Entropy of dynamical social networks
Human dynamical social networks encode information and are highly adaptive. To characterize the information encoded in the fast dynamics of social interactions, here we introduce the entropy of dynamical social networks. By analysing a large dataset of phone-call interactions we show evidence 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 duration of phone-call interactions that significantly deviates from the statistics of duration of face-to-face interactions. This adaptability of behavior corresponds to a different information content of the dynamics of social human interactions. We quantify this information by the use of the entropy of dynamical networks on realistic models of social interactions. 111Published in PLoS ONE 6(12): e28116 (2011).
Networks Dorogovtsev:2003 (); Newman:2003 (); Boccaletti:2006 (); Caldarelli:2007 (); Barrat:2008 () encode information in the topology of their interactions. This is the main reason why networks are ubiquitous in complexity theory and constitute the underlying structures of social, technological and biological systems. The information encoded in social networks Granovetter:1973 (); Wasserman:1994 () is essential to build strong collaborations Newman:2001 () that enhance the performance of a society, to build reputation trust and to navigate Kleinberg () efficiently the networks. For these reasons social networks are small world WS () with short average distance between the nodes but large clustering coefficient. Therefore to understand how social network evolve, adapt and respond to external stimuli, we need to develop a new information theory of complex social networks.
Recently, attention has been addressed to entropy measures applied to email correspondence Eckmann:2004 (), static networks Bianconi:2008 (); Bianconi:2008b (); Anand:2009 () and mobility patterns Song:2010 (). New network entropy measures quantify the information encoded in heterogenous static networks Bianconi:2008 (); Anand:2009 (). Information theory tools set the limit of predictability of human mobilitySong:2010 (). Still we lack methods to assess the information encoded in the dynamical social interaction networks.
Social networks are characterized by complex organizational structures revealed by network community and degree correlations Fortunato (). These structures are sometimes correlated with annotated features of the nodes or of the links such as age, gender, and other annotated features of the links such as shared interests, family ties or common work locations Palla:2007 (); Lehmann (). In a recent work Bianconi:2009 () it has been shown by studying social, technological and biological networks that the network entropy measures can assess how significant are the annotated features for the network structure.
Moreover social networks evolve on many different time-scales and relevant information is encoded in their dynamics. In fact social networks are highly adaptive. Indeed social ties can appear or disappear depending on the dynamical process occurring on the networks such as epidemic spreading or opinion dynamics. Several models for adaptive social evolution have been proposed showing phase transitions in different universality classes Bornholdt:2002 (); Marsili:2004 (); Holme:2006 (); MaxiSanMiguel:2008 (). Social ties have in addition to that a microscopic structure constituted by fast social interactions of the duration of a phone call or of a face-to-face interaction. Dynamical social networks characterize the social interaction at this fast time scale. For these dynamical networks new network measures are starting to be defined Latora:2009 () and recent works focus on the implication that the network dynamics has on percolation, epidemic spreading and opinion dynamics Holme:2005 (); Vazquez:2007 (); Havlin:2009 (); Isella:2011 (); Karsai:2011a ().
Thanks to the availability of new extensive data on a wide variety of human dynamics Barabasi:2005 (); Eagle:2006 (); Rybski:2009 (); Amaral:2008 (); Amaral:2009 (), human mobility Brockmann:2006 (); Gonzalez:2008 (); Song:2010 () and dynamical social networks Onnela:2007 (), it has been recently recognized that many human activities Vazquez:2007 () are bursty and not Poissonian. New data on social dynamical networks start to be collected with new technologies such as of Radio frequency Identification Devices Cattuto:2010 (); Isella:2011 () and Bluetooth Eagle:2006 (). These technologies are able to record the duration of social interactions and report evidence for a bursty nature of social interaction characterized by a fat tail distribution of the duration of face-to face interactions. This bursty behavior of social networks Hui:2005 (); Scherrer:2008 (); Cattuto:2010 (); Isella:2011 (); Stehle:2010 (); Zhao:2011 () is coexisting with modulations coming from periodic daily (circadian rhythms) or weakly patterns Karsai:2011b (). The fact that this bursty behavior is observed also in social interaction of simple animals, in the motion of rodents Chialvo (), or in the use of words Motter (), suggests that the underlying origin of this behavior is dictated by the biological and neurological processes underlying the dynamics of the social interaction. To our opinion this problem remains open: How much can humans 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?
In this paper we try to address this question by studying the dynamics of interactions through phone calls and comparing it with face-to-face interactions. We show that the entropy of dynamical networks is able to quantify the information encoded in the dynamics of phone-call interactions during a typical week-day. Moreover we show evidence that human social behavior is highly adaptive and that the duration of face-to-face interaction in a conference follows a different distribution than duration of phone-calls. We therefore have evidence of an intentional capability of humans to change statistically their behavior when interfacing with the technology of mobile phone communication. Finally we develop a model in order to quantify how much the entropy of dynamical networks changes if we allow modifications in the distribution of duration of the interactions.
Entropy of dynamical social networks
In this section we introduce the entropy of dynamical social networks as a measure of information encoded in their dynamics. Since we are interested in the dynamics of contacts we assume to have a quenched social network of friendships, collaborations or acquaintances formed by agents and we allow a dynamics of social interactions on this network. If two agents are linked in the network they can meet and interact at each given time giving rise to the dynamical social network under study in this paper. If a set of agents of size is connected through the social network the agents can interact in a group of size . Therefore at any given time the static network will be partitioned in connected components or groups of interacting agents as shown in Fig 1. In order to indicate that a social interaction is occurring at time in the group of agents and that these agents are not interacting with other agents, we write otherwise we put . Therefore each agent is interacting with one group of size or non interacting (interacting with a group of size ). Therefore at any given time
where we indicate with an arbitrary connected subgraph of . The history of the dynamical social network is given by . If we indicated by the probability that given the story , the likelihood that at time the dynamical networks has a group configuration is given by
The entropy characterizes the logarithm of the typical number of different group configurations that can be expected in the dynamical network model at time and is given by that we can explicitly express as
According to the information theory results Cover:2006 (), if the entropy is vanishing, i.e. the network dynamics is regular and perfectly predictable, if the entropy is larger the number of future possible configurations is growing and the system is less predictable. If we model face-to-face interactions we have to allow the possible formation of groups of any size, on the contrary, if we model the mobile phone communication, we need to allow only for pairwise interactions. Therefore, if we define the adjacency matrix of the network as the matrix , the log likelihood takes the very simple expression given by
for every time . The entropy is then given by
Social dynamics and entropy of phone call interactions
We have analyzed the call sequence of subscribers of a major euroepan mobile service provider. We considered calls between users who at least once called each other during the examined months period in order to examine calls only reflecting trusted social interactions. The resulted event list consists of calls between users. For the entropy calculation we selected users who executed at least one call per a day during a week period. First of all we have studied how the entropy of this dynamical network is affected by circadian rhythms. We assign to each agent a number indicating the size of the group where he/she belongs. If an agent has coordination number he/she is isolated, and if he/she is interacting with a group of agents. We also assign to each agent the variable indicating the last time at which the coordination number has changed. If we neglect the feature of the nodes, the most simple transition probabilities that includes for some memory effects present in the data, is given by a probability for an agent in state at time to change his/her state given that he has been in his/her current state for a duration .
We have estimated the probability in a typical week-day. Using the data on the probabilities we have calculated the entropy, estimated by a mean-field evaluation (Check Text S1) of the dynamical network as a function of time in a typical week-day. The entropy of the dynamical social network is reported in Fig. 2. It significantly changes during the day describing the fact that the predictability of the phone-call networks change 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.
Adaptive dynamics face-to face interactions and phone call durations
In this section we report evidence of adaptive human behavior by showing that the duration of phone calls, a binary social interactions mediated by technology, show different statistical features respect to face-to-face interactions. The distributions of the times describing human activities are typically broad Barabasi:2005 (); Vazquez:2007 (); Hui:2005 (); Rybski:2009 (); Cattuto:2010 (); Isella:2011 (), and are closer to power-laws, which lack a characteristic time scale, than to exponentials. In particular in Cattuto:2010 () there is reported data on Radio Frequency Identification devices, with temporal resolution of 20s, showing that both distribution duration of face-to-face contacts and inter-contact periods is fat tailed during conference venues.
|Weight of the link||Typical time in seconds (s)|
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 phone-call durations strongly deviates from a fat-tail distribution. In Fig. 3 we report this distributions and show that these distributions depend on the strength of the interactions (total duration of contacts in the observed period) but do not depend on the age, gender or type of contract in a significant way. The distribution of duration of contacts within agents with strenght is well fitted by a Weibull distribution
with . The typical times used for the data collapse of Figure 3 are listed in Table 1. The origin of this significant change in behavior of humans interactions could be due to the consideration of the cost of the interactions (although we are not in the position to draw these conclusions (See Fig. 4 in which we compare distribution of duration of calls for people with different type of contract) or might depend on the different nature of the communication. The duration of a phone call is quite short and is not affected significantly by the circadian rhythms of the population. On the contrary the duration of no-interaction periods is strongly affected by periodic daily of weekly rhythms. The distribution of no-interaction periods can be fitted by a double power-law but also a single Weibull distribution can give a first approximation to describe In Fig. 5 we report the distribution of duration of no-interaction periods in the day periods between 7AM and 2AM next day. The typical times used in Figure 5 are listed in Table 2.
The entropy of a realistic model of cell-phone interactions
The data on face-to-face and mobile-phone interactions show that a reinforcement dynamics is taking place during the human social interaction. Disregarding for the moment the effects of circadian rhythms and weakly patterns, a possible explanation of such results is given by mechanisms in which the decisions of the agents to form or leave a group are driven by memory effects dictated by reinforcement dynamics, that can be summarized in the following statements: i) the longer an agent is interacting in a group the smaller is the probability that he/she will leave the group; ii) the longer an agent is isolated the smaller is the probability that he/she will form a new group. In particular, such reinforcement principle implies that the probabilities that an agent with coordination number changes his/her state depends on the time elapsed since his/her last change of state, i.e., . To ensure the reinforcement dynamics any function which is a decreasing function of its argument can be taken. In two recent papersStehle:2010 (); Zhao:2011 () the face-to-face interactions have been realistically modelled with the use of the reinforcement dynamics, by choosing
with good agreement with the data when we took for and , .
In order to model the phone-call data studied in this paper we can always adopt the reinforcement dynamics but we need to modify the probability by a parametrization with an additional parameter . In order to be specific in our model of mobile-phone communication, we consider a system that consists of agents. Corresponding to the mechanism of daily cellphone communication, the agents can call each other to form a binary interaction if they are neighbor in the social network. The social network is characterized by a given degree distribution and a given weight distribution . Each agent is characterized by the size of the group he/she belongs to and the last time he/she has changed his/her state. Starting from random initial conditions, at each timestep we take a random agent. If the agent is isolated he/she will change his/her state with probability
with and . If he/she change his/her state he/she will call one of his/her neighbor in the social network which is still not engaged in a telephone call. A non-interacting neighbor agent will pick up the phone with probability where is the time he/she has not been interacting.
|Connectivity||Typical time in seconds (s)|
If, on the contrary the agent is interacting, he/she will change his/her state with probability depending on the weight of the link and on the duration of the phone call. We will take in particular
where and is a decreasing function of the weight of the link. The distributions and are parametrized by the parameter . As increases, the distribution of duration of contacts and duration of intercontact time become broader. These probabilities give rise to either Weibull distribution of duration of interactions (if ) or power-law distribution of duration of interaction . Indeed for , the probability that a conversation between two nodes with link weight ends after a duration is given by the Weibull distribution (See Text S1 for the details of the derivation)
with . This distribution well capture the distribution observed in mobile phone data and reported in Fig. 3 (for a discussion of the validity of the annealed approximation for predictions on a quenched network see the Text S1 ).
If, instead of having we have the probability distribution for duration of contacts is given by a power-law
This distribution is comparable with the distribution observed in face-to-face interaction during conference venues Stehle:2010 (); Zhao:2011 (). The adaptability of human behavior, evident when comparing the distribution of duration of phone-calls with the duration of face-to-face interactions, can be understood as a possibility to change the exponent regulating the duration of social interactions.
Changes in the parameter correspond to a different entropy of the dynamical social network. Solving analytically this model we are able to evaluate the dynamical entropy as a function of and . In Fig. 6 we report the entropy of the dynamical social network a function of and in the annealed approximation and the large network limit. In particular we have taken a network of size with exponential degree distribution of average degree , weight distribution and function and . Our aim in Fig. 6 is to show only the effects on the entropy due to the different distributions of duration of contacts and non-interaction periods. Therefore we have normalized the entropy with the entropy of a null model of social interactions in which the duration of groups are Poisson distributed but the average time of interaction and non interaction time are the same as in the model of cell-phone communication. From Fig. 6 we observe that if we keep constant, the ratio is a decreasing function of the parameter indicating that the broader are the distribution of probability of duration of contacts the higher is the information encoded in the dynamics of the networks. Therefore the heterogeneity in the distribution of duration of contacts and no-interaction periods implies higher level of information in the social network. The human adaptive behavior by changing the exponent in face-to-face interactions and mobile phone communication effectively change the entropy of the dynamical network.
In conclusion, in the last ten years it has been recognized that the vast majority of complex systems can be described by networks of interacting units. Network theory has made tremendous progresses in this period and we have gained important insight into the microscopic properties of complex networks. Key statistical properties have been found to occur universally in the networks, such as the small world properties and broad degree distributions. Moreover the local structure of networks has been characterized by degree correlations, clustering coefficient, loop structure, cliques, motifs and communities. The level of information present in these characteristic of the network can be now studied with the tools of information theory. An additional fundamental aspect of social networks is their dynamics. This dynamics encode for information and can be modulated by adaptive human behavior. In this paper we have introduced the entropy of social dynamical networks and we have evaluated the information present in dynamical data of phone-call communication. By analysing the phone-call interaction networks we have shown that the entropy of the network depends on the circadian rhythms. Moreover we have shown that social networks are extremely adaptive and are modified by the use of technologies. The statistics of duration of phone-call indeed is described by a Weibull distribution that strongly differ from the distribution of face-to-face interactions in a conference. Finally we have evaluated how the information encoded in social dynamical networks change if we allow a parametrization of the duration of contacts mimicking the adaptability of human behavior. Therefore the entropy of social dynamical networks is able to quantify how the social networks dynamically change during the day and how they dynamically adapt to different technologies.
Material and Methods
In order to describe the model of mobile phone communication, we consider a system consisting of agents representing the mobile phone users. The agents are interacting in a social network representing social ties such as friendships, collaborations or acquaintances. The network is weighted with the weights indicating the strength of the social ties between agents. We use to denote the number of agents with degree that at time are not interacting and have not interacted with another agent since time . Similarly we denote by the number of connected agents (with degree respectively and and weight of the link ) that at time are interacting in phone call started at time . The mean-field equation for this model read,
where the constant is given by
In Eqs. the rates indicate the average number of agents changing from state to state at time . These rates can be also expressed in a self-consistent way and the full system solved for any given choice of and (See Text S1 for details).
The definition of the entropy of dynamical social networks of a pairwise communication model, is given by Eq. (6). To evaluate the entropy of dynamical social network explicitly, we have to carry out the summations in Eq. . These sums, will in general depend on the particular history of the dynamical social network, but in the framework of the model we study, in the large network limit will be dominated by their average value. In the following therefore we perform these sum in the large network limit. The first summation in Eq. denotes the average loglikelihood of finding at time a non-interacting agent given a history . We can distinguish between two eventual situations occurring at time : (i) the agent has been non-interacting since a time , and at time remains non-interacting; (ii) the agent has been interacting with another agent since time , and at time the conversation is terminated by one of the two interacting agents.The second term in the right hand side of Eq. , denotes the average loglikelihood of finding two agents in a connected pair at time given a history . There are two possible situations that might occur for two interacting agents at time : (iii) these two agents have been non-interacting, and to time one of them decides to form a connection with the other one; (iv) the two agents have been interacting with each other since a time , and they remain interacting at time . Taking 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 (For further details on the calculation see the Text S1).
The dataset used in this study only involved de-identified information and no details about the subscribers were made available to us.
We thank A.-L. Barabási for his useful comments and for the mobile call data used in this research. MK acknowledges the financial support from EUâs 7th Framework Programâs FET-Open to ICTeCollective project no. 238597
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Entropy of dynamical networks
K. Zhao, M. Karsai and G. Bianconi
Appendix A The proposed model of cellphone communication
a.1 Dynamical social network for pairwise communication
We consider a system consisting of agents representing the mobile phone users. The agents are interacting in a social network representing social ties such as friendships, collaborations or acquaintances. The network 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 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 a coordination number to indicate his/her state. If the agent is non-interacting, and if the agent is in a mobile phone connection with another agent. The dynamical process of the model at each time step can be described explicitly by the following algorithm:
An agent is selected randomly at time .
The subsequent action of agent depends on his/her current state (i.e. ):
If , he/she will call one of his/her non-interacting neighbors of with probability where denotes the last time at which agent has changed his/her state. Once he/she decides to call, agent will be chosen randomly in between the neighbors of with probability proportional to , therefore the coordination numbers of agent and are updated according to the rule and .
If , he/she will terminate his/her current connection with probability where is the weight of the link between and the neighbor that is interacting with . Once he/she decides to terminate the connection, the coordination numbers are then updated according to the rule and .
Time is updated as (initially ) and the process is iterated until .
a.2 General solution to the model
In order to solve the model analytically, we assume the quenched network to be annealed and uncorrelated. Therefore we assume that at each time the network is rewired keeping the degree distribution and the weight distribution constant. Moreover we solve the model in the continuous time limit.Therefore we always approximate the sum over time-steps of size by integrals over time. We use to denote the number of agents with degree that at time are not interacting and have not interacted with another agent since time . Similarly we denote by the number of connected agents (with degree respectively and and weight of the link ) that at time are interacting in phone call started at time . Consistently with the annealed approximation the probability that an agent with degree is called is proportional to its degree. Therefore the rate equations of the model are given by
where the constant is given by
In Eqs. the rates indicate the average number of agents changing from state to state at time . These rates can be also expressed in a self-consistent way as
where the constant is given by
The solution to Eqs. (15) is given by
which must satisfy the self-consistent constraints Eqs. (17) and the conservation of the number of agents with different degree
In the following we will denote by the probability distribution that an agent with degree is non-interacting for a period from to and by the probability that a connection of weight at time is active since time . It is immediate to see that these distributions are given by the number of individual in a state multiplied by the probability of having a change of state, i.e.
a.3 Stationary solution with specific and
In order to capture the behavior of the empirical data with a realistic model, we have chosen
with parameters , , and arbitrary positive function . In Eqs. , is the duration time elapsed since the agent has changed his/her state for the last time (i.e. ). The functions of and are decreasing function of their argument reflecting the reinforcement dynamics discussed in the main body of the paper. The function is generally chosen as a decreasing function of , indicating that connected agents with a stronger weight of link interact typically for a longer time. We are especially interested in the stationary state solution of the dynamics. In this regime we have that for large times the distribution of the number of agents is only dependent on . Moreover the transition rates also converge to a constant independent of in the stationary state. Therefore the solution of the stationary state will satisfy
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 , , and the choice of function . In particular, when , the convergence is always satisfied. In the following subsections, we will characterize further the stationary state solution of this model in different limiting cases.
The expression for the number of agent in a given state and can be obtained by substituting Eqs. (22) into the general solution Eqs. (19), using the stationary conditions Eqs. (23). In this way we get the stationary solution given by
To complete the solution is necessary to determine the constants and in a self-consistent type of solution.To find the expression of as a function of we substitute Eqs. in Eq. and we get
Performing explicitly the last two integrals using the dynamical solution given by Eqs. , this equation can be simplified as
Finally the self-consistent solution of the dynamics is solved by expressing Eq. by
Therefore we can use Eqs. and to compute the numerical value of and . Inserting in these equations the expressions for given by Eqs. and the solutions given by Eqs. we get
The probability distributions and , can be manipulating performing a data collapse of the distributions, i.e.
with and defined as
where and are the normalization factors. The data collapse defined by Eqs. (30) of the curves , and are both described by Weibull distributions.
a.4 Comparisons with quenched simulations
To check the validity of our annealed approximation versus quenched simulations, we performed a computer simulation according to the dynamical process on a quenched network. In Fig. 7 we compare the results of the simulation with the prediction of the analytical solution. In particular in the reported simulation we have chosen , , and , the simulation is based on a number of agent and for a period of , finally the data are averaged over realizations and the network is Poisson with average and weight distribution . In Fig. 7, we show evidence that the Weibull distribution and the data collapse of well capture the empirical behavior observed in the mobile phone data (Fig. 3). The distribution of the non-interaction periods in the model is by construction unaffected by circadian rhythms but follow a similar data collapse as observed in the real data (Fig. 5). The simulated data are also in good agreement with the analytical prediction predicted in the annealed approximation for the parameter choosen in the figure. As the network becomes more busy and many agents are in a telephone call, the quenched simulation and the annealed prediction of differs more significantly.
For the functions and given by Eqs. reduce to constants, therefore the process of creation of an interaction is a Poisson process and no reinforcement dynamics is taking place in the network. Assigning to Eqs. (19), we get the solution
and consequently the distributions of duration of given states Eqs. are given by
Therefore the probability distributions and are exponentials as expected in a Poisson process.
In this section, we discuss the case for such that and . Using Eqs. (15) we get the solution
and consequently the distributions of duration of given states Eqs. are given by
The probability distributions are power-laws.This result remains valid for every value of the parameters (See Ref. Zhao:2011 () for a full account of the detailed solution of this model) nevertheless the stationary condition is only valid for
a.5 Solution of the mean-field model on a fully connected network
Finally, we discuss the mean-field limit on the model in which every agent can interact with every other agent. In this case, social network is a fully connected network. Therefore we use and to denote the number of agents of the two different states respectively and the rate equations are then revised to
Since we will refer to this model only in the framework of a null model, we will only discuss the case in which the dynamics of the network is Poissonian, i.e. when
The stationary solution of this model is given by exponentials, i.e.
Finally the distributions of duration of given states expressed by Eqs. are given by
which are exponential distributions as expected in a Poisson process.
Appendix B Entropy of the dynamical social networks
b.1 Entropy of the dynamical social networks of pairwise communication
The definition of the entropy of dynamical social networks of a pairwise communication model, is given by Eq. (6) of the main body of the article that we repeat here for convenience,
In this equation the matrix is the adjacency matrix of the social network and indicates that at time the agents and are interacting while indicates that agent is non-interacting.Finally indicates the dynamical evolution of the social network. In this section, we will evaluate the entropy of dynamical social networks in the framework of the annealed model of pairwise communication explained in detail in the previous section of this supplementary material. To evaluate the entropy of dynamical social network explicitly, we have to carry out the summations in Eq. . These sums, will in general depend on the particular history of the dynamical social network, but in the framework of the model we study, in the large network limit will be dominated by their average value. In the following therefore we perform these sum in the large network limit. The first summation in Eq. denotes the average loglikelihood of finding at time a non-interacting agent given a history . We can distinguish between two eventual situations occurring at time : (i) the agent has been non-interacting since a time , and at time remains non-interacting; (ii) the agent has been interacting with another agent since time , and at time the conversation is terminated by one of the two interacting agents. In order to characterize situation (i) we indicate by the probability that a non-interacting agent with degree in the social network, that has not interacted since a time , doesn’t change state. Similarly, in order to characterize situation (ii), we indicate by the probability that a connected pair of agents (with degrees and respectively, and weight of the link ) have interacted since time and terminate their conversation at time . Given the stationary solution of the pairwise communication model, performed in the annealed approximation, the rates and are given by