103018 2010

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014