Research ArticleSocial Networks through the Prism of Cognition

Radosław Michalski 1 Boleslaw K Szymanski 23 Przemysław Kazienko 1

Christian Lebiere 4 Omar Lizardo 5 and Marcin Kulisiewicz 1

1Department of Computational Intelligence Faculty of Computer Science and ManagementWrocław University of Science and Technology Wrocław Poland2Department of Computer Science Rensselaer Polytechnic Institute Troy NY USA3Społeczna Akademia Nauk Łodz Poland4Carnegie Mellon University Pittsburgh PA USA5University of California Los Angeles CA USA

Correspondence should be addressed to Radosław Michalski radoslawmichalskipwredupl

Received 11 June 2020 Revised 18 November 2020 Accepted 24 December 2020 Published 8 January 2021

Academic Editor Fei Xiong

Copyright copy 2021 Radosław Michalski et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Human relations are driven by social eventsmdashpeople interact exchange information share knowledge and emotions and gathernews from mass media +ese events leave traces in human memory the strength of which depends on cognitive factors such asemotions or attention span Each trace continuously weakens over time unless another related event activity strengthens it Herewe introduce a novel cognition-driven social network (CogSNet) model that accounts for cognitive aspects of social perception+emodel explicitly represents each social interaction as a trace in humanmemory with its corresponding dynamics+e strengthof the trace is the only measure of the influence that the interactions had on a person For validation we apply our model toNetSense data on social interactions among university students +e results show that CogSNet significantly improves the qualityof modeling of human interactions in social networks

1 Introduction

+ere is a fundamental difference between the ways socialevents are currently represented by network analysts and theway they are perceived and cognitively processed byhumans In the former the discrete nature of events isretained and the weights of social network edges are updatedonce per each relevant event In human memory by con-trast perception of events changes continuously over timeMoreover the initial strength of a trace depends on cognitivefactors defined by states of mind of participants and specificaspects of their interactions Decisions about whether toinitiate maintain or discontinue social relations involvecognitive processes operating on all relevant informationstored in memory traces over specific time-scales

To date however there are no models that accuratelyrepresent the dynamics of social relations strictly throughtheir corresponding traces in human memory To address

this gap we introduce a novel cognition-driven socialnetwork (CogSNet) model that captures impact of humanmemory on perception of accumulated events and on de-cisions to form maintain or dissolve social relations +emodel explicitly represents some of the human memorydynamics such as the gradual decay of memory traces overtime With suitable data it can be extended to includeadditional cognitive aspects such as individual levels ofsensitivity to relevant events emotions or distractionsduring perception of events Hence the model is capable ofcapturing the dynamics of social interactions in naturalsettings from the cognitive perspective of each participant

To evaluate the empirical performance of the proposedmodel we compare modelrsquos results obtained with param-eters fitted using behavioral communication event data tothe ground truth perceptual data collected from humanparticipants with regard to their most important relations+e comparisons reveal that the perception of the depth of

HindawiComplexityVolume 2021 Article ID 4963903 13 pageshttpsdoiorg10115520214963903

interactions between people is well captured by the CogSNetmodel At any given point in time the model can computethe current strength of memory traces including the impactof discrete events creating or reinforcing these traces +ecomputed state of each social relation (eg salient versusdecayed) can then be compared against the externalizationof cognition in the form of self-reports +e CogSNet modelwith parameters fitted using recorded event data achieveshigh accuracy in fitting ground truth data for the NetSensenetwork +is success demonstrates the importance of in-corporating cognitive processes and memory dynamics foradequate modeling of the dynamics of social relationships

Memory is considered to be one of the most importantcomponents of human cognition +is is especially the casegiven the necessity to efficiently retrieve large amounts ofknowledge and to select information from a noisy envi-ronment Hence one of the fundamental challenges forcognitive science has been to understand the mechanismsinvolved in managing information in human memory [1]+e exact details of these mechanisms are difficult to firmlyestablish since the human brain is a highly complex systemfeaturing strong differences across individuals based onexperimental tuning [2] Yet there have been significantadvances in the direction of developing good qualityworking models of memory and other core cognitive pro-cesses For example the ACT-R cognitive architecture does agood job of modeling core features of human declarativememory successfully replicating a large number of well-known effects For our purposes the most important amongthese empirical regularities are the well-established primacyand recency effects for list memory [3]

Limited capacity and the gradual decay of traces inmemory over time have been confirmed by many studiesanalyzing the relationship between time and recall [4ndash6]+eforgetting mechanism is modeled by a parametric functiondescribing how well a given item will be recalled as afunction of time It appears that humans tend to underes-timate the number of events they experience ie they ac-tually forget faster than they think they do [5] Obviouslyany reminiscence of a particular event primed by an externalsituation allows people to remember the information longerYet ultimately the limited capacity paradigm and the for-getting function are chiefly responsible for controlling thelifetime of such reminiscence in memory [7] +ere arerelatively many papers within the broader domain of link

predictions but they treat human communication as a seriesof fixed time widows that reflect neither human cognitionnor continuity of forgetting For example Reader et al [8]identified over a dozen of predictive factors quantifyingphone communication like time of the last call or frequencyof calls within a 4-week period that would enable to forecastpersistence of social relationships ie communication alsoin the following 4-week period +e influences amongculture cognition and network thinking are mentioned in[9] but in a very qualitative way without any hint how toquantify such influence into a model which is essential forour approach Some other works also study the problem ofrecalling the names of their peers in a variety of activities[10ndash12] but they focus on a detailed analysis of personalattributes differentiating subjects in terms of forgettingwhile here we concentrate on the models of forgetting

2 Materials and Methods

+e human brain records events as they arrive but only asmall fraction of the incoming information is stored in long-term memory due to its limited capacity +e forgettingmechanism dictates that the chance to recall a given eventdecreases gradually as we move forward from the time offirst exposure [6] In some sense it is similar to graph-streaming [13] and feed-based social media network cas-cades [14] scenarios where incoming events are ordered bytheir arrival time but only some of them are kept

Accordingly the CogSNet model uses a forgettingfunction f to account for the decreasing probability ofkeeping aging of memory traces over time Forgetting is thusa monotonically nonincreasing function of time with f(0)

1 and f(t)ge 0 for all tgt 0 It is defined by three parametersreinforcement peak 0lt μle 1 the forgetting intensity λ (bothused in the ACT-R model) and additional third parameterforgetting threshold 0lt θlt μ

For clarity we present here the basic CogSNet model amore general version is defined in Appendix and Appendixequations (A1) and (A2) +is model processes an eventhappening at time t as follows If the event involves a pair ofunconnected nodes i and j an edge (ij) linking these nodesis created +is edge is assigned weight wij(t) equal to thereinforcement peak value μ Otherwise if the nodes involvedare already connected the forgetting function f is used to setthe weight of the edge connecting these nodes

wij(t) μ if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μ + wij tij1113872 1113873f t minus tij1113872 1113873(1 minus μ) otherwise

⎧⎪⎨

⎪⎩(1)

where tij is the time of the preceding event for this edgeProcessing of each event ends with advancing tij to t Ini-tially tij and wij(0) are set to 0 Equation (1) and limit onvalue of μ ensure that a weight of any edge is at most 1

+e weight wij(t) of an edge (ij) between two nodes atany user selected time t is computed as follows Once all the

relevant events up to time t are processed we simply setwij(t) wij(tij)lowastf(t minus tij) If the result is less than theforgetting threshold θ wij(t) is reduced to zero and the edgeis removed A threshold is needed with forgetting functionssuch as power and exponential forgetting that are positivefor nonnegative arguments Otherwise an edge would get

2 Complexity

positive weight at creation and would always stay positiveie all createdmemory traces would never cease to exist+ereinforcement peak μ defines the impact of an event on theweight of the edge relevant to this event +is peak is a globalmodel parameter here In principle the peak can be adjustedaccording to the event or node type to allow for individu-alized event perception

In Figure 1 we compare the CogSNet model with theprevious proposals for representing temporal network dy-namics +e most common approach for representing socialnetwork dynamics is to use interaction sequences [15]Under this method each event is time-stamped and theweights are added to the edges connecting nodes involved inthis event cf Figure 1 ((I) and (IIa)) Moreover a given edgeis active (exists) only at a given time t +is is the mostgranular approach as it is capable of tracking all the eventsoccurring between nodes while preserving the temporalorder of events

In contrast a static binary network representation asshown in Figure 1 (IIb) aggregates all events by making alledges time-independent Consequently an edge existingbetween a pair of nodes corresponds to an event betweenthese nodes occurred at least once in the whole observedperiod [16] Such an edge representation throws away in-formation on the temporal ordering of events making itimpossible to study dynamic processes in static networks

+e incremental network solution accumulates eventsonly up to the current time t of analysis +e classical ap-proach used early in [17ndash19] views a dynamic network as aseries of time-ordered sequences of static graphs (see Fig-ure 1 (IIc)) More recently this method was applied tomodeling network and community evolution [20 21] +edrawback of this approach is that it does not preserve theordering of interactions within time slices Applying asimple frequency-based aggregation creates a frequency-based (FQ) metric cf Figure 1 (IId) Taking into accountonly a given number of the most recent events leads to therecency-based (RC) model cf Figure 1 (IIe) Both of thesemodels are used here as baseline models

Figure 1 (IIf ) shows an example of dynamic socialnetwork generated using the CogSNet model All othersocial network models presented in Figure 1 ((IIa)ndash(IIe)) canalso be represented by CogSNet by setting appropriateparameter combinations to achieve as needed no decayinstant decay and so forth In this way CogSNet can bethought of as a universal generative dynamic model fortemporal social networks encompassing previous ap-proaches as special cases

In general the forgetting function f(Δt) over time in-terval Δt can be of any type (linear power and logarithmic)but here informed by work in the cognitive psychology ofmemory [5] we evaluate only two such functions the ex-ponential function fexp and the power function fpow definedas follows

fexp

(Δt) eminusλΔt

(2)

fpow

(Δt) max(1Δt)minusλ (3)

where λ denotes the forgetting intensity typically λ isin [0 1]+e use of max in the power function ensures that per-ception of events that happened less than a time unit ago isnot changed by forgetting +e time unit in which theforgetting function is expressed scales the values of theparameters Our experiments use one hour as the time unit

To simplify optimal parameters search we aggregate allthree parameters into the trace lifetime L defining the timeafter which an unreinforced memory trace is forgotten ieis too weak to be recalled In the model L is the time overwhich the forgetting function reduces the edge weight fromμ to θ causing the edge to be removed cf Figure 2 For theexponential forgetting function (equation (2)) trace lifetimeLexp is

Lexp

1λln

μθ

1113874 1113875 (4)

while for the power function the formula is

Lpow

μθ

1113874 11138751λ

(5)

Please note that according to the CogSNet model therelationship weights belong to range [θ 1)cup 0 Edgersquosweight value tends to 1 when the node encounters largenumber of tightly successive contacts+e larger the numberof events is and the closer to each other they are the closer to1 the weight is

3 Results and Discussion

In this section we assess the empirical validity of the proposemodel by estimating the accuracy with which the CogSNetmodel reproduces the dynamics of social relations in adynamic social network Our computational experimentsuse the NetSense dataset introduced in [22]

+e data contain 7096844 human mobile phonecommunication events including both calls and text mes-sages +ese are augmented by 578 surveys containing self-reports on top contacts It is worth underlining that thisdataset provides a unique opportunity to evaluate how thecommunication frequencies are recalled by the study par-ticipants because students reported their perception of thefrequencymdashdetailed information on how this informationhas been reported and how data were processed is providedin Appendix

We use this dataset to study the evolution of two coupledsocial networks of university students +e first is a be-havioral network representing interactions between indi-viduals in the form of the records of their mobile calls andtext messages +e second one has perceptual edges definedby the personal nominations +ese nominations are basedon studentsrsquo perception of the corresponding relations asone of the top twenty most interacting peers in the surveysadministered to participants +ese surveys cover the firstfour semesters of the studentrsquos college experience (beginningof freshman year to the end of the sophomore year) Wecompare the list of nominations predicted from the CogSNetnetwork model purely from the communication event datawith the list of nominations collected in a given survey

Complexity 3

A

B

C

D

Time

(b) Staticnetwork

(c) Windows

(d)Incremental

A B

CD

11

3 232

A B

CD

11

2

A B

CD

13 2

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CD03 06

03

Linkingevents

(a) No contactin interactionsequence at t

A B

CD

A B

DA

B D

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02

02

02

02

03

06

03

I Interaction sequence

(f) Cognition-driven social

network(CogSNet)

II Social network at time t

A B

CD2

1

(e) Recent events

t

t0 t1 t2 t3 t4 t5 t6 t7 t8 t9t10 t11

Figure 1 Various approaches tomodeling dynamic networks and edge weighting (relation strengths) for the 4-node network at a given timet (I) and (IIa) interaction sequence (IIb) static (time-aggregated) network (IIc) sliding windows (IId) incremental network all events fromthe beginning time t0 to the current time t are considered and the frequency of interaction in the period constitutes the frequency-based(FQ) reference (IIe) network based on n 3 recent events used for recency-based reference RC (IIf ) cognition-based social networkmodel(CogSNet) introduced here

4 Complexity

Figure 3 shows an example of dynamic social networkgenerated from a subset of NetSense data using the CogSNetmodel

For comparison we also implement three baselinemodels using the mobile phone communication data +efirst is a frequency-based (FQ) model which orders the peersby the number of interactions regardless of their time orderUnder this model the highest ranking peers whose numberequals the number of the nominations listed in the corre-sponding survey are selected on the basis of their com-munication frequency+e recency-based (RC) model selectsthe top interacting peers who for each individual given anumber of recent events Finally the random sampling(RND) model creates the list by randomly selecting peersfrom those who are recorded in the history of interactions ofa given node While random baseline can be considered as asimplistic lower bound the two aforementioned are actuallyregarded to be state of the art since they express two cog-nitive phenomena +e recency effect was uncovered by theexperiments of Hermann Ebbinghaus on recall +e for-getting curve he proposed is based on studies in [23] andrecently confirmed in [24] In these papers the authorsdemonstrate that the learning curve rises the chances forrecalling at the beginning and at the end of the contactperiod while the chances to recall events are lower in themiddle of it However as the question in surveys mainlyrefers to the present we focused on the recency instead ofprimacy +e second baseline frequency-based is animplementation of scientific results taken from the domainof lexical memory [25] +e authors of this study show thatthe frequently repeated words are being recalled fasterUnfortunately to the best of our knowledge there are noother approaches suitable to be used in similar tasks

To compare the performance of all the models we use aJaccardmetric equation (C1) in Appendix whichmeasuresthe ratio of the number of nominations produced by themodel that are also ground truth nominations listed in thecorresponding survey divided by the number of uniquenominations on both lists Due to the nature of the questionformulation in the survey it was impossible to use anyapproach based on ordering such as Kendallrsquos or Pearsonrsquosranks since we had to consider the list as unordered

Figure 4 shows the results of this comparison over therange of parameters corresponding to reported values forforgetting of one to 14 weeks As reported in [26] the abilityto recall information about social interactions starts todegrade after about one week +e experiments usingNetSense dataset reveal that the performance is the highestwhen the forgetting of unreinforced memory traces happensafter two weeks +e results remain satisfactory for forget-ting thresholds as low as one week With the two-weekthreshold the CogSNet model is with either power or ex-ponential forgetting statistically significantly better than anybaseline model In the literature the power forgetting wasfound to perform significantly better than the exponentialforgetting when a range of parameters was limited [3] Wedo not observe such superior performance of power for-getting here Both functions have similar peak of Jaccardmetric albeit for the different lifetime values

When comparing the results of surveys with the states ofthe CogSNet network at the times of the surveys the Jaccardmetric is as high as 2998 for exponential function for L 3days +e distant second is recency-based RC model whichdelivers much lower Jaccard metric of 178 Neverthelesswhen looking at relative simplicity of the RC metric one canalso say that it performs quite goodmdashthis is due to the fact

L

θ

μ

Forgetting modelExponentialPower

00

02

04

06

Wei

ght o

f rel

atio

nshi

p10 20 30 400

Time (days)

Figure 2 Dynamics of relations in CogSNet network with exponential and power functions and with parameters set to μ 04 θ 01 andL 10 days

Complexity 5

CogSNet relationship strengths

Oct Nov Dec Jan Feb

Oct Nov Dec Jan Feb

Oct Nov Dec Jan Feb

t1 t2 t3 t4 ts

000

025

050

075

100A

minusB

000

025

050

075

100

AminusC

000

025

050

075

100

BminusC

000

025

050

075

100

AminusD

000

025

050

075

100

BminusD

Nov Dec Jan FebOct

FebDec JanNovOctTime

(a)

t1 t2 t3 t4 ts

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

(b)

Figure 3 Four-node CogSNet network for the sample of real NetSense data μ 04 θ 01 and L 10 days nodes A B C and Dcorrespond respectively to participants with ids 40997 11360 10841 and 1232 (a) Relation strengths according to the CogSNet model over4-month period (one term) (b) network snapshots at four timestamps t1ndasht4 and at the survey time ts

6 Complexity

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 201 100 20050Lifetime L in days

μ = 03 θ = 02μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

(a)

μ = 03 θ = 02 μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 20 50 100 2001Life time L in days

(b)

Figure 4 +e plot of the Jaccard metric as a function of lifetime L +e metric measures the overlap between the two sets of peers oneidentified by CogSNet and the other listed by students in the survey +ese plots are compared to the results achieved by the three baselinemodels recency-based with the best results obtained with the number of recent events set to 400 frequency-based and random+e resultsare plotted for the CogSNet running with various parameters for (a) exponential and (b) power forgetting functions

Complexity 7

Table 1 Numerical and symbolic p values adjusted by SimesndashHochberg step-up method of nonparametric pairwise post hoc Nemenyi testcomparing values of Jaccard metric for results obtained by each method with parameters fitted using recorded event data with the groundtruth based on surveys taken over semesters 1ndash4

Power Exponential Recency Frequency RandomPower mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast lowastlowastlowast

Exponential 00000 mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast

Recency 00000 00000 mdash lowastlowastlowast lowastlowastlowast

Frequency 00000 00000 00000 mdash lowastlowastlowast

Random 00000 00000 00000 00000 mdash+ese results show how each method in the leftmost column performs versus the method in the first row +e part of table under the diagonal showsnumerical values of the test while the part above the diagonal presents p values coded as follows lowastlowastlowastplt 000005 represents high confidence in theperformance differences between the methods since typically values of pgt 005 are considered indicative of nonsignificance

Table 2 Setting of parameters for power forgetting function in computational experiments

λ dependence on μ θ and L for power forgettingLifetime L days μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 043621 030863 0654312 03581 025337 0537163 032415 022934 0486234 030372 021489 0455585 028957 020487 0434356 027894 019736 0418417 027055 019142 0405838 026368 018656 0395529 02579 018247 03868510 025294 017896 03794211 024862 01759 03729312 02448 01732 0367213 024139 017079 03620814 023831 016861 03574715 023552 016663 03532816 023297 016483 03494517 023062 016317 03459218 022844 016163 03426719 022643 01602 03396420 022455 015887 03368221 022278 015762 03341822 022113 015645 0331723 021957 015535 03293624 02181 015431 03271625 021671 015333 03250726 021539 015239 03230927 021414 015151 0321228 021294 015066 03194129 02118 014985 0317730 021071 014908 03160640 020188 014283 03028250 019553 013834 02932960 019062 013487 02859480 018337 012974 027506100 017811 012602 026717

8 Complexity

Table 3 Setting of parameters for exponential forgetting function in computational experiments

λ dependence on μ θ and L for exponential forgettingLifetime L (days) μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 005776 004087 0086642 002888 002043 0043323 001925 001362 0028884 001444 001022 0021665 001155 000817 0017336 000963 000681 0014447 000825 000584 0012388 000722 000511 0010839 000642 000454 00096310 000578 000409 00086611 000525 000372 00078812 000481 000341 00072213 000444 000314 00066614 000413 000292 00061915 000385 000272 00057816 000361 000255 00054217 00034 00024 0005118 000321 000227 00048119 000304 000215 00045620 000289 000204 00043321 000275 000195 00041322 000263 000186 00039423 000251 000178 00037724 000241 00017 00036125 000231 000163 00034726 000222 000157 00033327 000214 000151 00032128 000206 000146 00030929 000199 000141 00029930 000193 000136 00028940 000144 000102 00021750 000116 000082 00017360 000096 000068 00014480 000072 000051 000108100 000058 000041 000087

000000

000002

000004

Den

sity

20000 40000 600000Number of communication events

(a)

0000000

0000005

0000010

0000015

0000020

Den

sity

100000 200000000000Time of day

(b)

Figure 5 Continued

Complexity 9

that primacy and recency effects come into play when listingpeers [27]

4 Conclusions

+e results demonstrate that accounting for event percep-tion and memory dynamics is essential for faithfullymodeling the interaction dynamics in social networks +eCogSNet performed statistically significantly better than anyof the base temporal models (see Table 1 in Appendix) Withsuitable data the model can be extended to represent in-dividual sensitivity to events and the impact of emotions anddistractions on event perception to further improve mod-eling quality

+e dataset on which we operate contains data limited toa relatively small fraction of human social activities Henceit cannot be stated that there were no other events happeningthat could have affected participant memory about othersExamples of such events are face to face interactions indirectreferences to someone else and personal reminiscencesabout others In addition emotions can also play a signif-icant role in these processes and they can be potentiallyidentified with some confidence from available data [28]

As a result the reinforcement peak value can be adjustedfor a given person and for an individual event cf alsoequation (A1) in Appendix What is observed here is apartial manifestation from human memory However eventaking into account that the model was built based on a

single data source over 6 million telephone calls andmessages among the NetSense study participants the modelfitted with the recorded event data yields results wellmatching the salience of social contacts over all 578 surveyscompleted by 184 participants It is observed that even whenusing the same values of model parameters for all partici-pants the results are satisfactory Nevertheless this accuracymost likely could have been increased if the parameters hadbeen individually adjusted for each participant Apart fromthat one should remember that the recall task is highlycontextual so the individuals will provide different sets ofpeers depending on the situation [11 12]

+e findings of this work can be also considered as animportant building block on how to maintain healthy re-lationships or how to discover anomalies For instancebased on the results it is anticipated that in order to keep aliving relationship with a peer the contacts should not besparser than lifetime L preferably even more frequent Onthe contrary these findings also demonstrate on how long agrieving period after breaking the relationship could po-tentially last and how to handle the recovery process af-terwards in order to make it effective Lastly smallmodifications making the model closer to ACT-R wouldenable its use for instance in the case of Alzheimer diseasewhere one could monitor whether the peers recalled bypatients are in line with what a CogSNet model generatesbased on the contacts or visits If there will be any dis-crepancy found between the model and observations based

1e minus 04

1e minus 03

1e minus 02

1e minus 01

1e + 00Em

piric

al C

CDF

1e + 04 1e + 071e + 01Interevent time (hours)

(c)

DirectionOutgoingIncoming

00000

00025

00050

00075

00100

00125

Den

sity

100 2000Number of peers

(d)

Figure 5 +e exploratory analysis of the communication data for students participating in the NetSense study at the University of NotreDame (a) density plot for the number of communication events for the students (b) the distribution of phone activity through the course ofthe day (c) the cumulative distribution function of the interevent times in hours (d) the number of incoming and outgoing communicationpeers for the students in NetSense where there are at least twenty communication events with the peer during the analyzed period [30]

10 Complexity

on interview this could be an early sign or disease emergingor becoming more active Another interesting case would beto see whether the strengths of new links are decaying slowerthan the old ones

In future work we plan to extend the model in thedirections outlined in Appendix including accounting fordistractions during interactions individualized strengthasymmetric of interactions of significance to participants(eg hierarchical relationships) and the impact of forms ofinteractions and of associated emotions Hence the CogSNetmodel represents an important first step towards modelingsocial network dynamics through the prism of humancognition In literature there are three approaches that areused widely to understand human cognition cognitivebandwidth (CB) dual-process morality (DPM) and implicitassociation tests (IAT) In [29] those methods are comparedand their common drawback is pointed outmdashthe lack ofperspective that takes a cultural background into accountOur approach makes such considerations possible and wesee this challenge as one of the future work directions Lastlywe would love to see more datasets that are so extensive asthe NetSense dataset that could allow us to evaluate ourmodel

Appendix

(A) Formal Definition of the CogSNet Model

A social network can be represented by a graph G (V E)where V v1 vn1113864 1113865 denotes the set of n nodes and E

e1 em1113864 1113865 is the set of directed m edges between pairs ofnodes Each edge eij from node vi to node vj ine j is assignedweight wij(t) and represented by a triple (vi vj wij(t))

+e model evolves in discrete steps as follows For eachpair of nodes (vi vj) the system maintains two variables tijwhich represents the time of the most recent event for thispair of nodes and cij which holds the count of eventsprocessed for this pair of nodes Initially both tij and cij areset to 0 as are the weights of all edges ie for all pairs ofnodes (vi vj) wij(0) 0

When an event happens at time t in the modeled socialnetwork it is processed in chronological order by the modelFirst the weight of the corresponding edge is updatedaccording to the following equation

wij(t)

μijcij+1 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μijcij+1 + wij tij1113872 1113873f t minus tij1113872 1113873 1 minus μijcij+11113874 1113875 otherwise

⎧⎪⎪⎨

⎪⎪⎩(A1)

where μijk is the value of reinforcement peak that resultsfrom the kth event that impacts the edge (vi vj) Here thevalue of reinforcement peak μijk depends on the engagementand emotions invoked by the event that is either directly orindirectly related to the edge (vi vj) An example of an eventindirectly related to this edge could be node vi talking aboutnode vj or any situation that reminds node vi about node vj+e values of μ can be individualized to node vi perception ofrelation with node vj at event k +e μrsquos values may also bedependent on event types μijk isin μ1 μ2 μ3 μ4 1113864 1113865 eg μ1

05 for emails μ2 055 for phone calls μ3 08 formeetings and μ4 09 for joint collaboration in projects

Finally the processing of the current event updates bothvariables associated with the updated edge (vi vj) as followstij t cij cij + 1

At any time t of the model evolution the user can obtainthe value of the weight of an arbitrary edge (vi vj) bycomputing the following equation

wij(t) 0 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

wij tij1113872 1113873f t minus tij1113872 1113873 otherwise

⎧⎪⎨

⎪⎩

(A2)

(B) Experimental Parameter Space

To analyze the CogSNet model and compare its results tothose obtained with the reference models many combina-tions of the CogSNet parameters were tested First differentvalues of μ θ and L were specified and then λ coefficientwas calculated using equations (4) and (5) which were alsoused to define CogSNet parameters for experimental veri-fication +e complete list of parameters used is listed inTables 2 and 3

(C) Quality Measures

+e values of Jaccard metric for a single surveyed studentparticipant vi have been computed as follows

Jaccard vi( 1113857 V

CogSNeti capV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868

VCogSNeti cupV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868 (C1)

where Vsurveyi is the set of up to 20 peers enumerated in the

survey by the participating student vi and VCogSNeti is the set

of neighbors of this student in the CogSNet network with the

Complexity 11

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

positive weight at creation and would always stay positiveie all createdmemory traces would never cease to exist+ereinforcement peak μ defines the impact of an event on theweight of the edge relevant to this event +is peak is a globalmodel parameter here In principle the peak can be adjustedaccording to the event or node type to allow for individu-alized event perception

In Figure 1 we compare the CogSNet model with theprevious proposals for representing temporal network dy-namics +e most common approach for representing socialnetwork dynamics is to use interaction sequences [15]Under this method each event is time-stamped and theweights are added to the edges connecting nodes involved inthis event cf Figure 1 ((I) and (IIa)) Moreover a given edgeis active (exists) only at a given time t +is is the mostgranular approach as it is capable of tracking all the eventsoccurring between nodes while preserving the temporalorder of events

In contrast a static binary network representation asshown in Figure 1 (IIb) aggregates all events by making alledges time-independent Consequently an edge existingbetween a pair of nodes corresponds to an event betweenthese nodes occurred at least once in the whole observedperiod [16] Such an edge representation throws away in-formation on the temporal ordering of events making itimpossible to study dynamic processes in static networks

+e incremental network solution accumulates eventsonly up to the current time t of analysis +e classical ap-proach used early in [17ndash19] views a dynamic network as aseries of time-ordered sequences of static graphs (see Fig-ure 1 (IIc)) More recently this method was applied tomodeling network and community evolution [20 21] +edrawback of this approach is that it does not preserve theordering of interactions within time slices Applying asimple frequency-based aggregation creates a frequency-based (FQ) metric cf Figure 1 (IId) Taking into accountonly a given number of the most recent events leads to therecency-based (RC) model cf Figure 1 (IIe) Both of thesemodels are used here as baseline models

Figure 1 (IIf ) shows an example of dynamic socialnetwork generated using the CogSNet model All othersocial network models presented in Figure 1 ((IIa)ndash(IIe)) canalso be represented by CogSNet by setting appropriateparameter combinations to achieve as needed no decayinstant decay and so forth In this way CogSNet can bethought of as a universal generative dynamic model fortemporal social networks encompassing previous ap-proaches as special cases

In general the forgetting function f(Δt) over time in-terval Δt can be of any type (linear power and logarithmic)but here informed by work in the cognitive psychology ofmemory [5] we evaluate only two such functions the ex-ponential function fexp and the power function fpow definedas follows

fexp

(Δt) eminusλΔt

(2)

fpow

(Δt) max(1Δt)minusλ (3)

where λ denotes the forgetting intensity typically λ isin [0 1]+e use of max in the power function ensures that per-ception of events that happened less than a time unit ago isnot changed by forgetting +e time unit in which theforgetting function is expressed scales the values of theparameters Our experiments use one hour as the time unit

To simplify optimal parameters search we aggregate allthree parameters into the trace lifetime L defining the timeafter which an unreinforced memory trace is forgotten ieis too weak to be recalled In the model L is the time overwhich the forgetting function reduces the edge weight fromμ to θ causing the edge to be removed cf Figure 2 For theexponential forgetting function (equation (2)) trace lifetimeLexp is

Lexp

1λln

μθ

1113874 1113875 (4)

while for the power function the formula is

Lpow

μθ

1113874 11138751λ

(5)

Please note that according to the CogSNet model therelationship weights belong to range [θ 1)cup 0 Edgersquosweight value tends to 1 when the node encounters largenumber of tightly successive contacts+e larger the numberof events is and the closer to each other they are the closer to1 the weight is

3 Results and Discussion

In this section we assess the empirical validity of the proposemodel by estimating the accuracy with which the CogSNetmodel reproduces the dynamics of social relations in adynamic social network Our computational experimentsuse the NetSense dataset introduced in [22]

+e data contain 7096844 human mobile phonecommunication events including both calls and text mes-sages +ese are augmented by 578 surveys containing self-reports on top contacts It is worth underlining that thisdataset provides a unique opportunity to evaluate how thecommunication frequencies are recalled by the study par-ticipants because students reported their perception of thefrequencymdashdetailed information on how this informationhas been reported and how data were processed is providedin Appendix

We use this dataset to study the evolution of two coupledsocial networks of university students +e first is a be-havioral network representing interactions between indi-viduals in the form of the records of their mobile calls andtext messages +e second one has perceptual edges definedby the personal nominations +ese nominations are basedon studentsrsquo perception of the corresponding relations asone of the top twenty most interacting peers in the surveysadministered to participants +ese surveys cover the firstfour semesters of the studentrsquos college experience (beginningof freshman year to the end of the sophomore year) Wecompare the list of nominations predicted from the CogSNetnetwork model purely from the communication event datawith the list of nominations collected in a given survey

Complexity 3

A

B

C

D

Time

(b) Staticnetwork

(c) Windows

(d)Incremental

A B

CD

11

3 232

A B

CD

11

2

A B

CD

13 2

2

A B

CD03 06

03

Linkingevents

(a) No contactin interactionsequence at t

A B

CD

A B

DA

B D

C D

02

02

02

02

03

06

03

I Interaction sequence

(f) Cognition-driven social

network(CogSNet)

II Social network at time t

A B

CD2

1

(e) Recent events

t

t0 t1 t2 t3 t4 t5 t6 t7 t8 t9t10 t11

Figure 1 Various approaches tomodeling dynamic networks and edge weighting (relation strengths) for the 4-node network at a given timet (I) and (IIa) interaction sequence (IIb) static (time-aggregated) network (IIc) sliding windows (IId) incremental network all events fromthe beginning time t0 to the current time t are considered and the frequency of interaction in the period constitutes the frequency-based(FQ) reference (IIe) network based on n 3 recent events used for recency-based reference RC (IIf ) cognition-based social networkmodel(CogSNet) introduced here

4 Complexity

Figure 3 shows an example of dynamic social networkgenerated from a subset of NetSense data using the CogSNetmodel

For comparison we also implement three baselinemodels using the mobile phone communication data +efirst is a frequency-based (FQ) model which orders the peersby the number of interactions regardless of their time orderUnder this model the highest ranking peers whose numberequals the number of the nominations listed in the corre-sponding survey are selected on the basis of their com-munication frequency+e recency-based (RC) model selectsthe top interacting peers who for each individual given anumber of recent events Finally the random sampling(RND) model creates the list by randomly selecting peersfrom those who are recorded in the history of interactions ofa given node While random baseline can be considered as asimplistic lower bound the two aforementioned are actuallyregarded to be state of the art since they express two cog-nitive phenomena +e recency effect was uncovered by theexperiments of Hermann Ebbinghaus on recall +e for-getting curve he proposed is based on studies in [23] andrecently confirmed in [24] In these papers the authorsdemonstrate that the learning curve rises the chances forrecalling at the beginning and at the end of the contactperiod while the chances to recall events are lower in themiddle of it However as the question in surveys mainlyrefers to the present we focused on the recency instead ofprimacy +e second baseline frequency-based is animplementation of scientific results taken from the domainof lexical memory [25] +e authors of this study show thatthe frequently repeated words are being recalled fasterUnfortunately to the best of our knowledge there are noother approaches suitable to be used in similar tasks

To compare the performance of all the models we use aJaccardmetric equation (C1) in Appendix whichmeasuresthe ratio of the number of nominations produced by themodel that are also ground truth nominations listed in thecorresponding survey divided by the number of uniquenominations on both lists Due to the nature of the questionformulation in the survey it was impossible to use anyapproach based on ordering such as Kendallrsquos or Pearsonrsquosranks since we had to consider the list as unordered

Figure 4 shows the results of this comparison over therange of parameters corresponding to reported values forforgetting of one to 14 weeks As reported in [26] the abilityto recall information about social interactions starts todegrade after about one week +e experiments usingNetSense dataset reveal that the performance is the highestwhen the forgetting of unreinforced memory traces happensafter two weeks +e results remain satisfactory for forget-ting thresholds as low as one week With the two-weekthreshold the CogSNet model is with either power or ex-ponential forgetting statistically significantly better than anybaseline model In the literature the power forgetting wasfound to perform significantly better than the exponentialforgetting when a range of parameters was limited [3] Wedo not observe such superior performance of power for-getting here Both functions have similar peak of Jaccardmetric albeit for the different lifetime values

When comparing the results of surveys with the states ofthe CogSNet network at the times of the surveys the Jaccardmetric is as high as 2998 for exponential function for L 3days +e distant second is recency-based RC model whichdelivers much lower Jaccard metric of 178 Neverthelesswhen looking at relative simplicity of the RC metric one canalso say that it performs quite goodmdashthis is due to the fact

L

θ

μ

Forgetting modelExponentialPower

00

02

04

06

Wei

ght o

f rel

atio

nshi

p10 20 30 400

Time (days)

Figure 2 Dynamics of relations in CogSNet network with exponential and power functions and with parameters set to μ 04 θ 01 andL 10 days

Complexity 5

CogSNet relationship strengths

Oct Nov Dec Jan Feb

Oct Nov Dec Jan Feb

Oct Nov Dec Jan Feb

t1 t2 t3 t4 ts

000

025

050

075

100A

minusB

000

025

050

075

100

AminusC

000

025

050

075

100

BminusC

000

025

050

075

100

AminusD

000

025

050

075

100

BminusD

Nov Dec Jan FebOct

FebDec JanNovOctTime

(a)

t1 t2 t3 t4 ts

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

(b)

Figure 3 Four-node CogSNet network for the sample of real NetSense data μ 04 θ 01 and L 10 days nodes A B C and Dcorrespond respectively to participants with ids 40997 11360 10841 and 1232 (a) Relation strengths according to the CogSNet model over4-month period (one term) (b) network snapshots at four timestamps t1ndasht4 and at the survey time ts

6 Complexity

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 201 100 20050Lifetime L in days

μ = 03 θ = 02μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

(a)

μ = 03 θ = 02 μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 20 50 100 2001Life time L in days

(b)

Figure 4 +e plot of the Jaccard metric as a function of lifetime L +e metric measures the overlap between the two sets of peers oneidentified by CogSNet and the other listed by students in the survey +ese plots are compared to the results achieved by the three baselinemodels recency-based with the best results obtained with the number of recent events set to 400 frequency-based and random+e resultsare plotted for the CogSNet running with various parameters for (a) exponential and (b) power forgetting functions

Complexity 7

Table 1 Numerical and symbolic p values adjusted by SimesndashHochberg step-up method of nonparametric pairwise post hoc Nemenyi testcomparing values of Jaccard metric for results obtained by each method with parameters fitted using recorded event data with the groundtruth based on surveys taken over semesters 1ndash4

Power Exponential Recency Frequency RandomPower mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast lowastlowastlowast

Exponential 00000 mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast

Recency 00000 00000 mdash lowastlowastlowast lowastlowastlowast

Frequency 00000 00000 00000 mdash lowastlowastlowast

Random 00000 00000 00000 00000 mdash+ese results show how each method in the leftmost column performs versus the method in the first row +e part of table under the diagonal showsnumerical values of the test while the part above the diagonal presents p values coded as follows lowastlowastlowastplt 000005 represents high confidence in theperformance differences between the methods since typically values of pgt 005 are considered indicative of nonsignificance

Table 2 Setting of parameters for power forgetting function in computational experiments

λ dependence on μ θ and L for power forgettingLifetime L days μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 043621 030863 0654312 03581 025337 0537163 032415 022934 0486234 030372 021489 0455585 028957 020487 0434356 027894 019736 0418417 027055 019142 0405838 026368 018656 0395529 02579 018247 03868510 025294 017896 03794211 024862 01759 03729312 02448 01732 0367213 024139 017079 03620814 023831 016861 03574715 023552 016663 03532816 023297 016483 03494517 023062 016317 03459218 022844 016163 03426719 022643 01602 03396420 022455 015887 03368221 022278 015762 03341822 022113 015645 0331723 021957 015535 03293624 02181 015431 03271625 021671 015333 03250726 021539 015239 03230927 021414 015151 0321228 021294 015066 03194129 02118 014985 0317730 021071 014908 03160640 020188 014283 03028250 019553 013834 02932960 019062 013487 02859480 018337 012974 027506100 017811 012602 026717

8 Complexity

Table 3 Setting of parameters for exponential forgetting function in computational experiments

λ dependence on μ θ and L for exponential forgettingLifetime L (days) μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 005776 004087 0086642 002888 002043 0043323 001925 001362 0028884 001444 001022 0021665 001155 000817 0017336 000963 000681 0014447 000825 000584 0012388 000722 000511 0010839 000642 000454 00096310 000578 000409 00086611 000525 000372 00078812 000481 000341 00072213 000444 000314 00066614 000413 000292 00061915 000385 000272 00057816 000361 000255 00054217 00034 00024 0005118 000321 000227 00048119 000304 000215 00045620 000289 000204 00043321 000275 000195 00041322 000263 000186 00039423 000251 000178 00037724 000241 00017 00036125 000231 000163 00034726 000222 000157 00033327 000214 000151 00032128 000206 000146 00030929 000199 000141 00029930 000193 000136 00028940 000144 000102 00021750 000116 000082 00017360 000096 000068 00014480 000072 000051 000108100 000058 000041 000087

000000

000002

000004

Den

sity

20000 40000 600000Number of communication events

(a)

0000000

0000005

0000010

0000015

0000020

Den

sity

100000 200000000000Time of day

(b)

Figure 5 Continued

Complexity 9

that primacy and recency effects come into play when listingpeers [27]

4 Conclusions

+e results demonstrate that accounting for event percep-tion and memory dynamics is essential for faithfullymodeling the interaction dynamics in social networks +eCogSNet performed statistically significantly better than anyof the base temporal models (see Table 1 in Appendix) Withsuitable data the model can be extended to represent in-dividual sensitivity to events and the impact of emotions anddistractions on event perception to further improve mod-eling quality

+e dataset on which we operate contains data limited toa relatively small fraction of human social activities Henceit cannot be stated that there were no other events happeningthat could have affected participant memory about othersExamples of such events are face to face interactions indirectreferences to someone else and personal reminiscencesabout others In addition emotions can also play a signif-icant role in these processes and they can be potentiallyidentified with some confidence from available data [28]

As a result the reinforcement peak value can be adjustedfor a given person and for an individual event cf alsoequation (A1) in Appendix What is observed here is apartial manifestation from human memory However eventaking into account that the model was built based on a

single data source over 6 million telephone calls andmessages among the NetSense study participants the modelfitted with the recorded event data yields results wellmatching the salience of social contacts over all 578 surveyscompleted by 184 participants It is observed that even whenusing the same values of model parameters for all partici-pants the results are satisfactory Nevertheless this accuracymost likely could have been increased if the parameters hadbeen individually adjusted for each participant Apart fromthat one should remember that the recall task is highlycontextual so the individuals will provide different sets ofpeers depending on the situation [11 12]

+e findings of this work can be also considered as animportant building block on how to maintain healthy re-lationships or how to discover anomalies For instancebased on the results it is anticipated that in order to keep aliving relationship with a peer the contacts should not besparser than lifetime L preferably even more frequent Onthe contrary these findings also demonstrate on how long agrieving period after breaking the relationship could po-tentially last and how to handle the recovery process af-terwards in order to make it effective Lastly smallmodifications making the model closer to ACT-R wouldenable its use for instance in the case of Alzheimer diseasewhere one could monitor whether the peers recalled bypatients are in line with what a CogSNet model generatesbased on the contacts or visits If there will be any dis-crepancy found between the model and observations based

1e minus 04

1e minus 03

1e minus 02

1e minus 01

1e + 00Em

piric

al C

CDF

1e + 04 1e + 071e + 01Interevent time (hours)

(c)

DirectionOutgoingIncoming

00000

00025

00050

00075

00100

00125

Den

sity

100 2000Number of peers

(d)

Figure 5 +e exploratory analysis of the communication data for students participating in the NetSense study at the University of NotreDame (a) density plot for the number of communication events for the students (b) the distribution of phone activity through the course ofthe day (c) the cumulative distribution function of the interevent times in hours (d) the number of incoming and outgoing communicationpeers for the students in NetSense where there are at least twenty communication events with the peer during the analyzed period [30]

10 Complexity

on interview this could be an early sign or disease emergingor becoming more active Another interesting case would beto see whether the strengths of new links are decaying slowerthan the old ones

In future work we plan to extend the model in thedirections outlined in Appendix including accounting fordistractions during interactions individualized strengthasymmetric of interactions of significance to participants(eg hierarchical relationships) and the impact of forms ofinteractions and of associated emotions Hence the CogSNetmodel represents an important first step towards modelingsocial network dynamics through the prism of humancognition In literature there are three approaches that areused widely to understand human cognition cognitivebandwidth (CB) dual-process morality (DPM) and implicitassociation tests (IAT) In [29] those methods are comparedand their common drawback is pointed outmdashthe lack ofperspective that takes a cultural background into accountOur approach makes such considerations possible and wesee this challenge as one of the future work directions Lastlywe would love to see more datasets that are so extensive asthe NetSense dataset that could allow us to evaluate ourmodel

Appendix

(A) Formal Definition of the CogSNet Model

A social network can be represented by a graph G (V E)where V v1 vn1113864 1113865 denotes the set of n nodes and E

e1 em1113864 1113865 is the set of directed m edges between pairs ofnodes Each edge eij from node vi to node vj ine j is assignedweight wij(t) and represented by a triple (vi vj wij(t))

+e model evolves in discrete steps as follows For eachpair of nodes (vi vj) the system maintains two variables tijwhich represents the time of the most recent event for thispair of nodes and cij which holds the count of eventsprocessed for this pair of nodes Initially both tij and cij areset to 0 as are the weights of all edges ie for all pairs ofnodes (vi vj) wij(0) 0

When an event happens at time t in the modeled socialnetwork it is processed in chronological order by the modelFirst the weight of the corresponding edge is updatedaccording to the following equation

wij(t)

μijcij+1 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μijcij+1 + wij tij1113872 1113873f t minus tij1113872 1113873 1 minus μijcij+11113874 1113875 otherwise

⎧⎪⎪⎨

⎪⎪⎩(A1)

where μijk is the value of reinforcement peak that resultsfrom the kth event that impacts the edge (vi vj) Here thevalue of reinforcement peak μijk depends on the engagementand emotions invoked by the event that is either directly orindirectly related to the edge (vi vj) An example of an eventindirectly related to this edge could be node vi talking aboutnode vj or any situation that reminds node vi about node vj+e values of μ can be individualized to node vi perception ofrelation with node vj at event k +e μrsquos values may also bedependent on event types μijk isin μ1 μ2 μ3 μ4 1113864 1113865 eg μ1

05 for emails μ2 055 for phone calls μ3 08 formeetings and μ4 09 for joint collaboration in projects

Finally the processing of the current event updates bothvariables associated with the updated edge (vi vj) as followstij t cij cij + 1

At any time t of the model evolution the user can obtainthe value of the weight of an arbitrary edge (vi vj) bycomputing the following equation

wij(t) 0 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

wij tij1113872 1113873f t minus tij1113872 1113873 otherwise

⎧⎪⎨

⎪⎩

(A2)

(B) Experimental Parameter Space

To analyze the CogSNet model and compare its results tothose obtained with the reference models many combina-tions of the CogSNet parameters were tested First differentvalues of μ θ and L were specified and then λ coefficientwas calculated using equations (4) and (5) which were alsoused to define CogSNet parameters for experimental veri-fication +e complete list of parameters used is listed inTables 2 and 3

(C) Quality Measures

+e values of Jaccard metric for a single surveyed studentparticipant vi have been computed as follows

Jaccard vi( 1113857 V

CogSNeti capV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868

VCogSNeti cupV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868 (C1)

where Vsurveyi is the set of up to 20 peers enumerated in the

survey by the participating student vi and VCogSNeti is the set

of neighbors of this student in the CogSNet network with the

Complexity 11

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

A

B

C

D

Time

(b) Staticnetwork

(c) Windows

(d)Incremental

A B

CD

11

3 232

A B

CD

11

2

A B

CD

13 2

2

A B

CD03 06

03

Linkingevents

(a) No contactin interactionsequence at t

A B

CD

A B

DA

B D

C D

02

02

02

02

03

06

03

I Interaction sequence

(f) Cognition-driven social

network(CogSNet)

II Social network at time t

A B

CD2

1

(e) Recent events

t

t0 t1 t2 t3 t4 t5 t6 t7 t8 t9t10 t11

Figure 1 Various approaches tomodeling dynamic networks and edge weighting (relation strengths) for the 4-node network at a given timet (I) and (IIa) interaction sequence (IIb) static (time-aggregated) network (IIc) sliding windows (IId) incremental network all events fromthe beginning time t0 to the current time t are considered and the frequency of interaction in the period constitutes the frequency-based(FQ) reference (IIe) network based on n 3 recent events used for recency-based reference RC (IIf ) cognition-based social networkmodel(CogSNet) introduced here

4 Complexity

Figure 3 shows an example of dynamic social networkgenerated from a subset of NetSense data using the CogSNetmodel

For comparison we also implement three baselinemodels using the mobile phone communication data +efirst is a frequency-based (FQ) model which orders the peersby the number of interactions regardless of their time orderUnder this model the highest ranking peers whose numberequals the number of the nominations listed in the corre-sponding survey are selected on the basis of their com-munication frequency+e recency-based (RC) model selectsthe top interacting peers who for each individual given anumber of recent events Finally the random sampling(RND) model creates the list by randomly selecting peersfrom those who are recorded in the history of interactions ofa given node While random baseline can be considered as asimplistic lower bound the two aforementioned are actuallyregarded to be state of the art since they express two cog-nitive phenomena +e recency effect was uncovered by theexperiments of Hermann Ebbinghaus on recall +e for-getting curve he proposed is based on studies in [23] andrecently confirmed in [24] In these papers the authorsdemonstrate that the learning curve rises the chances forrecalling at the beginning and at the end of the contactperiod while the chances to recall events are lower in themiddle of it However as the question in surveys mainlyrefers to the present we focused on the recency instead ofprimacy +e second baseline frequency-based is animplementation of scientific results taken from the domainof lexical memory [25] +e authors of this study show thatthe frequently repeated words are being recalled fasterUnfortunately to the best of our knowledge there are noother approaches suitable to be used in similar tasks

To compare the performance of all the models we use aJaccardmetric equation (C1) in Appendix whichmeasuresthe ratio of the number of nominations produced by themodel that are also ground truth nominations listed in thecorresponding survey divided by the number of uniquenominations on both lists Due to the nature of the questionformulation in the survey it was impossible to use anyapproach based on ordering such as Kendallrsquos or Pearsonrsquosranks since we had to consider the list as unordered

Figure 4 shows the results of this comparison over therange of parameters corresponding to reported values forforgetting of one to 14 weeks As reported in [26] the abilityto recall information about social interactions starts todegrade after about one week +e experiments usingNetSense dataset reveal that the performance is the highestwhen the forgetting of unreinforced memory traces happensafter two weeks +e results remain satisfactory for forget-ting thresholds as low as one week With the two-weekthreshold the CogSNet model is with either power or ex-ponential forgetting statistically significantly better than anybaseline model In the literature the power forgetting wasfound to perform significantly better than the exponentialforgetting when a range of parameters was limited [3] Wedo not observe such superior performance of power for-getting here Both functions have similar peak of Jaccardmetric albeit for the different lifetime values

When comparing the results of surveys with the states ofthe CogSNet network at the times of the surveys the Jaccardmetric is as high as 2998 for exponential function for L 3days +e distant second is recency-based RC model whichdelivers much lower Jaccard metric of 178 Neverthelesswhen looking at relative simplicity of the RC metric one canalso say that it performs quite goodmdashthis is due to the fact

L

θ

μ

Forgetting modelExponentialPower

00

02

04

06

Wei

ght o

f rel

atio

nshi

p10 20 30 400

Time (days)

Figure 2 Dynamics of relations in CogSNet network with exponential and power functions and with parameters set to μ 04 θ 01 andL 10 days

Complexity 5

CogSNet relationship strengths

Oct Nov Dec Jan Feb

Oct Nov Dec Jan Feb

Oct Nov Dec Jan Feb

t1 t2 t3 t4 ts

000

025

050

075

100A

minusB

000

025

050

075

100

AminusC

000

025

050

075

100

BminusC

000

025

050

075

100

AminusD

000

025

050

075

100

BminusD

Nov Dec Jan FebOct

FebDec JanNovOctTime

(a)

t1 t2 t3 t4 ts

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

(b)

Figure 3 Four-node CogSNet network for the sample of real NetSense data μ 04 θ 01 and L 10 days nodes A B C and Dcorrespond respectively to participants with ids 40997 11360 10841 and 1232 (a) Relation strengths according to the CogSNet model over4-month period (one term) (b) network snapshots at four timestamps t1ndasht4 and at the survey time ts

6 Complexity

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 201 100 20050Lifetime L in days

μ = 03 θ = 02μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

(a)

μ = 03 θ = 02 μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 20 50 100 2001Life time L in days

(b)

Figure 4 +e plot of the Jaccard metric as a function of lifetime L +e metric measures the overlap between the two sets of peers oneidentified by CogSNet and the other listed by students in the survey +ese plots are compared to the results achieved by the three baselinemodels recency-based with the best results obtained with the number of recent events set to 400 frequency-based and random+e resultsare plotted for the CogSNet running with various parameters for (a) exponential and (b) power forgetting functions

Complexity 7

Table 1 Numerical and symbolic p values adjusted by SimesndashHochberg step-up method of nonparametric pairwise post hoc Nemenyi testcomparing values of Jaccard metric for results obtained by each method with parameters fitted using recorded event data with the groundtruth based on surveys taken over semesters 1ndash4

Power Exponential Recency Frequency RandomPower mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast lowastlowastlowast

Exponential 00000 mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast

Recency 00000 00000 mdash lowastlowastlowast lowastlowastlowast

Frequency 00000 00000 00000 mdash lowastlowastlowast

Random 00000 00000 00000 00000 mdash+ese results show how each method in the leftmost column performs versus the method in the first row +e part of table under the diagonal showsnumerical values of the test while the part above the diagonal presents p values coded as follows lowastlowastlowastplt 000005 represents high confidence in theperformance differences between the methods since typically values of pgt 005 are considered indicative of nonsignificance

Table 2 Setting of parameters for power forgetting function in computational experiments

λ dependence on μ θ and L for power forgettingLifetime L days μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 043621 030863 0654312 03581 025337 0537163 032415 022934 0486234 030372 021489 0455585 028957 020487 0434356 027894 019736 0418417 027055 019142 0405838 026368 018656 0395529 02579 018247 03868510 025294 017896 03794211 024862 01759 03729312 02448 01732 0367213 024139 017079 03620814 023831 016861 03574715 023552 016663 03532816 023297 016483 03494517 023062 016317 03459218 022844 016163 03426719 022643 01602 03396420 022455 015887 03368221 022278 015762 03341822 022113 015645 0331723 021957 015535 03293624 02181 015431 03271625 021671 015333 03250726 021539 015239 03230927 021414 015151 0321228 021294 015066 03194129 02118 014985 0317730 021071 014908 03160640 020188 014283 03028250 019553 013834 02932960 019062 013487 02859480 018337 012974 027506100 017811 012602 026717

8 Complexity

Table 3 Setting of parameters for exponential forgetting function in computational experiments

λ dependence on μ θ and L for exponential forgettingLifetime L (days) μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 005776 004087 0086642 002888 002043 0043323 001925 001362 0028884 001444 001022 0021665 001155 000817 0017336 000963 000681 0014447 000825 000584 0012388 000722 000511 0010839 000642 000454 00096310 000578 000409 00086611 000525 000372 00078812 000481 000341 00072213 000444 000314 00066614 000413 000292 00061915 000385 000272 00057816 000361 000255 00054217 00034 00024 0005118 000321 000227 00048119 000304 000215 00045620 000289 000204 00043321 000275 000195 00041322 000263 000186 00039423 000251 000178 00037724 000241 00017 00036125 000231 000163 00034726 000222 000157 00033327 000214 000151 00032128 000206 000146 00030929 000199 000141 00029930 000193 000136 00028940 000144 000102 00021750 000116 000082 00017360 000096 000068 00014480 000072 000051 000108100 000058 000041 000087

000000

000002

000004

Den

sity

20000 40000 600000Number of communication events

(a)

0000000

0000005

0000010

0000015

0000020

Den

sity

100000 200000000000Time of day

(b)

Figure 5 Continued

Complexity 9

that primacy and recency effects come into play when listingpeers [27]

4 Conclusions

+e results demonstrate that accounting for event percep-tion and memory dynamics is essential for faithfullymodeling the interaction dynamics in social networks +eCogSNet performed statistically significantly better than anyof the base temporal models (see Table 1 in Appendix) Withsuitable data the model can be extended to represent in-dividual sensitivity to events and the impact of emotions anddistractions on event perception to further improve mod-eling quality

+e dataset on which we operate contains data limited toa relatively small fraction of human social activities Henceit cannot be stated that there were no other events happeningthat could have affected participant memory about othersExamples of such events are face to face interactions indirectreferences to someone else and personal reminiscencesabout others In addition emotions can also play a signif-icant role in these processes and they can be potentiallyidentified with some confidence from available data [28]

As a result the reinforcement peak value can be adjustedfor a given person and for an individual event cf alsoequation (A1) in Appendix What is observed here is apartial manifestation from human memory However eventaking into account that the model was built based on a

single data source over 6 million telephone calls andmessages among the NetSense study participants the modelfitted with the recorded event data yields results wellmatching the salience of social contacts over all 578 surveyscompleted by 184 participants It is observed that even whenusing the same values of model parameters for all partici-pants the results are satisfactory Nevertheless this accuracymost likely could have been increased if the parameters hadbeen individually adjusted for each participant Apart fromthat one should remember that the recall task is highlycontextual so the individuals will provide different sets ofpeers depending on the situation [11 12]

+e findings of this work can be also considered as animportant building block on how to maintain healthy re-lationships or how to discover anomalies For instancebased on the results it is anticipated that in order to keep aliving relationship with a peer the contacts should not besparser than lifetime L preferably even more frequent Onthe contrary these findings also demonstrate on how long agrieving period after breaking the relationship could po-tentially last and how to handle the recovery process af-terwards in order to make it effective Lastly smallmodifications making the model closer to ACT-R wouldenable its use for instance in the case of Alzheimer diseasewhere one could monitor whether the peers recalled bypatients are in line with what a CogSNet model generatesbased on the contacts or visits If there will be any dis-crepancy found between the model and observations based

1e minus 04

1e minus 03

1e minus 02

1e minus 01

1e + 00Em

piric

al C

CDF

1e + 04 1e + 071e + 01Interevent time (hours)

(c)

DirectionOutgoingIncoming

00000

00025

00050

00075

00100

00125

Den

sity

100 2000Number of peers

(d)

Figure 5 +e exploratory analysis of the communication data for students participating in the NetSense study at the University of NotreDame (a) density plot for the number of communication events for the students (b) the distribution of phone activity through the course ofthe day (c) the cumulative distribution function of the interevent times in hours (d) the number of incoming and outgoing communicationpeers for the students in NetSense where there are at least twenty communication events with the peer during the analyzed period [30]

10 Complexity

on interview this could be an early sign or disease emergingor becoming more active Another interesting case would beto see whether the strengths of new links are decaying slowerthan the old ones

In future work we plan to extend the model in thedirections outlined in Appendix including accounting fordistractions during interactions individualized strengthasymmetric of interactions of significance to participants(eg hierarchical relationships) and the impact of forms ofinteractions and of associated emotions Hence the CogSNetmodel represents an important first step towards modelingsocial network dynamics through the prism of humancognition In literature there are three approaches that areused widely to understand human cognition cognitivebandwidth (CB) dual-process morality (DPM) and implicitassociation tests (IAT) In [29] those methods are comparedand their common drawback is pointed outmdashthe lack ofperspective that takes a cultural background into accountOur approach makes such considerations possible and wesee this challenge as one of the future work directions Lastlywe would love to see more datasets that are so extensive asthe NetSense dataset that could allow us to evaluate ourmodel

Appendix

(A) Formal Definition of the CogSNet Model

A social network can be represented by a graph G (V E)where V v1 vn1113864 1113865 denotes the set of n nodes and E

e1 em1113864 1113865 is the set of directed m edges between pairs ofnodes Each edge eij from node vi to node vj ine j is assignedweight wij(t) and represented by a triple (vi vj wij(t))

+e model evolves in discrete steps as follows For eachpair of nodes (vi vj) the system maintains two variables tijwhich represents the time of the most recent event for thispair of nodes and cij which holds the count of eventsprocessed for this pair of nodes Initially both tij and cij areset to 0 as are the weights of all edges ie for all pairs ofnodes (vi vj) wij(0) 0

When an event happens at time t in the modeled socialnetwork it is processed in chronological order by the modelFirst the weight of the corresponding edge is updatedaccording to the following equation

wij(t)

μijcij+1 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μijcij+1 + wij tij1113872 1113873f t minus tij1113872 1113873 1 minus μijcij+11113874 1113875 otherwise

⎧⎪⎪⎨

⎪⎪⎩(A1)

where μijk is the value of reinforcement peak that resultsfrom the kth event that impacts the edge (vi vj) Here thevalue of reinforcement peak μijk depends on the engagementand emotions invoked by the event that is either directly orindirectly related to the edge (vi vj) An example of an eventindirectly related to this edge could be node vi talking aboutnode vj or any situation that reminds node vi about node vj+e values of μ can be individualized to node vi perception ofrelation with node vj at event k +e μrsquos values may also bedependent on event types μijk isin μ1 μ2 μ3 μ4 1113864 1113865 eg μ1

05 for emails μ2 055 for phone calls μ3 08 formeetings and μ4 09 for joint collaboration in projects

Finally the processing of the current event updates bothvariables associated with the updated edge (vi vj) as followstij t cij cij + 1

At any time t of the model evolution the user can obtainthe value of the weight of an arbitrary edge (vi vj) bycomputing the following equation

wij(t) 0 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

wij tij1113872 1113873f t minus tij1113872 1113873 otherwise

⎧⎪⎨

⎪⎩

(A2)

(B) Experimental Parameter Space

To analyze the CogSNet model and compare its results tothose obtained with the reference models many combina-tions of the CogSNet parameters were tested First differentvalues of μ θ and L were specified and then λ coefficientwas calculated using equations (4) and (5) which were alsoused to define CogSNet parameters for experimental veri-fication +e complete list of parameters used is listed inTables 2 and 3

(C) Quality Measures

+e values of Jaccard metric for a single surveyed studentparticipant vi have been computed as follows

Jaccard vi( 1113857 V

CogSNeti capV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868

VCogSNeti cupV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868 (C1)

where Vsurveyi is the set of up to 20 peers enumerated in the

survey by the participating student vi and VCogSNeti is the set

of neighbors of this student in the CogSNet network with the

Complexity 11

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

Figure 3 shows an example of dynamic social networkgenerated from a subset of NetSense data using the CogSNetmodel

For comparison we also implement three baselinemodels using the mobile phone communication data +efirst is a frequency-based (FQ) model which orders the peersby the number of interactions regardless of their time orderUnder this model the highest ranking peers whose numberequals the number of the nominations listed in the corre-sponding survey are selected on the basis of their com-munication frequency+e recency-based (RC) model selectsthe top interacting peers who for each individual given anumber of recent events Finally the random sampling(RND) model creates the list by randomly selecting peersfrom those who are recorded in the history of interactions ofa given node While random baseline can be considered as asimplistic lower bound the two aforementioned are actuallyregarded to be state of the art since they express two cog-nitive phenomena +e recency effect was uncovered by theexperiments of Hermann Ebbinghaus on recall +e for-getting curve he proposed is based on studies in [23] andrecently confirmed in [24] In these papers the authorsdemonstrate that the learning curve rises the chances forrecalling at the beginning and at the end of the contactperiod while the chances to recall events are lower in themiddle of it However as the question in surveys mainlyrefers to the present we focused on the recency instead ofprimacy +e second baseline frequency-based is animplementation of scientific results taken from the domainof lexical memory [25] +e authors of this study show thatthe frequently repeated words are being recalled fasterUnfortunately to the best of our knowledge there are noother approaches suitable to be used in similar tasks

To compare the performance of all the models we use aJaccardmetric equation (C1) in Appendix whichmeasuresthe ratio of the number of nominations produced by themodel that are also ground truth nominations listed in thecorresponding survey divided by the number of uniquenominations on both lists Due to the nature of the questionformulation in the survey it was impossible to use anyapproach based on ordering such as Kendallrsquos or Pearsonrsquosranks since we had to consider the list as unordered

Figure 4 shows the results of this comparison over therange of parameters corresponding to reported values forforgetting of one to 14 weeks As reported in [26] the abilityto recall information about social interactions starts todegrade after about one week +e experiments usingNetSense dataset reveal that the performance is the highestwhen the forgetting of unreinforced memory traces happensafter two weeks +e results remain satisfactory for forget-ting thresholds as low as one week With the two-weekthreshold the CogSNet model is with either power or ex-ponential forgetting statistically significantly better than anybaseline model In the literature the power forgetting wasfound to perform significantly better than the exponentialforgetting when a range of parameters was limited [3] Wedo not observe such superior performance of power for-getting here Both functions have similar peak of Jaccardmetric albeit for the different lifetime values

When comparing the results of surveys with the states ofthe CogSNet network at the times of the surveys the Jaccardmetric is as high as 2998 for exponential function for L 3days +e distant second is recency-based RC model whichdelivers much lower Jaccard metric of 178 Neverthelesswhen looking at relative simplicity of the RC metric one canalso say that it performs quite goodmdashthis is due to the fact

L

θ

μ

Forgetting modelExponentialPower

00

02

04

06

Wei

ght o

f rel

atio

nshi

p10 20 30 400

Time (days)

Figure 2 Dynamics of relations in CogSNet network with exponential and power functions and with parameters set to μ 04 θ 01 andL 10 days

Complexity 5

CogSNet relationship strengths

Oct Nov Dec Jan Feb

Oct Nov Dec Jan Feb

Oct Nov Dec Jan Feb

t1 t2 t3 t4 ts

000

025

050

075

100A

minusB

000

025

050

075

100

AminusC

000

025

050

075

100

BminusC

000

025

050

075

100

AminusD

000

025

050

075

100

BminusD

Nov Dec Jan FebOct

FebDec JanNovOctTime

(a)

t1 t2 t3 t4 ts

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

(b)

Figure 3 Four-node CogSNet network for the sample of real NetSense data μ 04 θ 01 and L 10 days nodes A B C and Dcorrespond respectively to participants with ids 40997 11360 10841 and 1232 (a) Relation strengths according to the CogSNet model over4-month period (one term) (b) network snapshots at four timestamps t1ndasht4 and at the survey time ts

6 Complexity

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 201 100 20050Lifetime L in days

μ = 03 θ = 02μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

(a)

μ = 03 θ = 02 μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 20 50 100 2001Life time L in days

(b)

Figure 4 +e plot of the Jaccard metric as a function of lifetime L +e metric measures the overlap between the two sets of peers oneidentified by CogSNet and the other listed by students in the survey +ese plots are compared to the results achieved by the three baselinemodels recency-based with the best results obtained with the number of recent events set to 400 frequency-based and random+e resultsare plotted for the CogSNet running with various parameters for (a) exponential and (b) power forgetting functions

Complexity 7

Table 1 Numerical and symbolic p values adjusted by SimesndashHochberg step-up method of nonparametric pairwise post hoc Nemenyi testcomparing values of Jaccard metric for results obtained by each method with parameters fitted using recorded event data with the groundtruth based on surveys taken over semesters 1ndash4

Power Exponential Recency Frequency RandomPower mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast lowastlowastlowast

Exponential 00000 mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast

Recency 00000 00000 mdash lowastlowastlowast lowastlowastlowast

Frequency 00000 00000 00000 mdash lowastlowastlowast

Random 00000 00000 00000 00000 mdash+ese results show how each method in the leftmost column performs versus the method in the first row +e part of table under the diagonal showsnumerical values of the test while the part above the diagonal presents p values coded as follows lowastlowastlowastplt 000005 represents high confidence in theperformance differences between the methods since typically values of pgt 005 are considered indicative of nonsignificance

Table 2 Setting of parameters for power forgetting function in computational experiments

λ dependence on μ θ and L for power forgettingLifetime L days μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 043621 030863 0654312 03581 025337 0537163 032415 022934 0486234 030372 021489 0455585 028957 020487 0434356 027894 019736 0418417 027055 019142 0405838 026368 018656 0395529 02579 018247 03868510 025294 017896 03794211 024862 01759 03729312 02448 01732 0367213 024139 017079 03620814 023831 016861 03574715 023552 016663 03532816 023297 016483 03494517 023062 016317 03459218 022844 016163 03426719 022643 01602 03396420 022455 015887 03368221 022278 015762 03341822 022113 015645 0331723 021957 015535 03293624 02181 015431 03271625 021671 015333 03250726 021539 015239 03230927 021414 015151 0321228 021294 015066 03194129 02118 014985 0317730 021071 014908 03160640 020188 014283 03028250 019553 013834 02932960 019062 013487 02859480 018337 012974 027506100 017811 012602 026717

8 Complexity

Table 3 Setting of parameters for exponential forgetting function in computational experiments

λ dependence on μ θ and L for exponential forgettingLifetime L (days) μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 005776 004087 0086642 002888 002043 0043323 001925 001362 0028884 001444 001022 0021665 001155 000817 0017336 000963 000681 0014447 000825 000584 0012388 000722 000511 0010839 000642 000454 00096310 000578 000409 00086611 000525 000372 00078812 000481 000341 00072213 000444 000314 00066614 000413 000292 00061915 000385 000272 00057816 000361 000255 00054217 00034 00024 0005118 000321 000227 00048119 000304 000215 00045620 000289 000204 00043321 000275 000195 00041322 000263 000186 00039423 000251 000178 00037724 000241 00017 00036125 000231 000163 00034726 000222 000157 00033327 000214 000151 00032128 000206 000146 00030929 000199 000141 00029930 000193 000136 00028940 000144 000102 00021750 000116 000082 00017360 000096 000068 00014480 000072 000051 000108100 000058 000041 000087

000000

000002

000004

Den

sity

20000 40000 600000Number of communication events

(a)

0000000

0000005

0000010

0000015

0000020

Den

sity

100000 200000000000Time of day

(b)

Figure 5 Continued

Complexity 9

that primacy and recency effects come into play when listingpeers [27]

4 Conclusions

+e results demonstrate that accounting for event percep-tion and memory dynamics is essential for faithfullymodeling the interaction dynamics in social networks +eCogSNet performed statistically significantly better than anyof the base temporal models (see Table 1 in Appendix) Withsuitable data the model can be extended to represent in-dividual sensitivity to events and the impact of emotions anddistractions on event perception to further improve mod-eling quality

+e dataset on which we operate contains data limited toa relatively small fraction of human social activities Henceit cannot be stated that there were no other events happeningthat could have affected participant memory about othersExamples of such events are face to face interactions indirectreferences to someone else and personal reminiscencesabout others In addition emotions can also play a signif-icant role in these processes and they can be potentiallyidentified with some confidence from available data [28]

As a result the reinforcement peak value can be adjustedfor a given person and for an individual event cf alsoequation (A1) in Appendix What is observed here is apartial manifestation from human memory However eventaking into account that the model was built based on a

single data source over 6 million telephone calls andmessages among the NetSense study participants the modelfitted with the recorded event data yields results wellmatching the salience of social contacts over all 578 surveyscompleted by 184 participants It is observed that even whenusing the same values of model parameters for all partici-pants the results are satisfactory Nevertheless this accuracymost likely could have been increased if the parameters hadbeen individually adjusted for each participant Apart fromthat one should remember that the recall task is highlycontextual so the individuals will provide different sets ofpeers depending on the situation [11 12]

+e findings of this work can be also considered as animportant building block on how to maintain healthy re-lationships or how to discover anomalies For instancebased on the results it is anticipated that in order to keep aliving relationship with a peer the contacts should not besparser than lifetime L preferably even more frequent Onthe contrary these findings also demonstrate on how long agrieving period after breaking the relationship could po-tentially last and how to handle the recovery process af-terwards in order to make it effective Lastly smallmodifications making the model closer to ACT-R wouldenable its use for instance in the case of Alzheimer diseasewhere one could monitor whether the peers recalled bypatients are in line with what a CogSNet model generatesbased on the contacts or visits If there will be any dis-crepancy found between the model and observations based

1e minus 04

1e minus 03

1e minus 02

1e minus 01

1e + 00Em

piric

al C

CDF

1e + 04 1e + 071e + 01Interevent time (hours)

(c)

DirectionOutgoingIncoming

00000

00025

00050

00075

00100

00125

Den

sity

100 2000Number of peers

(d)

Figure 5 +e exploratory analysis of the communication data for students participating in the NetSense study at the University of NotreDame (a) density plot for the number of communication events for the students (b) the distribution of phone activity through the course ofthe day (c) the cumulative distribution function of the interevent times in hours (d) the number of incoming and outgoing communicationpeers for the students in NetSense where there are at least twenty communication events with the peer during the analyzed period [30]

10 Complexity

on interview this could be an early sign or disease emergingor becoming more active Another interesting case would beto see whether the strengths of new links are decaying slowerthan the old ones

In future work we plan to extend the model in thedirections outlined in Appendix including accounting fordistractions during interactions individualized strengthasymmetric of interactions of significance to participants(eg hierarchical relationships) and the impact of forms ofinteractions and of associated emotions Hence the CogSNetmodel represents an important first step towards modelingsocial network dynamics through the prism of humancognition In literature there are three approaches that areused widely to understand human cognition cognitivebandwidth (CB) dual-process morality (DPM) and implicitassociation tests (IAT) In [29] those methods are comparedand their common drawback is pointed outmdashthe lack ofperspective that takes a cultural background into accountOur approach makes such considerations possible and wesee this challenge as one of the future work directions Lastlywe would love to see more datasets that are so extensive asthe NetSense dataset that could allow us to evaluate ourmodel

Appendix

(A) Formal Definition of the CogSNet Model

A social network can be represented by a graph G (V E)where V v1 vn1113864 1113865 denotes the set of n nodes and E

e1 em1113864 1113865 is the set of directed m edges between pairs ofnodes Each edge eij from node vi to node vj ine j is assignedweight wij(t) and represented by a triple (vi vj wij(t))

+e model evolves in discrete steps as follows For eachpair of nodes (vi vj) the system maintains two variables tijwhich represents the time of the most recent event for thispair of nodes and cij which holds the count of eventsprocessed for this pair of nodes Initially both tij and cij areset to 0 as are the weights of all edges ie for all pairs ofnodes (vi vj) wij(0) 0

When an event happens at time t in the modeled socialnetwork it is processed in chronological order by the modelFirst the weight of the corresponding edge is updatedaccording to the following equation

wij(t)

μijcij+1 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μijcij+1 + wij tij1113872 1113873f t minus tij1113872 1113873 1 minus μijcij+11113874 1113875 otherwise

⎧⎪⎪⎨

⎪⎪⎩(A1)

where μijk is the value of reinforcement peak that resultsfrom the kth event that impacts the edge (vi vj) Here thevalue of reinforcement peak μijk depends on the engagementand emotions invoked by the event that is either directly orindirectly related to the edge (vi vj) An example of an eventindirectly related to this edge could be node vi talking aboutnode vj or any situation that reminds node vi about node vj+e values of μ can be individualized to node vi perception ofrelation with node vj at event k +e μrsquos values may also bedependent on event types μijk isin μ1 μ2 μ3 μ4 1113864 1113865 eg μ1

05 for emails μ2 055 for phone calls μ3 08 formeetings and μ4 09 for joint collaboration in projects

Finally the processing of the current event updates bothvariables associated with the updated edge (vi vj) as followstij t cij cij + 1

At any time t of the model evolution the user can obtainthe value of the weight of an arbitrary edge (vi vj) bycomputing the following equation

wij(t) 0 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

wij tij1113872 1113873f t minus tij1113872 1113873 otherwise

⎧⎪⎨

⎪⎩

(A2)

(B) Experimental Parameter Space

To analyze the CogSNet model and compare its results tothose obtained with the reference models many combina-tions of the CogSNet parameters were tested First differentvalues of μ θ and L were specified and then λ coefficientwas calculated using equations (4) and (5) which were alsoused to define CogSNet parameters for experimental veri-fication +e complete list of parameters used is listed inTables 2 and 3

(C) Quality Measures

+e values of Jaccard metric for a single surveyed studentparticipant vi have been computed as follows

Jaccard vi( 1113857 V

CogSNeti capV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868

VCogSNeti cupV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868 (C1)

where Vsurveyi is the set of up to 20 peers enumerated in the

survey by the participating student vi and VCogSNeti is the set

of neighbors of this student in the CogSNet network with the

Complexity 11

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

CogSNet relationship strengths

Oct Nov Dec Jan Feb

Oct Nov Dec Jan Feb

Oct Nov Dec Jan Feb

t1 t2 t3 t4 ts

000

025

050

075

100A

minusB

000

025

050

075

100

AminusC

000

025

050

075

100

BminusC

000

025

050

075

100

AminusD

000

025

050

075

100

BminusD

Nov Dec Jan FebOct

FebDec JanNovOctTime

(a)

t1 t2 t3 t4 ts

A B

C D

A B

C D

A B

C D

A B

C D

A B

C D

(b)

Figure 3 Four-node CogSNet network for the sample of real NetSense data μ 04 θ 01 and L 10 days nodes A B C and Dcorrespond respectively to participants with ids 40997 11360 10841 and 1232 (a) Relation strengths according to the CogSNet model over4-month period (one term) (b) network snapshots at four timestamps t1ndasht4 and at the survey time ts

6 Complexity

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 201 100 20050Lifetime L in days

μ = 03 θ = 02μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

(a)

μ = 03 θ = 02 μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 20 50 100 2001Life time L in days

(b)

Figure 4 +e plot of the Jaccard metric as a function of lifetime L +e metric measures the overlap between the two sets of peers oneidentified by CogSNet and the other listed by students in the survey +ese plots are compared to the results achieved by the three baselinemodels recency-based with the best results obtained with the number of recent events set to 400 frequency-based and random+e resultsare plotted for the CogSNet running with various parameters for (a) exponential and (b) power forgetting functions

Complexity 7

Table 1 Numerical and symbolic p values adjusted by SimesndashHochberg step-up method of nonparametric pairwise post hoc Nemenyi testcomparing values of Jaccard metric for results obtained by each method with parameters fitted using recorded event data with the groundtruth based on surveys taken over semesters 1ndash4

Power Exponential Recency Frequency RandomPower mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast lowastlowastlowast

Exponential 00000 mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast

Recency 00000 00000 mdash lowastlowastlowast lowastlowastlowast

Frequency 00000 00000 00000 mdash lowastlowastlowast

Random 00000 00000 00000 00000 mdash+ese results show how each method in the leftmost column performs versus the method in the first row +e part of table under the diagonal showsnumerical values of the test while the part above the diagonal presents p values coded as follows lowastlowastlowastplt 000005 represents high confidence in theperformance differences between the methods since typically values of pgt 005 are considered indicative of nonsignificance

Table 2 Setting of parameters for power forgetting function in computational experiments

λ dependence on μ θ and L for power forgettingLifetime L days μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 043621 030863 0654312 03581 025337 0537163 032415 022934 0486234 030372 021489 0455585 028957 020487 0434356 027894 019736 0418417 027055 019142 0405838 026368 018656 0395529 02579 018247 03868510 025294 017896 03794211 024862 01759 03729312 02448 01732 0367213 024139 017079 03620814 023831 016861 03574715 023552 016663 03532816 023297 016483 03494517 023062 016317 03459218 022844 016163 03426719 022643 01602 03396420 022455 015887 03368221 022278 015762 03341822 022113 015645 0331723 021957 015535 03293624 02181 015431 03271625 021671 015333 03250726 021539 015239 03230927 021414 015151 0321228 021294 015066 03194129 02118 014985 0317730 021071 014908 03160640 020188 014283 03028250 019553 013834 02932960 019062 013487 02859480 018337 012974 027506100 017811 012602 026717

8 Complexity

Table 3 Setting of parameters for exponential forgetting function in computational experiments

λ dependence on μ θ and L for exponential forgettingLifetime L (days) μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 005776 004087 0086642 002888 002043 0043323 001925 001362 0028884 001444 001022 0021665 001155 000817 0017336 000963 000681 0014447 000825 000584 0012388 000722 000511 0010839 000642 000454 00096310 000578 000409 00086611 000525 000372 00078812 000481 000341 00072213 000444 000314 00066614 000413 000292 00061915 000385 000272 00057816 000361 000255 00054217 00034 00024 0005118 000321 000227 00048119 000304 000215 00045620 000289 000204 00043321 000275 000195 00041322 000263 000186 00039423 000251 000178 00037724 000241 00017 00036125 000231 000163 00034726 000222 000157 00033327 000214 000151 00032128 000206 000146 00030929 000199 000141 00029930 000193 000136 00028940 000144 000102 00021750 000116 000082 00017360 000096 000068 00014480 000072 000051 000108100 000058 000041 000087

000000

000002

000004

Den

sity

20000 40000 600000Number of communication events

(a)

0000000

0000005

0000010

0000015

0000020

Den

sity

100000 200000000000Time of day

(b)

Figure 5 Continued

Complexity 9

that primacy and recency effects come into play when listingpeers [27]

4 Conclusions

+e results demonstrate that accounting for event percep-tion and memory dynamics is essential for faithfullymodeling the interaction dynamics in social networks +eCogSNet performed statistically significantly better than anyof the base temporal models (see Table 1 in Appendix) Withsuitable data the model can be extended to represent in-dividual sensitivity to events and the impact of emotions anddistractions on event perception to further improve mod-eling quality

+e dataset on which we operate contains data limited toa relatively small fraction of human social activities Henceit cannot be stated that there were no other events happeningthat could have affected participant memory about othersExamples of such events are face to face interactions indirectreferences to someone else and personal reminiscencesabout others In addition emotions can also play a signif-icant role in these processes and they can be potentiallyidentified with some confidence from available data [28]

As a result the reinforcement peak value can be adjustedfor a given person and for an individual event cf alsoequation (A1) in Appendix What is observed here is apartial manifestation from human memory However eventaking into account that the model was built based on a

single data source over 6 million telephone calls andmessages among the NetSense study participants the modelfitted with the recorded event data yields results wellmatching the salience of social contacts over all 578 surveyscompleted by 184 participants It is observed that even whenusing the same values of model parameters for all partici-pants the results are satisfactory Nevertheless this accuracymost likely could have been increased if the parameters hadbeen individually adjusted for each participant Apart fromthat one should remember that the recall task is highlycontextual so the individuals will provide different sets ofpeers depending on the situation [11 12]

+e findings of this work can be also considered as animportant building block on how to maintain healthy re-lationships or how to discover anomalies For instancebased on the results it is anticipated that in order to keep aliving relationship with a peer the contacts should not besparser than lifetime L preferably even more frequent Onthe contrary these findings also demonstrate on how long agrieving period after breaking the relationship could po-tentially last and how to handle the recovery process af-terwards in order to make it effective Lastly smallmodifications making the model closer to ACT-R wouldenable its use for instance in the case of Alzheimer diseasewhere one could monitor whether the peers recalled bypatients are in line with what a CogSNet model generatesbased on the contacts or visits If there will be any dis-crepancy found between the model and observations based

1e minus 04

1e minus 03

1e minus 02

1e minus 01

1e + 00Em

piric

al C

CDF

1e + 04 1e + 071e + 01Interevent time (hours)

(c)

DirectionOutgoingIncoming

00000

00025

00050

00075

00100

00125

Den

sity

100 2000Number of peers

(d)

Figure 5 +e exploratory analysis of the communication data for students participating in the NetSense study at the University of NotreDame (a) density plot for the number of communication events for the students (b) the distribution of phone activity through the course ofthe day (c) the cumulative distribution function of the interevent times in hours (d) the number of incoming and outgoing communicationpeers for the students in NetSense where there are at least twenty communication events with the peer during the analyzed period [30]

10 Complexity

on interview this could be an early sign or disease emergingor becoming more active Another interesting case would beto see whether the strengths of new links are decaying slowerthan the old ones

In future work we plan to extend the model in thedirections outlined in Appendix including accounting fordistractions during interactions individualized strengthasymmetric of interactions of significance to participants(eg hierarchical relationships) and the impact of forms ofinteractions and of associated emotions Hence the CogSNetmodel represents an important first step towards modelingsocial network dynamics through the prism of humancognition In literature there are three approaches that areused widely to understand human cognition cognitivebandwidth (CB) dual-process morality (DPM) and implicitassociation tests (IAT) In [29] those methods are comparedand their common drawback is pointed outmdashthe lack ofperspective that takes a cultural background into accountOur approach makes such considerations possible and wesee this challenge as one of the future work directions Lastlywe would love to see more datasets that are so extensive asthe NetSense dataset that could allow us to evaluate ourmodel

Appendix

(A) Formal Definition of the CogSNet Model

A social network can be represented by a graph G (V E)where V v1 vn1113864 1113865 denotes the set of n nodes and E

e1 em1113864 1113865 is the set of directed m edges between pairs ofnodes Each edge eij from node vi to node vj ine j is assignedweight wij(t) and represented by a triple (vi vj wij(t))

+e model evolves in discrete steps as follows For eachpair of nodes (vi vj) the system maintains two variables tijwhich represents the time of the most recent event for thispair of nodes and cij which holds the count of eventsprocessed for this pair of nodes Initially both tij and cij areset to 0 as are the weights of all edges ie for all pairs ofnodes (vi vj) wij(0) 0

When an event happens at time t in the modeled socialnetwork it is processed in chronological order by the modelFirst the weight of the corresponding edge is updatedaccording to the following equation

wij(t)

μijcij+1 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μijcij+1 + wij tij1113872 1113873f t minus tij1113872 1113873 1 minus μijcij+11113874 1113875 otherwise

⎧⎪⎪⎨

⎪⎪⎩(A1)

where μijk is the value of reinforcement peak that resultsfrom the kth event that impacts the edge (vi vj) Here thevalue of reinforcement peak μijk depends on the engagementand emotions invoked by the event that is either directly orindirectly related to the edge (vi vj) An example of an eventindirectly related to this edge could be node vi talking aboutnode vj or any situation that reminds node vi about node vj+e values of μ can be individualized to node vi perception ofrelation with node vj at event k +e μrsquos values may also bedependent on event types μijk isin μ1 μ2 μ3 μ4 1113864 1113865 eg μ1

05 for emails μ2 055 for phone calls μ3 08 formeetings and μ4 09 for joint collaboration in projects

Finally the processing of the current event updates bothvariables associated with the updated edge (vi vj) as followstij t cij cij + 1

At any time t of the model evolution the user can obtainthe value of the weight of an arbitrary edge (vi vj) bycomputing the following equation

wij(t) 0 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

wij tij1113872 1113873f t minus tij1113872 1113873 otherwise

⎧⎪⎨

⎪⎩

(A2)

(B) Experimental Parameter Space

To analyze the CogSNet model and compare its results tothose obtained with the reference models many combina-tions of the CogSNet parameters were tested First differentvalues of μ θ and L were specified and then λ coefficientwas calculated using equations (4) and (5) which were alsoused to define CogSNet parameters for experimental veri-fication +e complete list of parameters used is listed inTables 2 and 3

(C) Quality Measures

+e values of Jaccard metric for a single surveyed studentparticipant vi have been computed as follows

Jaccard vi( 1113857 V

CogSNeti capV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868

VCogSNeti cupV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868 (C1)

where Vsurveyi is the set of up to 20 peers enumerated in the

survey by the participating student vi and VCogSNeti is the set

of neighbors of this student in the CogSNet network with the

Complexity 11

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 201 100 20050Lifetime L in days

μ = 03 θ = 02μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

(a)

μ = 03 θ = 02 μ = 04 θ = 01

μ = 08 θ = 01μ = 08 θ = 03

Recency-based

Frequency-based

Random005

010

015

020

025

030

Jacc

ard

2 5 10 20 50 100 2001Life time L in days

(b)

Figure 4 +e plot of the Jaccard metric as a function of lifetime L +e metric measures the overlap between the two sets of peers oneidentified by CogSNet and the other listed by students in the survey +ese plots are compared to the results achieved by the three baselinemodels recency-based with the best results obtained with the number of recent events set to 400 frequency-based and random+e resultsare plotted for the CogSNet running with various parameters for (a) exponential and (b) power forgetting functions

Complexity 7

Table 1 Numerical and symbolic p values adjusted by SimesndashHochberg step-up method of nonparametric pairwise post hoc Nemenyi testcomparing values of Jaccard metric for results obtained by each method with parameters fitted using recorded event data with the groundtruth based on surveys taken over semesters 1ndash4

Power Exponential Recency Frequency RandomPower mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast lowastlowastlowast

Exponential 00000 mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast

Recency 00000 00000 mdash lowastlowastlowast lowastlowastlowast

Frequency 00000 00000 00000 mdash lowastlowastlowast

Random 00000 00000 00000 00000 mdash+ese results show how each method in the leftmost column performs versus the method in the first row +e part of table under the diagonal showsnumerical values of the test while the part above the diagonal presents p values coded as follows lowastlowastlowastplt 000005 represents high confidence in theperformance differences between the methods since typically values of pgt 005 are considered indicative of nonsignificance

Table 2 Setting of parameters for power forgetting function in computational experiments

λ dependence on μ θ and L for power forgettingLifetime L days μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 043621 030863 0654312 03581 025337 0537163 032415 022934 0486234 030372 021489 0455585 028957 020487 0434356 027894 019736 0418417 027055 019142 0405838 026368 018656 0395529 02579 018247 03868510 025294 017896 03794211 024862 01759 03729312 02448 01732 0367213 024139 017079 03620814 023831 016861 03574715 023552 016663 03532816 023297 016483 03494517 023062 016317 03459218 022844 016163 03426719 022643 01602 03396420 022455 015887 03368221 022278 015762 03341822 022113 015645 0331723 021957 015535 03293624 02181 015431 03271625 021671 015333 03250726 021539 015239 03230927 021414 015151 0321228 021294 015066 03194129 02118 014985 0317730 021071 014908 03160640 020188 014283 03028250 019553 013834 02932960 019062 013487 02859480 018337 012974 027506100 017811 012602 026717

8 Complexity

Table 3 Setting of parameters for exponential forgetting function in computational experiments

λ dependence on μ θ and L for exponential forgettingLifetime L (days) μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 005776 004087 0086642 002888 002043 0043323 001925 001362 0028884 001444 001022 0021665 001155 000817 0017336 000963 000681 0014447 000825 000584 0012388 000722 000511 0010839 000642 000454 00096310 000578 000409 00086611 000525 000372 00078812 000481 000341 00072213 000444 000314 00066614 000413 000292 00061915 000385 000272 00057816 000361 000255 00054217 00034 00024 0005118 000321 000227 00048119 000304 000215 00045620 000289 000204 00043321 000275 000195 00041322 000263 000186 00039423 000251 000178 00037724 000241 00017 00036125 000231 000163 00034726 000222 000157 00033327 000214 000151 00032128 000206 000146 00030929 000199 000141 00029930 000193 000136 00028940 000144 000102 00021750 000116 000082 00017360 000096 000068 00014480 000072 000051 000108100 000058 000041 000087

000000

000002

000004

Den

sity

20000 40000 600000Number of communication events

(a)

0000000

0000005

0000010

0000015

0000020

Den

sity

100000 200000000000Time of day

(b)

Figure 5 Continued

Complexity 9

that primacy and recency effects come into play when listingpeers [27]

4 Conclusions

+e results demonstrate that accounting for event percep-tion and memory dynamics is essential for faithfullymodeling the interaction dynamics in social networks +eCogSNet performed statistically significantly better than anyof the base temporal models (see Table 1 in Appendix) Withsuitable data the model can be extended to represent in-dividual sensitivity to events and the impact of emotions anddistractions on event perception to further improve mod-eling quality

+e dataset on which we operate contains data limited toa relatively small fraction of human social activities Henceit cannot be stated that there were no other events happeningthat could have affected participant memory about othersExamples of such events are face to face interactions indirectreferences to someone else and personal reminiscencesabout others In addition emotions can also play a signif-icant role in these processes and they can be potentiallyidentified with some confidence from available data [28]

As a result the reinforcement peak value can be adjustedfor a given person and for an individual event cf alsoequation (A1) in Appendix What is observed here is apartial manifestation from human memory However eventaking into account that the model was built based on a

single data source over 6 million telephone calls andmessages among the NetSense study participants the modelfitted with the recorded event data yields results wellmatching the salience of social contacts over all 578 surveyscompleted by 184 participants It is observed that even whenusing the same values of model parameters for all partici-pants the results are satisfactory Nevertheless this accuracymost likely could have been increased if the parameters hadbeen individually adjusted for each participant Apart fromthat one should remember that the recall task is highlycontextual so the individuals will provide different sets ofpeers depending on the situation [11 12]

+e findings of this work can be also considered as animportant building block on how to maintain healthy re-lationships or how to discover anomalies For instancebased on the results it is anticipated that in order to keep aliving relationship with a peer the contacts should not besparser than lifetime L preferably even more frequent Onthe contrary these findings also demonstrate on how long agrieving period after breaking the relationship could po-tentially last and how to handle the recovery process af-terwards in order to make it effective Lastly smallmodifications making the model closer to ACT-R wouldenable its use for instance in the case of Alzheimer diseasewhere one could monitor whether the peers recalled bypatients are in line with what a CogSNet model generatesbased on the contacts or visits If there will be any dis-crepancy found between the model and observations based

1e minus 04

1e minus 03

1e minus 02

1e minus 01

1e + 00Em

piric

al C

CDF

1e + 04 1e + 071e + 01Interevent time (hours)

(c)

DirectionOutgoingIncoming

00000

00025

00050

00075

00100

00125

Den

sity

100 2000Number of peers

(d)

Figure 5 +e exploratory analysis of the communication data for students participating in the NetSense study at the University of NotreDame (a) density plot for the number of communication events for the students (b) the distribution of phone activity through the course ofthe day (c) the cumulative distribution function of the interevent times in hours (d) the number of incoming and outgoing communicationpeers for the students in NetSense where there are at least twenty communication events with the peer during the analyzed period [30]

10 Complexity

on interview this could be an early sign or disease emergingor becoming more active Another interesting case would beto see whether the strengths of new links are decaying slowerthan the old ones

In future work we plan to extend the model in thedirections outlined in Appendix including accounting fordistractions during interactions individualized strengthasymmetric of interactions of significance to participants(eg hierarchical relationships) and the impact of forms ofinteractions and of associated emotions Hence the CogSNetmodel represents an important first step towards modelingsocial network dynamics through the prism of humancognition In literature there are three approaches that areused widely to understand human cognition cognitivebandwidth (CB) dual-process morality (DPM) and implicitassociation tests (IAT) In [29] those methods are comparedand their common drawback is pointed outmdashthe lack ofperspective that takes a cultural background into accountOur approach makes such considerations possible and wesee this challenge as one of the future work directions Lastlywe would love to see more datasets that are so extensive asthe NetSense dataset that could allow us to evaluate ourmodel

Appendix

(A) Formal Definition of the CogSNet Model

A social network can be represented by a graph G (V E)where V v1 vn1113864 1113865 denotes the set of n nodes and E

e1 em1113864 1113865 is the set of directed m edges between pairs ofnodes Each edge eij from node vi to node vj ine j is assignedweight wij(t) and represented by a triple (vi vj wij(t))

+e model evolves in discrete steps as follows For eachpair of nodes (vi vj) the system maintains two variables tijwhich represents the time of the most recent event for thispair of nodes and cij which holds the count of eventsprocessed for this pair of nodes Initially both tij and cij areset to 0 as are the weights of all edges ie for all pairs ofnodes (vi vj) wij(0) 0

When an event happens at time t in the modeled socialnetwork it is processed in chronological order by the modelFirst the weight of the corresponding edge is updatedaccording to the following equation

wij(t)

μijcij+1 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μijcij+1 + wij tij1113872 1113873f t minus tij1113872 1113873 1 minus μijcij+11113874 1113875 otherwise

⎧⎪⎪⎨

⎪⎪⎩(A1)

where μijk is the value of reinforcement peak that resultsfrom the kth event that impacts the edge (vi vj) Here thevalue of reinforcement peak μijk depends on the engagementand emotions invoked by the event that is either directly orindirectly related to the edge (vi vj) An example of an eventindirectly related to this edge could be node vi talking aboutnode vj or any situation that reminds node vi about node vj+e values of μ can be individualized to node vi perception ofrelation with node vj at event k +e μrsquos values may also bedependent on event types μijk isin μ1 μ2 μ3 μ4 1113864 1113865 eg μ1

05 for emails μ2 055 for phone calls μ3 08 formeetings and μ4 09 for joint collaboration in projects

Finally the processing of the current event updates bothvariables associated with the updated edge (vi vj) as followstij t cij cij + 1

At any time t of the model evolution the user can obtainthe value of the weight of an arbitrary edge (vi vj) bycomputing the following equation

wij(t) 0 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

wij tij1113872 1113873f t minus tij1113872 1113873 otherwise

⎧⎪⎨

⎪⎩

(A2)

(B) Experimental Parameter Space

To analyze the CogSNet model and compare its results tothose obtained with the reference models many combina-tions of the CogSNet parameters were tested First differentvalues of μ θ and L were specified and then λ coefficientwas calculated using equations (4) and (5) which were alsoused to define CogSNet parameters for experimental veri-fication +e complete list of parameters used is listed inTables 2 and 3

(C) Quality Measures

+e values of Jaccard metric for a single surveyed studentparticipant vi have been computed as follows

Jaccard vi( 1113857 V

CogSNeti capV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868

VCogSNeti cupV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868 (C1)

where Vsurveyi is the set of up to 20 peers enumerated in the

survey by the participating student vi and VCogSNeti is the set

of neighbors of this student in the CogSNet network with the

Complexity 11

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

Table 1 Numerical and symbolic p values adjusted by SimesndashHochberg step-up method of nonparametric pairwise post hoc Nemenyi testcomparing values of Jaccard metric for results obtained by each method with parameters fitted using recorded event data with the groundtruth based on surveys taken over semesters 1ndash4

Power Exponential Recency Frequency RandomPower mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast lowastlowastlowast

Exponential 00000 mdash lowastlowastlowast lowastlowastlowast lowastlowastlowast

Recency 00000 00000 mdash lowastlowastlowast lowastlowastlowast

Frequency 00000 00000 00000 mdash lowastlowastlowast

Random 00000 00000 00000 00000 mdash+ese results show how each method in the leftmost column performs versus the method in the first row +e part of table under the diagonal showsnumerical values of the test while the part above the diagonal presents p values coded as follows lowastlowastlowastplt 000005 represents high confidence in theperformance differences between the methods since typically values of pgt 005 are considered indicative of nonsignificance

Table 2 Setting of parameters for power forgetting function in computational experiments

λ dependence on μ θ and L for power forgettingLifetime L days μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 043621 030863 0654312 03581 025337 0537163 032415 022934 0486234 030372 021489 0455585 028957 020487 0434356 027894 019736 0418417 027055 019142 0405838 026368 018656 0395529 02579 018247 03868510 025294 017896 03794211 024862 01759 03729312 02448 01732 0367213 024139 017079 03620814 023831 016861 03574715 023552 016663 03532816 023297 016483 03494517 023062 016317 03459218 022844 016163 03426719 022643 01602 03396420 022455 015887 03368221 022278 015762 03341822 022113 015645 0331723 021957 015535 03293624 02181 015431 03271625 021671 015333 03250726 021539 015239 03230927 021414 015151 0321228 021294 015066 03194129 02118 014985 0317730 021071 014908 03160640 020188 014283 03028250 019553 013834 02932960 019062 013487 02859480 018337 012974 027506100 017811 012602 026717

8 Complexity

Table 3 Setting of parameters for exponential forgetting function in computational experiments

λ dependence on μ θ and L for exponential forgettingLifetime L (days) μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 005776 004087 0086642 002888 002043 0043323 001925 001362 0028884 001444 001022 0021665 001155 000817 0017336 000963 000681 0014447 000825 000584 0012388 000722 000511 0010839 000642 000454 00096310 000578 000409 00086611 000525 000372 00078812 000481 000341 00072213 000444 000314 00066614 000413 000292 00061915 000385 000272 00057816 000361 000255 00054217 00034 00024 0005118 000321 000227 00048119 000304 000215 00045620 000289 000204 00043321 000275 000195 00041322 000263 000186 00039423 000251 000178 00037724 000241 00017 00036125 000231 000163 00034726 000222 000157 00033327 000214 000151 00032128 000206 000146 00030929 000199 000141 00029930 000193 000136 00028940 000144 000102 00021750 000116 000082 00017360 000096 000068 00014480 000072 000051 000108100 000058 000041 000087

000000

000002

000004

Den

sity

20000 40000 600000Number of communication events

(a)

0000000

0000005

0000010

0000015

0000020

Den

sity

100000 200000000000Time of day

(b)

Figure 5 Continued

Complexity 9

that primacy and recency effects come into play when listingpeers [27]

4 Conclusions

+e results demonstrate that accounting for event percep-tion and memory dynamics is essential for faithfullymodeling the interaction dynamics in social networks +eCogSNet performed statistically significantly better than anyof the base temporal models (see Table 1 in Appendix) Withsuitable data the model can be extended to represent in-dividual sensitivity to events and the impact of emotions anddistractions on event perception to further improve mod-eling quality

+e dataset on which we operate contains data limited toa relatively small fraction of human social activities Henceit cannot be stated that there were no other events happeningthat could have affected participant memory about othersExamples of such events are face to face interactions indirectreferences to someone else and personal reminiscencesabout others In addition emotions can also play a signif-icant role in these processes and they can be potentiallyidentified with some confidence from available data [28]

As a result the reinforcement peak value can be adjustedfor a given person and for an individual event cf alsoequation (A1) in Appendix What is observed here is apartial manifestation from human memory However eventaking into account that the model was built based on a

single data source over 6 million telephone calls andmessages among the NetSense study participants the modelfitted with the recorded event data yields results wellmatching the salience of social contacts over all 578 surveyscompleted by 184 participants It is observed that even whenusing the same values of model parameters for all partici-pants the results are satisfactory Nevertheless this accuracymost likely could have been increased if the parameters hadbeen individually adjusted for each participant Apart fromthat one should remember that the recall task is highlycontextual so the individuals will provide different sets ofpeers depending on the situation [11 12]

+e findings of this work can be also considered as animportant building block on how to maintain healthy re-lationships or how to discover anomalies For instancebased on the results it is anticipated that in order to keep aliving relationship with a peer the contacts should not besparser than lifetime L preferably even more frequent Onthe contrary these findings also demonstrate on how long agrieving period after breaking the relationship could po-tentially last and how to handle the recovery process af-terwards in order to make it effective Lastly smallmodifications making the model closer to ACT-R wouldenable its use for instance in the case of Alzheimer diseasewhere one could monitor whether the peers recalled bypatients are in line with what a CogSNet model generatesbased on the contacts or visits If there will be any dis-crepancy found between the model and observations based

1e minus 04

1e minus 03

1e minus 02

1e minus 01

1e + 00Em

piric

al C

CDF

1e + 04 1e + 071e + 01Interevent time (hours)

(c)

DirectionOutgoingIncoming

00000

00025

00050

00075

00100

00125

Den

sity

100 2000Number of peers

(d)

Figure 5 +e exploratory analysis of the communication data for students participating in the NetSense study at the University of NotreDame (a) density plot for the number of communication events for the students (b) the distribution of phone activity through the course ofthe day (c) the cumulative distribution function of the interevent times in hours (d) the number of incoming and outgoing communicationpeers for the students in NetSense where there are at least twenty communication events with the peer during the analyzed period [30]

10 Complexity

on interview this could be an early sign or disease emergingor becoming more active Another interesting case would beto see whether the strengths of new links are decaying slowerthan the old ones

In future work we plan to extend the model in thedirections outlined in Appendix including accounting fordistractions during interactions individualized strengthasymmetric of interactions of significance to participants(eg hierarchical relationships) and the impact of forms ofinteractions and of associated emotions Hence the CogSNetmodel represents an important first step towards modelingsocial network dynamics through the prism of humancognition In literature there are three approaches that areused widely to understand human cognition cognitivebandwidth (CB) dual-process morality (DPM) and implicitassociation tests (IAT) In [29] those methods are comparedand their common drawback is pointed outmdashthe lack ofperspective that takes a cultural background into accountOur approach makes such considerations possible and wesee this challenge as one of the future work directions Lastlywe would love to see more datasets that are so extensive asthe NetSense dataset that could allow us to evaluate ourmodel

Appendix

(A) Formal Definition of the CogSNet Model

A social network can be represented by a graph G (V E)where V v1 vn1113864 1113865 denotes the set of n nodes and E

e1 em1113864 1113865 is the set of directed m edges between pairs ofnodes Each edge eij from node vi to node vj ine j is assignedweight wij(t) and represented by a triple (vi vj wij(t))

+e model evolves in discrete steps as follows For eachpair of nodes (vi vj) the system maintains two variables tijwhich represents the time of the most recent event for thispair of nodes and cij which holds the count of eventsprocessed for this pair of nodes Initially both tij and cij areset to 0 as are the weights of all edges ie for all pairs ofnodes (vi vj) wij(0) 0

When an event happens at time t in the modeled socialnetwork it is processed in chronological order by the modelFirst the weight of the corresponding edge is updatedaccording to the following equation

wij(t)

μijcij+1 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μijcij+1 + wij tij1113872 1113873f t minus tij1113872 1113873 1 minus μijcij+11113874 1113875 otherwise

⎧⎪⎪⎨

⎪⎪⎩(A1)

where μijk is the value of reinforcement peak that resultsfrom the kth event that impacts the edge (vi vj) Here thevalue of reinforcement peak μijk depends on the engagementand emotions invoked by the event that is either directly orindirectly related to the edge (vi vj) An example of an eventindirectly related to this edge could be node vi talking aboutnode vj or any situation that reminds node vi about node vj+e values of μ can be individualized to node vi perception ofrelation with node vj at event k +e μrsquos values may also bedependent on event types μijk isin μ1 μ2 μ3 μ4 1113864 1113865 eg μ1

05 for emails μ2 055 for phone calls μ3 08 formeetings and μ4 09 for joint collaboration in projects

Finally the processing of the current event updates bothvariables associated with the updated edge (vi vj) as followstij t cij cij + 1

At any time t of the model evolution the user can obtainthe value of the weight of an arbitrary edge (vi vj) bycomputing the following equation

wij(t) 0 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

wij tij1113872 1113873f t minus tij1113872 1113873 otherwise

⎧⎪⎨

⎪⎩

(A2)

(B) Experimental Parameter Space

To analyze the CogSNet model and compare its results tothose obtained with the reference models many combina-tions of the CogSNet parameters were tested First differentvalues of μ θ and L were specified and then λ coefficientwas calculated using equations (4) and (5) which were alsoused to define CogSNet parameters for experimental veri-fication +e complete list of parameters used is listed inTables 2 and 3

(C) Quality Measures

+e values of Jaccard metric for a single surveyed studentparticipant vi have been computed as follows

Jaccard vi( 1113857 V

CogSNeti capV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868

VCogSNeti cupV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868 (C1)

where Vsurveyi is the set of up to 20 peers enumerated in the

survey by the participating student vi and VCogSNeti is the set

of neighbors of this student in the CogSNet network with the

Complexity 11

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

Table 3 Setting of parameters for exponential forgetting function in computational experiments

λ dependence on μ θ and L for exponential forgettingLifetime L (days) μ 04 θ 01 μ 08 θ 03 μ 08 θ 011 005776 004087 0086642 002888 002043 0043323 001925 001362 0028884 001444 001022 0021665 001155 000817 0017336 000963 000681 0014447 000825 000584 0012388 000722 000511 0010839 000642 000454 00096310 000578 000409 00086611 000525 000372 00078812 000481 000341 00072213 000444 000314 00066614 000413 000292 00061915 000385 000272 00057816 000361 000255 00054217 00034 00024 0005118 000321 000227 00048119 000304 000215 00045620 000289 000204 00043321 000275 000195 00041322 000263 000186 00039423 000251 000178 00037724 000241 00017 00036125 000231 000163 00034726 000222 000157 00033327 000214 000151 00032128 000206 000146 00030929 000199 000141 00029930 000193 000136 00028940 000144 000102 00021750 000116 000082 00017360 000096 000068 00014480 000072 000051 000108100 000058 000041 000087

000000

000002

000004

Den

sity

20000 40000 600000Number of communication events

(a)

0000000

0000005

0000010

0000015

0000020

Den

sity

100000 200000000000Time of day

(b)

Figure 5 Continued

Complexity 9

that primacy and recency effects come into play when listingpeers [27]

4 Conclusions

+e results demonstrate that accounting for event percep-tion and memory dynamics is essential for faithfullymodeling the interaction dynamics in social networks +eCogSNet performed statistically significantly better than anyof the base temporal models (see Table 1 in Appendix) Withsuitable data the model can be extended to represent in-dividual sensitivity to events and the impact of emotions anddistractions on event perception to further improve mod-eling quality

+e dataset on which we operate contains data limited toa relatively small fraction of human social activities Henceit cannot be stated that there were no other events happeningthat could have affected participant memory about othersExamples of such events are face to face interactions indirectreferences to someone else and personal reminiscencesabout others In addition emotions can also play a signif-icant role in these processes and they can be potentiallyidentified with some confidence from available data [28]

As a result the reinforcement peak value can be adjustedfor a given person and for an individual event cf alsoequation (A1) in Appendix What is observed here is apartial manifestation from human memory However eventaking into account that the model was built based on a

single data source over 6 million telephone calls andmessages among the NetSense study participants the modelfitted with the recorded event data yields results wellmatching the salience of social contacts over all 578 surveyscompleted by 184 participants It is observed that even whenusing the same values of model parameters for all partici-pants the results are satisfactory Nevertheless this accuracymost likely could have been increased if the parameters hadbeen individually adjusted for each participant Apart fromthat one should remember that the recall task is highlycontextual so the individuals will provide different sets ofpeers depending on the situation [11 12]

+e findings of this work can be also considered as animportant building block on how to maintain healthy re-lationships or how to discover anomalies For instancebased on the results it is anticipated that in order to keep aliving relationship with a peer the contacts should not besparser than lifetime L preferably even more frequent Onthe contrary these findings also demonstrate on how long agrieving period after breaking the relationship could po-tentially last and how to handle the recovery process af-terwards in order to make it effective Lastly smallmodifications making the model closer to ACT-R wouldenable its use for instance in the case of Alzheimer diseasewhere one could monitor whether the peers recalled bypatients are in line with what a CogSNet model generatesbased on the contacts or visits If there will be any dis-crepancy found between the model and observations based

1e minus 04

1e minus 03

1e minus 02

1e minus 01

1e + 00Em

piric

al C

CDF

1e + 04 1e + 071e + 01Interevent time (hours)

(c)

DirectionOutgoingIncoming

00000

00025

00050

00075

00100

00125

Den

sity

100 2000Number of peers

(d)

Figure 5 +e exploratory analysis of the communication data for students participating in the NetSense study at the University of NotreDame (a) density plot for the number of communication events for the students (b) the distribution of phone activity through the course ofthe day (c) the cumulative distribution function of the interevent times in hours (d) the number of incoming and outgoing communicationpeers for the students in NetSense where there are at least twenty communication events with the peer during the analyzed period [30]

10 Complexity

on interview this could be an early sign or disease emergingor becoming more active Another interesting case would beto see whether the strengths of new links are decaying slowerthan the old ones

In future work we plan to extend the model in thedirections outlined in Appendix including accounting fordistractions during interactions individualized strengthasymmetric of interactions of significance to participants(eg hierarchical relationships) and the impact of forms ofinteractions and of associated emotions Hence the CogSNetmodel represents an important first step towards modelingsocial network dynamics through the prism of humancognition In literature there are three approaches that areused widely to understand human cognition cognitivebandwidth (CB) dual-process morality (DPM) and implicitassociation tests (IAT) In [29] those methods are comparedand their common drawback is pointed outmdashthe lack ofperspective that takes a cultural background into accountOur approach makes such considerations possible and wesee this challenge as one of the future work directions Lastlywe would love to see more datasets that are so extensive asthe NetSense dataset that could allow us to evaluate ourmodel

Appendix

(A) Formal Definition of the CogSNet Model

A social network can be represented by a graph G (V E)where V v1 vn1113864 1113865 denotes the set of n nodes and E

e1 em1113864 1113865 is the set of directed m edges between pairs ofnodes Each edge eij from node vi to node vj ine j is assignedweight wij(t) and represented by a triple (vi vj wij(t))

+e model evolves in discrete steps as follows For eachpair of nodes (vi vj) the system maintains two variables tijwhich represents the time of the most recent event for thispair of nodes and cij which holds the count of eventsprocessed for this pair of nodes Initially both tij and cij areset to 0 as are the weights of all edges ie for all pairs ofnodes (vi vj) wij(0) 0

When an event happens at time t in the modeled socialnetwork it is processed in chronological order by the modelFirst the weight of the corresponding edge is updatedaccording to the following equation

wij(t)

μijcij+1 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μijcij+1 + wij tij1113872 1113873f t minus tij1113872 1113873 1 minus μijcij+11113874 1113875 otherwise

⎧⎪⎪⎨

⎪⎪⎩(A1)

where μijk is the value of reinforcement peak that resultsfrom the kth event that impacts the edge (vi vj) Here thevalue of reinforcement peak μijk depends on the engagementand emotions invoked by the event that is either directly orindirectly related to the edge (vi vj) An example of an eventindirectly related to this edge could be node vi talking aboutnode vj or any situation that reminds node vi about node vj+e values of μ can be individualized to node vi perception ofrelation with node vj at event k +e μrsquos values may also bedependent on event types μijk isin μ1 μ2 μ3 μ4 1113864 1113865 eg μ1

05 for emails μ2 055 for phone calls μ3 08 formeetings and μ4 09 for joint collaboration in projects

Finally the processing of the current event updates bothvariables associated with the updated edge (vi vj) as followstij t cij cij + 1

At any time t of the model evolution the user can obtainthe value of the weight of an arbitrary edge (vi vj) bycomputing the following equation

wij(t) 0 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

wij tij1113872 1113873f t minus tij1113872 1113873 otherwise

⎧⎪⎨

⎪⎩

(A2)

(B) Experimental Parameter Space

To analyze the CogSNet model and compare its results tothose obtained with the reference models many combina-tions of the CogSNet parameters were tested First differentvalues of μ θ and L were specified and then λ coefficientwas calculated using equations (4) and (5) which were alsoused to define CogSNet parameters for experimental veri-fication +e complete list of parameters used is listed inTables 2 and 3

(C) Quality Measures

+e values of Jaccard metric for a single surveyed studentparticipant vi have been computed as follows

Jaccard vi( 1113857 V

CogSNeti capV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868

VCogSNeti cupV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868 (C1)

where Vsurveyi is the set of up to 20 peers enumerated in the

survey by the participating student vi and VCogSNeti is the set

of neighbors of this student in the CogSNet network with the

Complexity 11

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

that primacy and recency effects come into play when listingpeers [27]

4 Conclusions

+e results demonstrate that accounting for event percep-tion and memory dynamics is essential for faithfullymodeling the interaction dynamics in social networks +eCogSNet performed statistically significantly better than anyof the base temporal models (see Table 1 in Appendix) Withsuitable data the model can be extended to represent in-dividual sensitivity to events and the impact of emotions anddistractions on event perception to further improve mod-eling quality

+e dataset on which we operate contains data limited toa relatively small fraction of human social activities Henceit cannot be stated that there were no other events happeningthat could have affected participant memory about othersExamples of such events are face to face interactions indirectreferences to someone else and personal reminiscencesabout others In addition emotions can also play a signif-icant role in these processes and they can be potentiallyidentified with some confidence from available data [28]

As a result the reinforcement peak value can be adjustedfor a given person and for an individual event cf alsoequation (A1) in Appendix What is observed here is apartial manifestation from human memory However eventaking into account that the model was built based on a

single data source over 6 million telephone calls andmessages among the NetSense study participants the modelfitted with the recorded event data yields results wellmatching the salience of social contacts over all 578 surveyscompleted by 184 participants It is observed that even whenusing the same values of model parameters for all partici-pants the results are satisfactory Nevertheless this accuracymost likely could have been increased if the parameters hadbeen individually adjusted for each participant Apart fromthat one should remember that the recall task is highlycontextual so the individuals will provide different sets ofpeers depending on the situation [11 12]

+e findings of this work can be also considered as animportant building block on how to maintain healthy re-lationships or how to discover anomalies For instancebased on the results it is anticipated that in order to keep aliving relationship with a peer the contacts should not besparser than lifetime L preferably even more frequent Onthe contrary these findings also demonstrate on how long agrieving period after breaking the relationship could po-tentially last and how to handle the recovery process af-terwards in order to make it effective Lastly smallmodifications making the model closer to ACT-R wouldenable its use for instance in the case of Alzheimer diseasewhere one could monitor whether the peers recalled bypatients are in line with what a CogSNet model generatesbased on the contacts or visits If there will be any dis-crepancy found between the model and observations based

1e minus 04

1e minus 03

1e minus 02

1e minus 01

1e + 00Em

piric

al C

CDF

1e + 04 1e + 071e + 01Interevent time (hours)

(c)

DirectionOutgoingIncoming

00000

00025

00050

00075

00100

00125

Den

sity

100 2000Number of peers

(d)

Figure 5 +e exploratory analysis of the communication data for students participating in the NetSense study at the University of NotreDame (a) density plot for the number of communication events for the students (b) the distribution of phone activity through the course ofthe day (c) the cumulative distribution function of the interevent times in hours (d) the number of incoming and outgoing communicationpeers for the students in NetSense where there are at least twenty communication events with the peer during the analyzed period [30]

10 Complexity

on interview this could be an early sign or disease emergingor becoming more active Another interesting case would beto see whether the strengths of new links are decaying slowerthan the old ones

In future work we plan to extend the model in thedirections outlined in Appendix including accounting fordistractions during interactions individualized strengthasymmetric of interactions of significance to participants(eg hierarchical relationships) and the impact of forms ofinteractions and of associated emotions Hence the CogSNetmodel represents an important first step towards modelingsocial network dynamics through the prism of humancognition In literature there are three approaches that areused widely to understand human cognition cognitivebandwidth (CB) dual-process morality (DPM) and implicitassociation tests (IAT) In [29] those methods are comparedand their common drawback is pointed outmdashthe lack ofperspective that takes a cultural background into accountOur approach makes such considerations possible and wesee this challenge as one of the future work directions Lastlywe would love to see more datasets that are so extensive asthe NetSense dataset that could allow us to evaluate ourmodel

Appendix

(A) Formal Definition of the CogSNet Model

A social network can be represented by a graph G (V E)where V v1 vn1113864 1113865 denotes the set of n nodes and E

e1 em1113864 1113865 is the set of directed m edges between pairs ofnodes Each edge eij from node vi to node vj ine j is assignedweight wij(t) and represented by a triple (vi vj wij(t))

+e model evolves in discrete steps as follows For eachpair of nodes (vi vj) the system maintains two variables tijwhich represents the time of the most recent event for thispair of nodes and cij which holds the count of eventsprocessed for this pair of nodes Initially both tij and cij areset to 0 as are the weights of all edges ie for all pairs ofnodes (vi vj) wij(0) 0

When an event happens at time t in the modeled socialnetwork it is processed in chronological order by the modelFirst the weight of the corresponding edge is updatedaccording to the following equation

wij(t)

μijcij+1 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μijcij+1 + wij tij1113872 1113873f t minus tij1113872 1113873 1 minus μijcij+11113874 1113875 otherwise

⎧⎪⎪⎨

⎪⎪⎩(A1)

where μijk is the value of reinforcement peak that resultsfrom the kth event that impacts the edge (vi vj) Here thevalue of reinforcement peak μijk depends on the engagementand emotions invoked by the event that is either directly orindirectly related to the edge (vi vj) An example of an eventindirectly related to this edge could be node vi talking aboutnode vj or any situation that reminds node vi about node vj+e values of μ can be individualized to node vi perception ofrelation with node vj at event k +e μrsquos values may also bedependent on event types μijk isin μ1 μ2 μ3 μ4 1113864 1113865 eg μ1

05 for emails μ2 055 for phone calls μ3 08 formeetings and μ4 09 for joint collaboration in projects

Finally the processing of the current event updates bothvariables associated with the updated edge (vi vj) as followstij t cij cij + 1

At any time t of the model evolution the user can obtainthe value of the weight of an arbitrary edge (vi vj) bycomputing the following equation

wij(t) 0 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

wij tij1113872 1113873f t minus tij1113872 1113873 otherwise

⎧⎪⎨

⎪⎩

(A2)

(B) Experimental Parameter Space

To analyze the CogSNet model and compare its results tothose obtained with the reference models many combina-tions of the CogSNet parameters were tested First differentvalues of μ θ and L were specified and then λ coefficientwas calculated using equations (4) and (5) which were alsoused to define CogSNet parameters for experimental veri-fication +e complete list of parameters used is listed inTables 2 and 3

(C) Quality Measures

+e values of Jaccard metric for a single surveyed studentparticipant vi have been computed as follows

Jaccard vi( 1113857 V

CogSNeti capV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868

VCogSNeti cupV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868 (C1)

where Vsurveyi is the set of up to 20 peers enumerated in the

survey by the participating student vi and VCogSNeti is the set

of neighbors of this student in the CogSNet network with the

Complexity 11

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

on interview this could be an early sign or disease emergingor becoming more active Another interesting case would beto see whether the strengths of new links are decaying slowerthan the old ones

In future work we plan to extend the model in thedirections outlined in Appendix including accounting fordistractions during interactions individualized strengthasymmetric of interactions of significance to participants(eg hierarchical relationships) and the impact of forms ofinteractions and of associated emotions Hence the CogSNetmodel represents an important first step towards modelingsocial network dynamics through the prism of humancognition In literature there are three approaches that areused widely to understand human cognition cognitivebandwidth (CB) dual-process morality (DPM) and implicitassociation tests (IAT) In [29] those methods are comparedand their common drawback is pointed outmdashthe lack ofperspective that takes a cultural background into accountOur approach makes such considerations possible and wesee this challenge as one of the future work directions Lastlywe would love to see more datasets that are so extensive asthe NetSense dataset that could allow us to evaluate ourmodel

Appendix

(A) Formal Definition of the CogSNet Model

A social network can be represented by a graph G (V E)where V v1 vn1113864 1113865 denotes the set of n nodes and E

e1 em1113864 1113865 is the set of directed m edges between pairs ofnodes Each edge eij from node vi to node vj ine j is assignedweight wij(t) and represented by a triple (vi vj wij(t))

+e model evolves in discrete steps as follows For eachpair of nodes (vi vj) the system maintains two variables tijwhich represents the time of the most recent event for thispair of nodes and cij which holds the count of eventsprocessed for this pair of nodes Initially both tij and cij areset to 0 as are the weights of all edges ie for all pairs ofnodes (vi vj) wij(0) 0

When an event happens at time t in the modeled socialnetwork it is processed in chronological order by the modelFirst the weight of the corresponding edge is updatedaccording to the following equation

wij(t)

μijcij+1 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

μijcij+1 + wij tij1113872 1113873f t minus tij1113872 1113873 1 minus μijcij+11113874 1113875 otherwise

⎧⎪⎪⎨

⎪⎪⎩(A1)

where μijk is the value of reinforcement peak that resultsfrom the kth event that impacts the edge (vi vj) Here thevalue of reinforcement peak μijk depends on the engagementand emotions invoked by the event that is either directly orindirectly related to the edge (vi vj) An example of an eventindirectly related to this edge could be node vi talking aboutnode vj or any situation that reminds node vi about node vj+e values of μ can be individualized to node vi perception ofrelation with node vj at event k +e μrsquos values may also bedependent on event types μijk isin μ1 μ2 μ3 μ4 1113864 1113865 eg μ1

05 for emails μ2 055 for phone calls μ3 08 formeetings and μ4 09 for joint collaboration in projects

Finally the processing of the current event updates bothvariables associated with the updated edge (vi vj) as followstij t cij cij + 1

At any time t of the model evolution the user can obtainthe value of the weight of an arbitrary edge (vi vj) bycomputing the following equation

wij(t) 0 if wij tij1113872 1113873f t minus tij1113872 1113873lt θ

wij tij1113872 1113873f t minus tij1113872 1113873 otherwise

⎧⎪⎨

⎪⎩

(A2)

(B) Experimental Parameter Space

To analyze the CogSNet model and compare its results tothose obtained with the reference models many combina-tions of the CogSNet parameters were tested First differentvalues of μ θ and L were specified and then λ coefficientwas calculated using equations (4) and (5) which were alsoused to define CogSNet parameters for experimental veri-fication +e complete list of parameters used is listed inTables 2 and 3

(C) Quality Measures

+e values of Jaccard metric for a single surveyed studentparticipant vi have been computed as follows

Jaccard vi( 1113857 V

CogSNeti capV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868

VCogSNeti cupV

surveyi

11138681113868111386811138681113868

11138681113868111386811138681113868 (C1)

where Vsurveyi is the set of up to 20 peers enumerated in the

survey by the participating student vi and VCogSNeti is the set

of neighbors of this student in the CogSNet network with the

Complexity 11

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

largest nonzero weights on edges to this student on the dayon which the given survey was administered

(D) Recency-Based Baseline

+e recency-based baseline model was tuned by finding thebest parameter representing the number of the most recentevents to be taken into account in computing the perfor-mance+e resulting value of 400 events was used in Figure 4for computing the results for the recency-based baselinemodel

(E) Statistical Tests

To confirm that there is a statistical difference between theproposed CogSNet model and baselines we performed anumber of statistical tests +e results are presented inTable 1 which shows that CogSNet approach is statisticallysignificantly better from all baseline methods Moreoverpower forgetting performs statistically significantly betterthan exponential forgetting

(F) NetSense Dataset

+e dataset consists of two parts +e first includes thetimestamps and durationlength of phone calls and textmessages collected for each student participating in thestudy Each student phone device recorded all connectionsmessages including those to the phones of people outside ofthe test group so recording was done on both sides ofcommunication by sending and receiving callsmessages ifboth belonged to the study In total the mobile phonecommunication data consist of 7575865 activities Out ofthese 7096844 have been text messages (937 of allevents) and 479021 phone calls (63) Due to the fact thatthe dataset covers millions of mobile phone interactions thathave been recorded by a special application installed onstudentsrsquo phones some issues discussed below arose duringthe preliminary data analysis phase

Since events involving each student possessing themobile phone were recorded independently from othermobile phones some inconsistencies arose among which themost important was that not in all cases the event wasrecorded on both sides Since usually a couple of secondspass between sending text message and receiving it by therecipients we identified the same message by its lengthHowever sometimes the recorded lengths were different soin such cases we were forced to make somewhat arbitrarydecision based on the length difference whether the messagewas the same or separate one for which the correspondingevent on the other side of communication was not recorded

Next when the students phoned each other and the callwas not answered the recipientrsquos voice mail was reached if itwas enabled Such case has been recorded as a regular phonecall even if it has not been actually answered by the recipientIn order to solve this problem we had to look for orphanrecords that do not have their corresponding records on therecipientrsquos side

+is part of data preprocessing removed 12482 recordsfrom our dataset Afterwards we performed an exploratoryanalysis of events In Figure 5 we show the basic statisticsregarding the communication patterns of the participants ofthe study

+e second part of the dataset includes surveys con-taining peers enumerated by the participants at the end ofeach term in response to the following question ldquoIn thespaces below please list up to 20 people (friends familymembers acquaintances or other people) with whom youspend time communicating or interactingrdquo To be morespecific on how the process of obtaining data looked like thestudents have been presented with a form containing twentyempty fields with the request in questionmdashthey had to fill thenames by themselves without any suggestions or generatorsAs already mentioned in the main part of the manuscriptthe ordering of the enumerated peers was not imposed buteven if the students would start with listing the most im-portant and not necessary most frequent contacts we couldnot assume that

Lastly the students were asked to enumerate in anyorder up to twenty peers with whom they interacted recentlyHowever the system recorded each name with the ac-companying number representing namersquos position on thelist For some surveys there were some holes in thisnumbering so either the student did not list the peers ac-cordingly (left some blank lines in between names) or partialresults were not recorded +is is why we consider listedpeers as the sets without any implied ordering and theJaccard sets comparison metric was used for evaluationpurposes instead of any other that are based on orderingeg Pearson or Spearman rank correlation coefficients

Data Availability

+e code for computing the state of the social network builtusing the CogSNet model has been made public A fast Cimplementation prepared by the authors of this manuscriptis available as a Code Ocean capsule (httpsdoiorg1024433CO6088785v1) +e NetSense dataset used for themodel validation is available for other researchers thereaders interested in accessing the data should contact thecoauthor Professor Omar Lizardo

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Authorsrsquo Contributions

All authors participated in the design of the research andcomputational experiments RM and MK ran the compu-tational experiments and collected the results All authorsparticipated in analysis of the results All authors wrote andedited the manuscript

Acknowledgments

+e authors would like to thank Prof Tomasz Kajdanowiczfor his valuable remarks regarding statistical tests +is work

12 Complexity

was partially supported by the National Science CentrePoland by the following projects 201517DST604046(RM) and 201621BST601463 (PK MK) Office of NavalResearch (ONR) (Grant no N00014-15-1ndash2640) ArmyResearch Office (ARO) (Grant no W911NF-16-1ndash0524)European UnionrsquosMarie Skłodowska-Curie Program (Grantno 691152) and Polish Ministry of Science and HigherEducation (Grant no 3628H202020162) Calculationshave been carried out using resources provided by WrocławCentre for Networking and Supercomputing (httpswcsspl) (Grant no 177) Collection of the NetSense data wassupported by the US National Science Foundation (Grantno IIS-0968529) Further support for data collection wasprovided through University of Notre Dame and Sprint

References

[1] D B Alan ldquo+e psychology of memoryrdquo 9e EssentialHandbook of Memory Disorders for Clinicians pp 1ndash13 JohnWiley amp Sons Hoboken NJ USA 2004

[2] P R Huttenlocher Neural Plasticity 9e Effects of Envi-ronment on the Development of the Cerebral Cortex HarvardUniversity Press Cambridge MA USA 2009

[3] J R Anderson D Bothell C Lebiere and M Matessa ldquoAnintegrated theory of list memoryrdquo Journal of Memory andLanguage vol 38 no 4 pp 341ndash380 1998

[4] R S Burt ldquoDecay functionsrdquo Social Networks vol 22 no 1pp 1ndash28 2000

[5] P Jenkins G Earle-Richardson D T Slingerland and J MayldquoTime dependent memory decayrdquo American Journal of In-dustrial Medicine vol 41 no 2 pp 98ndash101 2002

[6] J T Wixted and E B Ebbesen ldquoOn the form of forgettingrdquoPsychological Science vol 2 no 6 pp 409ndash415 1991

[7] D A Norman Memory and Attention An Introduction toHuman Information Processing JohnWiley amp Sons HobokenNJ USA 1976

[8] T Raeder O Lizardo D Hachen and N V Chawla ldquoPre-dictors of short-term decay of cell phone contacts in a largescale communication networkrdquo Social Networks vol 33 no 4pp 245ndash257 2011

[9] P McLean Culture in Networks John Wiley amp SonsHoboken NJ USA 2016

[10] D C Bell B Belli-McQueen and A Haider ldquoPartner namingand forgetting recall of network membersrdquo Social Networksvol 29 no 2 pp 279ndash299 2007

[11] M E Brashears E Hoagland and E Quintane ldquoSex andnetwork recall accuracyrdquo Social Networks vol 44 pp 74ndash842016

[12] M E Brashears and E Quintane ldquo+e microstructures ofnetwork recall how social networks are encoded and rep-resented in human memoryrdquo Social Networks vol 41pp 113ndash126 2015

[13] J Feigenbaum S Kannan A McGregor S Suri and J ZhangldquoOn graph problems in a semi-streaming modelrdquo in Pro-ceedings of the International Colloquium on AutomataLanguages and Programming pp 531ndash543 Springer TurkuFinland July 2004

[14] S Sreenivasan K S Chan A Swami G Korniss andB K Szymanski ldquoInformation cascades in feed-based net-works of users with limited attentionrdquo IEEE Transactions onNetwork Science and Engineering vol 4 no 2 pp 120ndash1282017

[15] P Holme and J Saramaki ldquoTemporal networksrdquo PhysicsReports vol 519 no 3 pp 97ndash125 2012

[16] P Holme ldquoNetwork dynamics of ongoing social relation-shipsrdquo Europhysics Letters (EPL) vol 64 no 3 p 427 2003

[17] T Y Berger-Wolf and J Saia ldquoA framework for analysis ofdynamic social networksrdquo in Proceedings of the 12th ACMSIGKDD International Conference on Knowledge Discoveryand Data Mining pp 523ndash528 ACM Philadelphia PA USAAugust 2006

[18] G Kossinets and D J Watts ldquoEmpirical analysis of anevolving social networkrdquo Science vol 311 no 5757pp 88ndash90 2006

[19] J Moody ldquo+e importance of relationship timing for diffu-sionrdquo Social Forces vol 81 no 1 pp 25ndash56 2002

[20] V Sekara A Stopczynski and S Lehmann ldquoFundamentalstructures of dynamic social networksrdquo Proceedings of theNational Academy of Sciences vol 113 no 36 pp 9977ndash99822016

[21] J Xie M Chen and K S Boleslaw ldquoLabelRankT incrementalcommunity detection in dynamic networks via label propa-gationrdquo in Proceedings of the Workshop on Dynamic NetworksManagement and Mining pp 25ndash32 ACM New York NYUSA May 2013

[22] A Striegel S Liu L Meng C Poellabauer D Hachen andO Lizardo ldquoLessons learned from the NetSense smartphonestudyrdquo ACM SIGCOMM Computer Communication Reviewvol 43 no 4 pp 51ndash56 2013

[23] H Ebbinghaus Uber Das Gedachtnis Untersuchungen ZurExperimentellen Psychologie Duncker amp Humblot BerlinGermany 1885

[24] J M J Murre and J Dros ldquoReplication and analysis ofebbinghausrsquo forgetting curverdquo PLoS One vol 10 no 7 ArticleID e0120644 2015

[25] D L Scarborough C Cortese and H S ScarboroughldquoFrequency and repetition effects in lexical memoryrdquo Journalof Experimental Psychology Human Perception and Perfor-mance vol 3 no 1 p 1 1977

[26] E Nathan A S Pentland and D Lazer ldquoInferring friendshipnetwork structure by usingmobile phone datardquo Proceedings ofthe National Academy of Sciences vol 106 no 36pp 15274ndash15278 2009

[27] N Miller and D T Campbell ldquoRecency and primacy inpersuasion as a function of the timing of speeches andmeasurementsrdquo 9e Journal of Abnormal and Social Psy-chology vol 59 no 1 1959

[28] R A Calvo and S DrsquoMello ldquoAffect detection an interdis-ciplinary review of models methods and their applicationsrdquoIEEE Transactions on Affective Computing vol 1 no 1pp 18ndash37 2010

[29] M Lamont L Adler B Y Park and X Xiang ldquoBridgingcultural sociology and cognitive psychology in three con-temporary research programmesrdquo Nature Human Behaviourvol 1 no 12 p 866 2017

[30] M Nurek R Michalski and R Marian-Andrei ldquoHawkes-modeled telecommunication patterns reveal relationshipdynamics and personality traitsrdquo 2020 httpsarxivorgabs200902032

Complexity 13