Table Of Content

Entropy of dynamical social networks Kun Zhao,1 Márton Karsai,2 and Ginestra Bianconi1,∗ 1Physics Department, Northeastern University, Boston, Massachusetts, United States of America 2BECS, School of Science, Aalto University, Aalto, Finland (Dated: January 17, 2012) Human dynamical social networks encode information and are highly adaptive. To characterize theinformationencodedinthefastdynamicsofsocialinteractions,hereweintroducetheentropyof dynamicalsocialnetworks. Byanalysingalargedatasetofphone-callinteractionsweshowevidence that the dynamical social network has an entropy that depends on the time of the day in a typical week-day. Moreover we show evidence for adaptability of human social behavior showing data on durationofphone-callinteractionsthatsignificantlydeviatesfromthestatisticsofdurationofface- 2 to-faceinteractions. Thisadaptabilityofbehaviorcorrespondstoadifferentinformationcontentof 1 the dynamics of social human interactions. We quantify this information by the use of the entropy 0 of dynamical networks on realistic models of social interactions. a 2 n INTRODUCTION dynamics. In fact social networks are highly adaptive. a Indeed social ties can appear or disappear depending on J 6 Networks [1–5] encode information in the topology of thedynamicalprocessoccurringonthenetworkssuchas 1 their interactions. This is the main reason why networks epidemic spreading or opinion dynamics. Several models are ubiquitous in complexity theory and constitute the foradaptivesocialevolutionhavebeenproposedshowing ] underlying structures of social, technological and biolog- phase transitions in different universality classes [20–23]. h p icalsystems. Theinformationencodedinsocialnetworks Social ties have in addition to that a microscopic struc- - [6, 7] is essential to build strong collaborations [8] that tureconstitutedbyfastsocialinteractionsoftheduration c enhancetheperformanceofasociety,tobuildreputation ofaphonecallorofaface-to-faceinteraction. Dynamical o trust and to navigate [9] efficiently the networks. For socialnetworkscharacterizethesocialinteractionatthis s . these reasons social networks are small world [10] with fast time scale. For these dynamical networks new net- s c short average distance between the nodes but large clus- work measures are starting to be defined [24] and recent i tering coefficient. Therefore to understand how social works focus on the implication that the network dynam- s y network evolve, adapt and respond to external stimuli, ics has on percolation, epidemic spreading and opinion h we need to develop a new information theory of complex dynamics [25–29]. p social networks. Thanks to the availability of new extensive data on [ Recently,attentionhasbeenaddressedtoentropymea- a wide variety of human dynamics [30–34], human mo- 1 sures applied to email correspondence [11], static net- bility [15, 35, 36] and dynamical social networks [37], it v works [12–14] and mobility patterns [15]. New network has been recently recognized that many human activi- 2 entropy measures quantify the information encoded in ties [26] are bursty and not Poissonian. New data on 9 heterogenousstaticnetworks[12,14]. Informationtheory socialdynamical networksstart tobecollected withnew 2 toolssetthelimitofpredictabilityofhumanmobility[15]. technologiessuchasofRadiofrequencyIdentificationDe- 3 Still we lack methods to assess the information encoded vices [28, 38] and Bluetooth [31]. These technologies are . 1 in the dynamical social interaction networks. able to record the duration of social interactions and re- 0 Social networks are characterized by complex organi- port evidence for a bursty nature of social interaction 2 zational structures revealed by network community and characterized by a fat tail distribution of the duration 1 degree correlations [16]. These structures are sometimes of face-to face interactions. This bursty behavior of so- : v correlated with annotated features of the nodes or of the cial networks [1, 28, 38–41] is coexisting with modula- Xi links such as age, gender, and other annotated features tions coming from periodic daily (circadian rhythms) or of the links such as shared interests, family ties or com- weakly patterns [43]. The fact that this bursty behavior r a mon work locations [17, 18]. In a recent work [19] it is observed also in social interaction of simple animals, has been shown by studying social, technological and bi- in the motion of rodents [44], or in the use of words [45], ologicalnetworksthatthenetworkentropymeasurescan suggeststhattheunderlyingoriginofthisbehaviorisdic- assess how significant are the annotated features for the tatedbythebiologicalandneurologicalprocessesunder- network structure. lyingthedynamicsofthesocialinteraction. Toouropin- Moreover social networks evolve on many different ion this problem remains open: How much can humans time-scales and relevant information is encoded in their intentionally change the statistics of social interactions and the level of information encoded in the dynamics of their social networks, when they are interfacing with a new technology? ∗ [email protected] Inthispaperwetrytoaddressthisquestionbystudy- a PublishedinPLoSONE6(12): e28116(2011). ingthedynamicsofinteractionsthroughphonecallsand 2 comparingitwithface-to-faceinteractions. Weshowthat theentropyofdynamicalnetworksisabletoquantifythe informationencodedinthedynamicsofphone-callinter- actions during a typical week-day. Moreover we show evidence that human social behavior is highly adaptive andthatthedurationofface-to-faceinteractioninacon- ference follows a different distribution than duration of phone-calls. Wethereforehaveevidenceofanintentional capability of humans to change statistically their behav- iorwheninterfacingwiththetechnologyofmobilephone FIG.1. The dynamical social networks are composed communication. Finally we develop a model in order to by different dynamically changing groups of interact- quantify how much the entropy of dynamical networks ing agents. In panel (A) we allow only for groups of size changes if we allow modifications in the distribution of one or two as it typically happens in mobile phone commu- duration of the interactions. nication. In panel (B) we allow for groups of any size as in face-to-face interactions. RESULTS expected in the dynamical network model at time t and isgivenbyS =−(cid:104)logL(cid:105) thatwecanexplicitlyexpress Entropy of dynamical social networks |St as In this section we introduce the entropy of dynamical S =−(cid:88)p(g (t)=1|S )logp(g (t)=1|S ). social networks as a measure of information encoded in i1,i2,...,in t i1,i2,...,in t G their dynamics. Since we are interested in the dynamics (3) ofcontactsweassumetohaveaquenchedsocialnetwork According to the information theory results [46], if the G of friendships, collaborations or acquaintances formed entropy is vanishing, i.e. S =0 the network dynamics is by N agents and we allow a dynamics of social interac- regular and perfectly predictable, if the entropy is larger tions on this network. If two agents i,j are linked in the the number of future possible configurations is growing network they can meet and interact at each given time and the system is less predictable. If we model face-to- giving rise to the dynamical social network under study face interactions we have to allow the possible formation in this paper. If a set of agents of size N is connected of groups of any size, on the contrary, if we model the through the social network G the agents i1,i2,...in can mobile phone communication, we need to allow only for interact in a group of size n. Therefore at any given pairwise interactions. Therefore, if we define the adja- time the static network G will be partitioned in con- cency matrix of the network G as the matrix a , the log ij nected components or groups of interacting agents as likelihood takes the very simple expression given by shown in Fig 1. In order to indicate that a social interaction is occurring at time t in the group of agents L=(cid:89)p(g (t)=1|S )gi(t) (cid:89) p(g (t)=1|S )gij(t) i t ij t i ,i ,...,i and that these agents are not interacting w1ith2 othernagents, we write g (t) = 1 otherwise i ij|aij=1 i1,i2,...,in (4) we put g (t) = 0. Therefore each agent is inter- i1,i2,...,in with acting with one group of size n > 1 or non interacting (cid:88) (interacting with a group of size n = 1). Therefore at g (t)+ a g (t)=1, (5) i ij ij any given time j (cid:88) gi,i2,...,in(t)=1. (1) for every time t. The entropy is then given by G=(i,i2,...,in)|i∈G (cid:88) S =− p(g (t)=1|S )logp(g (t)=1|S ) i t i t where we indicate with G an arbitrary connected sub- i graph of G. The history St of the dynamical social net- −(cid:88)a p(g (t)=1|S )logp(g (t)=1|S ). (6) work is given by S = {g (t(cid:48))∀t(cid:48) < t}. If we in- ij ij t ij t dicated by p(g t (t)i1=,i2,.1..|,Sin) the probability that ij i1,i2,...,in t g (t) = 1 given the story S , the likelihood that i1,i2,...,in t at time t the dynamical networks has a group configura- Social dynamics and entropy of phone call tion g (t) is given by i1,i2,...,in interactions (cid:89) L= p(gi1,i2,...,in(t)=1|St)gi1,i2,...,in(t) (2) We have analyzed the call sequence of subscribers of G a major euroepan mobile service provider. We consid- The entropy S characterizes the logarithm of the typi- ered calls between users who at least once called each cal number of different group configurations that can be other during the examined 6 months period in order to 3 examine calls only reflecting trusted social interactions. binary social interactions mediated by technology, show The resulted event list consists of 633,986,311 calls be- different statistical features respect to face-to-face inter- tween 6,243,322 users. For the entropy calculation we actions. Thedistributionsofthetimesdescribinghuman selected562,337userswhoexecutedatleastonecallper activities are typically broad [26, 28, 30, 32, 38, 39], and a day during a week period. First of all we have studied areclosertopower-laws,whichlackacharacteristictime howtheentropyofthisdynamicalnetworkisaffectedby scale, than to exponentials. In particular in [38] there is circadian rhythms. We assign to each agent i = 1,2 a reporteddataonRadioFrequencyIdentificationdevices, number n = 1,2 indicating the size of the group where withtemporalresolutionof20s,showingthatbothdistri- i he/she belongs. If an agent i has coordination number butiondurationofface-to-facecontactsandinter-contact n =1he/sheisisolated,andifn =2he/sheisinteract- periods is fat tailed during conference venues. i i ing with a group of n=2 agents. We also assign to each agent i the variable ti indicating the last time at which 102 the coordination number n has changed. If we neglect A i 100 thefeatureofthenodes,themostsimpletransitionprob- atahbgeeilnidttiaeitsnat,shtiasattgeiinvncelnuatdbetysimfaoerptsrootmboaecbhmilaientmygeoprnhyis=e/ffhepecnrt(ssτtp,atrt)eesfeognritvaeinnn ∗τ∆(w)P(t)int111000---642 www===wwwmmmaaaxxx(((024---248%%%))) that he has been in his/her current state for a duration w=w (8-16%) max τ =t−ti. 10-8 w=wmax(16-32%) 0.12 10-1100-3 10-2 10-1 100 101 102 103 Normalized call duration ∆t /τ∗(w) int 100 0.1 10-3 B 10-3 C 0.08 ∆P(t)int111000-1--851 aaaaagggggeeeee:::::1246800000-----24681000000 ∆P(t)intd1100--85 fmemaleale N 10-13 10-11 S/0.06 100C1a0ll1 du1r0a2tion1 0∆3t (1s0e4c) 105 100 C1a0l1l du1r0a2tion1 0∆3t (1se0c4) 105 int int 0.04 FIG.3. Probabilitydistributionofdurationofphone- 0.02 calls. (A)Probabilitydistributionofdurationofphone-calls between two given persons connected by a link of weight w. 00 4 8 12 16 20 24 Thedatadependonthetypicalscaleτ(cid:63)(w)ofdurationofthe t(hrs) phone-call. (B) Probability distribution of duration of phone calls for people of different age. (C) Probability distribution of duration of phone-calls for people of different gender. The distributions shown in the panel (B) and (C) do not signifi- FIG. 2. Mean-field evaluation of the entropy of the cantly depend on the attributes of the nodes. dynamical social networks of phone calls communication in a typical week-day. In the nights the social dynamical network is more predictable. TABLEI.Typicaltimesτ(cid:63)(w)usedinthedatacollapse We have estimated the probability p (τ,t) in a typical n of Fig. 3. week-day. Usingthedataontheprobabilitiesp (τ,t)we n Weight of the link Typical time τ(cid:63)(w) in seconds (s) have calculated the entropy, estimated by a mean-field (0-2%) w 111.6 evaluation (Check Text S1) of the dynamical network as max (2-4%) w 237.8 a function of time in a typical week-day. The entropy of max (4-8%) w 334.4 the dynamical social network is reported in Fig. 2. It max (8-16%) w 492.0 max significantly changes during the day describing the fact (16-32%) w 718.8 max thatthepredictabilityofthephone-callnetworkschange as a function of time. In fact, as if the entropy of the dynamical network is smaller and the network is an a more predictable state. Here we analysed the above defined mobile-call event sequence performing the measurements on all the users for the entire 6 months time period. The distribution of Adaptive dynamics face-to face interactions and phone-calldurationsstronglydeviatesfromafat-taildis- phone call durations tribution. InFig. 3wereportthisdistributionsandshow that these distributions depend on the strength w of the In this section we report evidence of adaptive human interactions (total duration of contacts in the observed behavior by showing that the duration of phone calls, a period) but do not depend on the age, gender or type of 4 contract in a significant way. The distribution Pw(∆t ) ing the human social interaction. Disregarding for the in of duration of contacts within agents with strenght w is momenttheeffectsofcircadianrhythmsandweaklypat- well fitted by a Weibull distribution terns, a possible explanation of such results is given by mechanisms in which the decisions of the agents to form (cid:18) (cid:19) τ∗(w)Pw(∆tin)=Wβ x= τ(cid:63)∆(wt) = x1βe−1−1βx1−β. obrylreeaivnefoarcgemroeunptadryendamrivicesn, bthyamt ceamnorbyeesffuemctmsadriiczteadteind (7) the following statements: i) the longer an agent is in- with β = 0.47... The typical times τ∗(w) used for the teracting in a group the smaller is the probability that data collapse of Figure 3 are listed in Table I. The origin he/she will leave the group; ii) the longer an agent is iso- of this significant change in behavior of humans inter- lated the smaller is the probability that he/she will form actions could be due to the consideration of the cost of a new group. In particular, such reinforcement principle the interactions (although we are not in the position to implies that the probabilities p (τ,t) that an agent with n draw these conclusions (See Fig. 4 in which we compare coordinationnumbernchangeshis/herstatedependson distribution of duration of calls for people with different the time elapsed since his/her last change of state, i.e., typeofcontract)ormightdependonthedifferentnature p (τ,t) = f (τ). To ensure the reinforcement dynamics n n of the communication. The duration of a phone call is any function f (τ) which is a decreasing function of its n quite short and is not affected significantly by the cir- argument can be taken. In two recent papers[1, 41] the cadian rhythms of the population. On the contrary the face-to-face interactions have been realistically modelled duration of no-interaction periods is strongly affected by with the use of the reinforcement dynamics, by choosing periodic daily of weekly rhythms. The distribution of no-interaction periods can be fitted by a double power- f (τ)= bn . (8) law but also a single Weibull distribution can give a first n (τ +1) approximation to describe P(∆t ). In Fig. 5 we report no the distribution of duration of no-interaction periods in withgoodagreementwiththedatawhenwetookbn =b2 the day periods between 7AM and 2AM next day. The for n≥2 and b1 >0, b2 >0. typical times τ∗(k) used in Figure 5 are listed in Table In order to model the phone-call data studied in this II. paper we can always adopt the reinforcement dynamics but we need to modify the probability f (τ) by a n 0 parametrization with an additional parameter β ≤1. In 10 ordertobespecificinourmodelofmobile-phonecommu- nication, weconsiderasystemthatconsistsofN agents. 10-3 Correspondingtothemechanismofdailycellphonecom- munication, the agents can call each other to form a bi- ) nary interaction if they are neighbor in the social net- nt ∆ti 10-6 work. The social network is characterized by a given ( degree distribution p(k) and a given weight distribution P p(w). Each agent i is characterized by the size n =1,2 i 10-9 post-payed ofthegrouphe/shebelongstoandthelasttimeti he/she pre-payed has changed his/her state. Starting from random initial conditions, at each timestep δt=1/N we take a random 10-12 agent. If the agent is isolated he/she will change his/her 0 1 2 3 4 5 10 10 10 10 10 10 state with probability Call duration ∆t (sec) int b fβ(τ)= 1 (9) 1 (τ +1)β FIG.4. Probabilitydistributionofdurationofphone- callsforpeoplewithdifferenttypesofcontract. Nosig- with τ = t−t and b > 0. If he/she change his/her i 1 nificant change is observed that modifies the functional form statehe/shewillcalloneofhis/herneighborinthesocial of the distribution. network which is still not engaged in a telephone call. A non-interacting neighbor agent will pick up the phone with probability fβ(τ(cid:48)) where τ(cid:48) is the time he/she has 1 not been interacting. DISCUSSION If,onthecontrarytheagentiisinteracting,he/shewill changehis/herstatewithprobabilityfβ(τ|w)depending 2 The entropy of a realistic model of cell-phone ontheweightofthelinkandonthedurationofthephone interactions call. We will take in particular Thedataonface-to-faceandmobile-phoneinteractions b g(w) fβ(τ|w)= 2 (10) show that a reinforcement dynamics is taking place dur- 2 (τ +1)β 5 10-2 104 s 2 yy 10 10-4 dada 1 2 )no 100 ∆t)no 10-6 k∆P(t 10-2 kk==12 P( 10-8 ∗τ(k) 10-4 kk==48 10-6 k=16 -10 k=32 10 -8 10 100 102 103 105 107 10-6 10-4 10-3 10-1 100 101 103 No-interaction time ∆t (sec) ∆t /τ∗(k) no no FIG.5. Distribution of non-interaction times in the phone-call data. Thedistributionstronglydependsoncircadian rhythms. Thedistributionofrescaledtimedependsstronglyontheconnectivityofeachnode. Nodeswithhigherconnectivity k are typically non-interacting for a shorter typical time scale τ(cid:63)(k). evident when comparing the distribution of duration of TABLEII.Typicaltimesτ(cid:63)(k)usedinthedatacollapse phone-callswiththedurationofface-to-faceinteractions, of Fig. 5. canbeunderstoodasapossibilitytochangetheexponent Connectivity Typical time τ(cid:63)(k) in seconds (s) β regulating the duration of social interactions. k=1 158,594 k=2 118,047 Changes in the parameter β correspond to a different k=4 69,741 entropyofthedynamicalsocialnetwork. Solvinganalyt- k=8 39,082 ically this model we are able to evaluate the dynamical k=16 22,824 entropy as a function of β and b1. In Fig. 6 we report k=32 13,451 theentropyS ofthedynamicalsocialnetworkafunction of β and b in the annealed approximation and the large 1 network limit. In particular we have taken a network of where b2 > 0 and g(w) is a decreasing function of sizeN =2000withexponentialdegreedistributionofav- the weight w of the link. The distributions fβ(τ) and erage degree (cid:104)k(cid:105) = 6, weight distribution P(w) = Cw−2 1 f2β(τ|w) are parametrized by the parameter β ≤1. As β and function g(w) = 1/w and b2 = 0.05. Our aim in increases, the distribution of duration of contacts and Fig. 6 is to show only the effects on the entropy due duration of intercontact time become broader. These to the different distributions of duration of contacts and probabilities give rise to either Weibull distribution of non-interaction periods. Therefore we have normalized duration of interactions (if β <1) or power-law distribu- the entropy S with the entropy SR of a null model of tion of duration of interaction β = 1. Indeed for β < 1, social interactions in which the duration of groups are the probability Pw(τ) that a conversation between two Poisson distributed but the average time of interaction 2 nodeswithlinkweightw endsafteradurationτ isgiven andnoninteractiontimearethesameasinthemodelof by the Weibull distribution (See Text S1 for the details cell-phone communication. From Fig. 6 we observe that of the derivation) if we keep b1 constant, the ratio S/SR is a decreasing function of the parameter β indicating that the broader τ∗(w)P2w(τ)∝Wβ((τ +1)/τ(cid:63)(w)) (11) arethedistributionofprobabilityofdurationofcontacts thehigheristheinformationencodedinthedynamicsof with τ(cid:63)(w) = [2b2g(w)]−1/(1−β). This distribution well the networks. Therefore the heterogeneity in the distri- capture the distribution observed in mobile phone data butionofdurationofcontactsandno-interactionperiods and reported in Fig. 3 (for a discussion of the valid- implies higher level of information in the social network. ity of the annealed approximation for predictions on a The human adaptive behavior by changing the exponent quenched network see the Text S1 ). β in face-to-face interactions and mobile phone commu- If, instead of having β < 1 we have β = 1 the proba- nication effectively change the entropy of the dynamical bility distribution for duration of contacts is given by a network. power-law In conclusion, in the last ten years it has been recog- Pw(τ)∝(τ +1)−[2b2g(w)+1]. (12) nized that the vast majority of complex systems can be 2 describedbynetworksofinteractingunits. Networkthe- This distribution is comparable with the distribution ory has made tremendous progresses in this period and observed in face-to-face interaction during conference we have gained important insight into the microscopic venues [1, 41]. The adaptability of human behavior, properties of complex networks. Key statistical proper- 6 MATERIAL AND METHODS In order to describe the model of mobile phone com- 1 munication, we consider a system consisting of N agents 0.8 representing the mobile phone users. The agents are 0.6 interacting in a social network G representing social S/S R ties such as friendships, collaborations or acquaintances. 0.4 The network G is weighted with the weights indicating 0.2 the strength of the social ties between agents. We use 00 0.2 0.4 β 0.6 0.8 1 0 0.2 0.4 0.b61 0.8 1 Nkac1ktthe(dta0t,wta)ittdhtt0iamtnoeotdtheaenrroetaengoetnthteinsntinuercmaecbtteiirnmgoefatan(cid:48)gd∈enh(tatsvw,etintho+td1ien/gNtreer)e-. 0 0 Similarly we denote by Nk,k(cid:48),w(t ,t)dt the number of 2 0 0 connected agents (with degree respectively k and k(cid:48) and FIG. 6. Entropy S of social dynamical network weight of the link w) that at time t are interacting in model of pairwise communication normalized with phone call started at time t(cid:48) ∈(t ,t +1/N). The mean- the entropy S of a null model in which the ex- 0 0 R field equation for this model read, pected average duration of phone-calls is the same but the distribution of duration of phone-calls and ∂Nk(t ,t) non-interaction time are Poisson distributed. Thenet- 1 0 =−(1+ck)Nk(t ,t)f (t ,t)+Nπk (t)δ ∂t 1 0 1 0 21 tt0 work size is N =2000 the degree distribution of the network is exponential with average (cid:104)k(cid:105) = 6, the weight distribution ibs2p=(w0).0=5.CTwh−e2vaanludego(fwS)/isSRtakisendetpoenbdeingg(wo)n=thbe2/twwowpitah- ∂N2k,k∂(cid:48),wt(t0,t) =−2N2k,k(cid:48),w(t0,t)f2(t0,t|w)+Nπ1k2,k(cid:48),w(t)δtt0 rameters β,b . For every value of b the normalized entropy 1 1 (13) is smaller for β →1. where the constant c is given by (cid:80) (cid:82)tdt Nk(cid:48)(t ,t)f (t ,t) c= k(cid:48) 0 0 1 0 1 0 . (14) (cid:80) k(cid:48)(cid:82)tdt Nk(cid:48)(t ,t)f (t ,t) k(cid:48) 0 0 1 0 1 0 InEqs. (15)theratesπ (t)indicatetheaveragenumber pq tieshavebeenfoundtooccuruniversallyinthenetworks, of agents changing from state p = 1,2 to state q = 1,2 such as the small world properties and broad degree dis- at time t. These rates can be also expressed in a self- tributions. Moreover the local structure of networks has consistent way and the full system solved for any given been characterized by degree correlations, clustering co- choiceoff (t ,t)andf (t ,t|w)(SeeTextS1fordetails). 1 0 2 0 efficient,loopstructure,cliques,motifsandcommunities. The definition of the entropy of dynamical social net- Thelevelofinformationpresentinthesecharacteristicof worksofapairwisecommunicationmodel,isgivenbyEq. thenetworkcanbenowstudiedwiththetoolsofinforma- (6). Toevaluatetheentropyofdynamicalsocialnetwork tion theory. An additional fundamental aspect of social explicitly, we have to carry out the summations in Eq. networks is their dynamics. This dynamics encode for (6). Thesesums,willingeneraldependontheparticular information and can be modulated by adaptive human historyofthedynamicalsocialnetwork,butintheframe- behavior. In this paper we have introduced the entropy work of the model we study, in the large network limit of social dynamical networks and we have evaluated the will be dominated by their average value. In the follow- informationpresentindynamicaldataofphone-callcom- ing therefore we perform these sum in the large network munication. Byanalysingthephone-callinteractionnet- limit. The first summation in Eq. (6) denotes the av- works we have shown that the entropy of the network erage loglikelihood of finding at time t a non-interacting depends on the circadian rhythms. Moreover we have agent given a history S . We can distinguish between t shown that social networks are extremely adaptive and twoeventualsituationsoccurringattimet: (i)theagent are modified by the use of technologies. The statistics of has been non-interacting since a time t−τ, and at time duration of phone-call indeed is described by a Weibull t remains non-interacting; (ii) the agent has been inter- distribution that strongly differ from the distribution of acting with another agent since time t−τ, and at time face-to-faceinteractionsina conference. Finally wehave t the conversation is terminated by one of the two inter- evaluated how the information encoded in social dynam- acting agents.The second term in the right hand side of icalnetworkschangeifweallowaparametrizationofthe Eq. (6), denotes the average loglikelihood of finding two duration of contacts mimicking the adaptability of hu- agents in a connected pair at time t given a history S . t man behavior. Therefore the entropy of social dynami- There are two possible situations that might occur for cal networks is able to quantify how the social networks two interacting agents at time t: (iii) these two agents dynamicallychangeduringthedayandhowtheydynam- havebeennon-interacting, andtotime toneofthemde- ically adapt to different technologies. cides to form a connection with the other one; (iv) the 7 two agents have been interacting with each other since a scribers were made available to us. time t−τ, and they remain interacting at time t. Tak- ing into account all these possibilities we have been able to use the transition probability form different state and the number of agents in each state to evaluate the entropy of dynamical networks in the large network limit ACKNOWLEDGMENTS (For further details on the calculation see the Text S1). We thank A.-L. Barabási for his useful comments and ETHICS STATEMENT for the mobile call data used in this research. MK ac- knowledges the financial support from EUs 7th Frame- The dataset used in this study only involved de- workProgramsFET-OpentoICTeCollectiveprojectno. identified information and no details about the sub- 238597 [1] Dorogovtsev SN, Mendes JFF (2003) Evolution of net- [20] Davidsen J, Ebel H, Bornholdt S (2002) Emergence of works: From biological nets to the Internet and WWW. a small world from local interactions: Modeling acquain- Oxford Univ Press. tance networks. Phys Rev Lett 88: 128701. [2] Newman MEJ (2003) The structure and function of com- [21] Marsili M, Vega-Redondo F, Slanina F (2004) The rise plex networks. SIAM Rev 45: 157-256. and fall of a networked society: A formal model. Proc [3] BoccalettiS,LatoraV,MorenoY,ChavezM,HwangDU, Natl Acad Sci USA 101: 1439-1442. etal.(2006)Complexnetworks: Structureanddynamics. [22] Holme P, Newman MEJ (2006) Nonequilibrium phase Phys Rep 424: 175-308. transition in the coevolution of networks and opinions. [4] Caldarelli G (2007) Scale-Free Networks. Oxford Univ Phys Rev E 74: 056108. Press. [23] Vazquez F, Egu´ıluz VM, San-Miguel M (2008) Generic [5] BarratA,BarthélemyM,VespignaniA(2008).Dynamical absorbing transition in coevolution dynamics. Phys Rev processes on complex networks. Lett 100: 108702. [6] Granovetter M (1973) The strength in weak ties. Am J [24] Tang J, Scellato S, Musolesi M, Mascolo C, Latora V Socio 78: 1360-1380. (2009)Small-worldbehaviorintime-varyinggraphs. Phys [7] Wasserman S, Faust K (1994) Social Network Analysis: Rev E 81: 055101. Methods and applications. Cambridge Univ Press. [25] Holme P (2005) Network reachability of real-world con- [8] Newman MEJ (2001) The structure of scientific collabo- tact sequences. Phys Rev E 71: 046119. ration networks. Proc Natl Acad Sci USA 98: 404-409. [26] Va´zquez A, Raćz B, Lukacs A, Baraba`si AL (2007) Im- [9] KleinbergJM(2000)Navigationinasmallworld. Nature pactofnon-poissonianactivitypatternsonspreadingpro- 406: 845. cesses. Phys Rev Lett 98: 158702. [10] Watts DJ, Strogatz SH (1998) Collective dynamics of [27] ParshaniR,DickisonM,CohenR,StanleyHE,HavlinS small-world networks. Nature 393: 440-442. (2009)Dynamicnetworksanddirectedpercolation. Euro- [11] Eckmann JP, Moses E, Sergi D (2004) Entropy of dia- phys Lett 90: 38004. logues creates coherent structures in e-mail traffic. Proc [28] IsellaL,StehléJ,BarratA,CattutoC,PintonJF,etal. Natl Acad Sci USA 101: 14333. (2011)What’sinacrowd? analysisofface-to-facebehav- [12] Bianconi G (2008) The entropy of randomized network ioral networks. J Theor Biol 271: 166-180. ensembles. Europhys Lett 81: 28005. [29] Karsai M, Kivela¨ M, Pan RK, Kaski K, Kertész J, et al. [13] Bianconi G, Coolen ACC, Perez-Vicente CJ (2008) (2011) Small but slow world: How network topology and Entropies of complex networks with hierarchically con- burstiness slow down spreading. Phys Rev E 83: 025102. strained topologies. Phys Rev E 78: 016114. [30] Baraba´si AL (2005) The origin of bursts and heavy tails [14] Anand K, Bianconi G (2009) Entropy measures for net- in humans dynamics. Nature 435: 207-211. works: Toward an information theory of complex topolo- [31] Eagle N, Pentland AS (2006) Reality mining: sensing gies. Phys Rev E 80: 045102. complex social systems. Personal Ubiquitous Comput 10: [15] Song C, Qu Z, Blumm N, Baraba´si AL (2010) Limits of 255-268. predictabilityinhumanmobility. Science327: 1018-1021. [32] RybskiD,BuldyrevSV,HavlinS,LiljerosF,MakseHA [16] Castellano C, Fortunato S, Loreto V (2009) Statistical (2009) Scaling laws of human interaction activity. Proc physics of social dynamics. Rev Mod Phys 81: 591-646. Natl Acad Sci USA 106: 12640-12645. [17] PallaG,Baraba´siAL,VicsekT(2007)Quantifyingsocial [33] Malmgren RD, Stouffer DB, Motter AR, Amaral LA group evolution. Nature 446: 664-667. (2008) A poissonian explanation for heavy tails in e-mail [18] AhnYY,BagrowJP,LehmannS(2010)Linkcommuni- communication. Proc Natl Acad Sci USA 105: 18153- tiesrevealmultiscalecomplexityinnetworks. Nature466: 18158. 761-764. [34] MalmgrenRD,StoufferDB,CampanharoASLO,Nunes- [19] Bianconi G, Pin P, Marsili M (2009) Assessing the rele- Amaral LA (2009) On universality in human correspon- vance of node features for network structure. Proc Natl dence activity. Science 325: 1696-1700. Acad Sci USA 106: 11433-11438. 8 [35] Brockmann D, Hufnagel L, Geisel T (2006) The scaling networks. Comp Net 52: 2842-2858. laws of human travel. Nature 439: 462-465. [41] Barrat JSA, Bianconi G (2010) Dynamical and bursty [36] Gonza´lez MC, Hidalgo AC, Baraba´si AL (2008) Under- interactions in social networks. Phys Rev E 81: 035101. standingindividualhumanmobilitypatterns. Nature453: [42] Zhao K, Stehlé J, Bianconi G, Barrat A (2010) Social 779-782. network dynamics of face-to-face interactions. Phys Rev [37] Onnela JP, Sarama¨ki J, Hyvo¨nen J, Szabo´ G, Lazer D, E 83: 056109. et al. (2007) Structure and tie strengths in mobile com- [43] Jo HH, Karsai M, Kertész J, Kaski K (2011) Circadian municationnetworks. ProcNatlAcadSciUSA104: 7332- pattern and burstiness in human communication activity. 7336. arXiv:11010377 . [38] CattutoC,denBroeckWV,BarratA,ColizzaV,Pinton [44] Anteneodo C, Chialvo DR (2009) Unraveling the fluctu- JF, et al. (2010) Dynamics of person-to-person interac- ations of animal motor activity. Chaos 19: 033123. tionsfromdistributedRFIDsensornetworks. PLoSONE [45] Altmann EG, Pierrehumbert JB, Motter AE (2009) Be- 5: e11596. yondwordfrequency: Bursts,lulls,andscalinginthetem- [39] HuiP,ChaintreauA,ScottJ,GassR,CrowcroftJ,etal. poral distributions of words. PLoS ONE 4: e7678. (2005) Pocket switched networks and human mobility in [46] Cover T, Thomas JA (2006) Elements of Information conference environments. In: Proceedings of the 2005 Theory. Wiley-Interscience. ACM SIGCOMM workshop on Delay-tolerant network- ing(Philadelphia, PA). pp. 244-251. [40] Borgnat ASP, Fleury E, Guillaume JL, Robardet C (2008) Description and simulation of dynamic mobility 9 Entropy of dynamical networks Supporting Informations K. Zhao, M. Karsai and G. Bianconi I. THE PROPOSED MODEL OF CELLPHONE COMMUNICATION A. Dynamical social network for pairwise communication We consider a system consisting of N agents representing the mobile phone users. The agents are interacting in a social network G representing social ties such as friendships, collaborations or acquaintances. The network G is weighted with the weights indicating the strength of the social ties between agents. To model the mechanism of cellphone communication, the agents can call their neighbors in the social network G forming groups of interacting agents of size two. Since at any given time a call can be initiated or terminated the network is highly dynamical. We assign to each agent i = 1,2,...,N a coordination number n to indicate his/her state. If n = 1 the agent is i i non-interacting, and if n =2 the agent is in a mobile phone connection with another agent. The dynamical process i of the model at each time step t can be described explicitly by the following algorithm: (1) An agent i is selected randomly at time t. (2) The subsequent action of agent i depends on his/her current state (i.e. n ): i (i) If n = 1, he/she will call one of his/her non-interacting neighbors j of G with probability f (t ,t) where i 1 i t denotes the last time at which agent i has changed his/her state. Once he/she decides to call, agent j i will be chosen randomly in between the neighbors of i with probability proportional to f (t ,t), therefore 1 j the coordination numbers of agent i and j are updated according to the rule n →2 and n →2. i j (ii) If n = 2, he/she will terminate his/her current connection with probability f (t ,t|w ) where w is the i 2 i ij ij weightofthelinkbetweeniandtheneighborj thatisinteractingwithi. Oncehe/shedecidestoterminate the connection, the coordination numbers are then updated according to the rule n →1 and n →1. i j (3) Time t is updated as t→t+1/N (initially t=0) and the process is iterated until t=T . max B. General solution to the model In order to solve the model analytically, we assume the quenched network G to be annealed and uncorrelated. Therefore we assume that at each time the network is rewired keeping the degree distribution p(k) and the weight distributionp(w)constant. Moreoverwesolvethemodelinthecontinuoustimelimit.Thereforewealwaysapproximate the sum over time-steps of size δt=1/N by integrals over time. We use Nk(t ,t)dt to denote the number of agents 1 0 0 withdegreekthatattimetarenotinteractingandhavenotinteractedwithanotheragentsincetimet(cid:48) ∈(t ,t +1/N). 0 0 SimilarlywedenotebyNk,k(cid:48),w(t ,t)dt thenumberofconnectedagents(withdegreerespectivelyk andk(cid:48) andweight 2 0 0 of the link w) that at time t are interacting in phone call started at time t(cid:48) ∈ (t ,t +1/N). Consistently with the 0 0 annealed approximation the probability that an agent with degree k is called is proportional to its degree. Therefore the rate equations of the model are given by ∂Nk(t ,t) 1 0 =−Nk(t ,t)f (t ,t)−ckNk(t ,t)f (t ,t)+Nπk (t)δ ∂t 1 0 1 0 1 0 1 0 21 tt0 ∂N2k,k(cid:48),w(t0,t) =−2Nk,k(cid:48),w(t ,t)f (t ,t|w)+Nπk,k(cid:48),w(t)δ (15) ∂t 2 0 2 0 12 tt0 where the constant c is given by (cid:80) (cid:82)tdt Nk(cid:48)(t ,t)f (t ,t) c= k(cid:48) 0 0 1 0 1 0 . (16) (cid:80) k(cid:48)(cid:82)tdt Nk(cid:48)(t ,t)f (t ,t) k(cid:48) 0 0 1 0 1 0 10 In Eqs. (15) the rates π (t) indicate the average number of agents changing from state p = 1,2 to state q = 1,2 at pq time t. These rates can be also expressed in a self-consistent way as πk (t)= 2 (cid:88)(cid:90) tdt f (t ,t|w)Nk,k(cid:48),w(t ,t) 21 N 0 2 0 2 0 k(cid:48),w 0 πk,k(cid:48),w(t)= P(w)(cid:90) tdt (cid:90) tdt(cid:48)Nk(t ,t)Nk(cid:48)(t(cid:48),t)f (t ,t)f (t(cid:48),t)(k+k(cid:48)) (17) 12 CN 0 0 1 0 1 0 1 0 1 0 0 0 where the constant C is given by (cid:90) t C =(cid:88) dt k(cid:48)Nk(cid:48)(t ,t)f (t ,t). (18) 0 1 0 1 0 k(cid:48) 0 The solution to Eqs. (15) is given by N1k(t0,t)=Nπ2k1(t0)e−(1+ck)(cid:82)tt0f1(t0,t)dt N2k,k(cid:48),w(t0,t)=Nπ1k2,k(cid:48),w(t0)e−2(cid:82)tt0f2(t0,t|w)dt (19) whichmustsatisfytheself-consistentconstraintsEqs. (17)andtheconservationofthenumberofagentswithdifferent degree (cid:90) dt (cid:2)Nk(t ,t)+(cid:88)Nk,k(cid:48),w(t ,t)(cid:3)=Np(k). (20) 0 1 0 2 0 k(cid:48),w InthefollowingwewilldenotebyPk(t ,t)theprobabilitydistributionthatanagentwithdegreek isnon-interacting 1 0 for a period from t to t and by Pw(t ,t) the probability that a connection of weight w at time t is active since time 0 2 0 t . It is immediate to see that these distributions are given by the number of individual in a state n=1,2 multiplied 0 by the probability of having a change of state, i.e. Pk(t ,t)=(1+ck)f (t ,t)Nk(t ,t) 1 0 1 0 1 0 Pw(t ,t)=2f (t ,t|w)(cid:88)Nk,k(cid:48),w(t ,t). (21) 2 0 2 0 2 0 k,k(cid:48) C. Stationary solution with specific f (t ,t) and f (t ,t) 1 0 2 0 In order to capture the behavior of the empirical data with a realistic model, we have chosen b f (t ,t)=f (τ)= 1 1 0 1 (1+τ)β b g(w) f (t ,t|w)=f (τ|w)= 2 (22) 2 0 2 (1+τ)β with parameters b > 0, b > 0, 0 ≤ β ≤ 1 and arbitrary positive function g(w). In Eqs. (22), τ is the duration 1 2 time elapsed since the agent has changed his/her state for the last time (i.e. τ = t−t ). The functions of f (τ) 0 1 and f (τ|w) are decreasing function of their argument τ reflecting the reinforcement dynamics discussed in the main 2 body of the paper. The function g(w) is generally chosen as a decreasing function of w, indicating that connected agentswithastrongerweightoflinkinteracttypicallyforalongertime. Weareespeciallyinterestedinthestationary state solution of the dynamics. In this regime we have that for large times t (cid:29) 1 the distribution of the number of agents is only dependent on τ. Moreover the transition rates π (t) also converge to a constant independent of t in pq the stationary state. Therefore the solution of the stationary state will satisfy Nk(t ,t)=Nk(τ) 1 0 1 Nk,k(cid:48),w(t ,t)=Nk,k(cid:48),w(τ) 2 0 2 π (t)=π . (23) pq pq The necessary condition for the stationary solution to exist is that the summation of self-consistent constraints given by Eq. (16) and Eq. (18) together with the conservation law Eq. (20) converge under the stationary assumptions Eqs. (23). The convergence depends on the value of the parameters b , b , β and the choice of function g(w). In 0 1 particular, when 0 ≤ β < 1, the convergence is always satisfied. In the following subsections, we will characterize further the stationary state solution of this model in different limiting cases.