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Dynamics of Social Interactions at Short Timescales G. Bianconi Department of Physics, Northeastern University SAMSI Workshop: Dynamics of networks SAMSI, January 10-12, 2011

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Complex networks describe the underlying structure of interacting complex Biological, Social and Technological systems.

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Dynamics on networks Scale-free degree distribution change the critical behavior of the Ising model, Percolation, disease spreading Spectral properties of the Laplacian matrix change the synchronization properties of networks with complex topologies Nishikawa et al.PRL 2003

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How do critical phenomena on complex networks change if we include the spatial interactions?

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Annealed uncorrelated complex networks In annealed uncorrelated complex networks, we assign to each node an expected degree Each link is present with probability p ij The degree k i a node i is a Poisson variable with mean i Boguna, Pastor-Satorras PRE 2003

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Ising model in annealed complex networks The Ising model on annealed complex networks has Hamiltonian given by The critical temperature is given by The magnetization is non-homogeneous G. Bianconi 2002,S.N. Dorogovtsev et al. 2002, Leone et al. 2002, Goltsev et al. 2003,Lee et al. 2009

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Critical exponents of the Ising model on complex topologies M C(T<T c ) >5 |T c -T| 1/2 Jump at T c |T c -T| -1 =5 |T-T c | 1/2 /(|ln|T c T||) 1/2 1/ln|T c -T||T c -T| -1 3< <5 |T c -T| 1/( |T c -T| |T c -T| -1 =3 e -2T/ T 2 e -4T/ T -1 2< <3 T T T -1 But the critical fluctuations still remain mean-field !

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Ensembles of spatial complex networks The function J(d) can be measured in real spatial networks The maximally entropic network with spatial structure has link probability given by Airport Network Bianconi et al. PNAS 2009 J(d)

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Annealead Ising model in spatial complex networks The linking probability of spatial complex networks is chosen to be The Ising model on spatial annealed complex networks has Hamiltonian given by We want to study the critical fluctuations in this model as a function of the typical range of the interactions

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Stability of the mean-field approximation The partition function is given by The magnetization in the mean field approximation is given by The susceptibility is then evaluated by stationary phase approximation

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Dynamics on spatial networks: the Ising model We assume that the spectrum is given by is the spectral gap and c the spectral edge. Anomalous critical fluctuations sets in only if the gap vanish in the thermodynamic limit, and S <1 For regular lattice S =(d-2)/2 S <1 only if d<4 The effective dimension of complex networks is d eff =2 S +2 c (S. Bradde, F. Caccioli, L. DallAsta and G. Bianconi PRL 2010)

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Dynamics of networks At any given time dynamical networks looks disconnected Protein complexes during the cell cycle of yeast Social networks (phone calls, small gathering of people) De Lichtenberg et al.2005 Barrat et al.2008

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Human social interaction are characterized by networks at different level of organization FriendshipsCitiesPolitical parties

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Human social interactions are organized at different time scales From long lasting friendships and collaborations To the duration of a single phone-call or the duration of a small gathering during the coffee break of a conference

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Bursty human activities Einstein and Darwin correspondence Olivera and Barabasi Nature (2005) Human dynamics is not described by Poisson processes

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Inter-event time of human activities Vazquez et al. PRE (2006)

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Queuing model for bursty human activities Priorities and random activities Queuing model (Barabasi Nature 2005,Vazquez PRL 2005) Only priorities

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New data: face-to-face interactions Bluetooth sensors (Infocom 2005 conference)IMOTE data set -Temporal resolution 120s MIT experiment 100 students for 9 months -Time resolution 300s Radio Frequence Identification Devices (RFID) -face-to face interactions at a distance of 1-2 meters - temporal resolution of 20s

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Distribution of contact lifetimes and intercontact duration Infocom 2005 conference 41 sensor for 3 days Sampling period 120s 100 MIT students for 9 months Sampling period 300s Chaintreaux et al. 2005 Eagle and Pentland Reality Mining 2006 Contact Intercontact IMOTE MIT

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Duration of contacts A.Barrat, C. Cattuto, V. Colizza, J.F. Pinton,W. Van den Broeck, A. Vespignani Arxiv:0811.4170

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Weighted social network A.Barrat, C. Cattuto, V. Colizza, J.F. Pinton,W. Van den Broeck, A. Vespignani Arxiv:0811.4170

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Cognitive Hebbian mechanisms Reinforcement dynamics in social interactions For the interacting individual The longer an individual interacts with a group the less is likely to leave the group For the isolated individual The longer and individual is isolated the less is likely to interact with a group

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Stochastic processes with reinforcement Polya urns Reinforced random walk Hebbian Learning Replicator Dynamics Chinese restaurant processes Preferential attachment in networks

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The ingredients of the dynamical pairwise model The individual i is associated to a state n i =0,1 indicating if he/she is isolated or interacting to a time t i which is the last time it has changed his state Reinforcement dynamics The more an individual is in a state the less likely it that he/she change his/her state Transition rates Only between 0 1 1 0

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The dynamical paiwise model Choose one random agent If n=0 with probability p 0 (t,t i ) he connects to another isolated agent chosen with probability p 0 (t,t i ) If n=1, with probability p 1 (t,t i ) there is a transition and he/she disconnects from his/her group

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Choice of p n (t,t i ) Absence of reinforcement Presence of reinforcement

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The dynamical equations with reinforcement The dynamical equations for the number of individuals N 0,1 (t,t i ) that at time t are in state 0,1 since time t i are given by

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Structure of the dynamical solution Where the transition rates are given in terms of N 0,1 (t,t)

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Self-consistent assumption and phase diagram Stationary phase (white) Non-stationary phase

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Transition rates: simulation vs. analytical results Green Stationary region Red b 0 0.5 Blue b 0 <0.5, b 1 <0.5

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Contact and inter-contact time distributions Stationary region Non-stationary region (b 1 <0.5 b 0 <0.5)

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The dynamical model with groups of any size The individual i is associated to a state n i =0,1,2… indicating the number of other individual in his/her group to a time t i which is the last time it has changed his state Reinforcement dynamics The more an individual is in a state the less likely it that he/she change his/her state Transition rates Only between n n+1 or n n-1 An individual in a group which is changing state can either detach himself/herself from his/her group with rate or introduce an insolated individual to its group with rate 1-

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The dynamical model Choose one random agent If n=0 with probability p 0 (t,t i ) he connects to another isolated agent chosen with probability p 0 (t,t i ) If n>0, with probability p n (t,t i ) there is a transition -with probability he/she connects to an insolated agent chosen with probability p 0 (t,t i ) -with probability 1- he/she disconnects from his/her group

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Choice of p n (t,t i ) In presence of reinforcement

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The dynamical equations with reinforcement The dynamical equations for the number of individuals N n (t,t i ) that at time t are in state n since time t i are given by with (t) given by

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Phase diagram of the model (I) Stationary region (II) Non-stationary region (III) Self-consistent assumption breaks-down

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Lifetime of a group of size n+1 Langer groups are more unstable (J. Stehle, A. Barrat and G. Bianconi PRE 2010)

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Real data versus model The model well capture the distribution of lifetime of different group sizes of small human gatherings (Sociopatterns,data from Berlin conference )

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Instability for the formation of a large group in region I The model present an instability for the formation of a large group of the order of magnitude of N

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Strong finite size effects in region III

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Features of the nodes In complex networks nodes are generally heterogeneous and they are characterized by specific features In Social networks nodes have specific features: age, gender, type of jobs, drinking and smoking habits, nationality Specific feature might affects the social inclination of different people, therefore a natural first generalization of the model would describe heterogeneous social behavior

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Heterogeneous model The agents are assigned a parameter i drawn from a uniform distribution in (0,1) that describe their social behavior and we call sociability The larger is the more social is the agent behavior

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Pairwise heterogeneous model: The duration of contacts of agents with sociability

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Self-consistent solution

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Aggregated data for the pairwise heterogeneous model: simulations versus analytical results

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Heterogeneous model with groups of any size

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Conclusions Human social interaction on a fast timescales are characterized by a dynamics with reinforcement that is able to predict both power- law distribution of durations of contacts and inter-contact times. The model show a rich phase diagram with the power-law lifetime of groups persisting also in the non-stationary region The model can be easily generalized to include for heterogeneous sociability of the agents The model is a perfect platform to perform simulation of social behavior on the fast time scale

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Many thanks go to my collaborators Kun Zhao (Northeastern University, USA) Alain Barrat, Juliette Stehle (Universite de Marseille, France,SOCIOPATTERNS) Ciro Cattuto, Wouter Van den Broeck, Jean-Francois Pinton (ISI Foudation,Tourin, SOCIOPATTERNS)

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