Dynamics of Social Interactions at Short Timescales

G. Bianconi Department of Physics, Northeastern University

SAMSI Workshop: Dynamics of networksSAMSI, January 10-12, 2011

Complex networks

describe

the underlying structure of interacting complex

Biological, Social and Technological systems.

Dynamics on networks

Scale-free degree distribution change the critical behavior of the

Ising model, Percolation,

disease spreading

Spectral propertiesof the Laplacian matrix change the

synchronization properties

of networks with complex topologiesNishikawa et al.PRL 2003

How do critical phenomenaon complex networks change if we include

the spatial interactions?

Annealed uncorrelatedcomplex networks

In annealed uncorrelated complex networks, we assign to each node an expected degree

Each link is present with probability pij

The degree ki a node i is a Poisson variable with mean i

€

pij =θ iθ j

θ N

€

= k

θ 2 = k(k −1)

Boguna, Pastor-Satorras PRE 2003

Ising model in annealedcomplex networks

The Ising model on annealed complex networks has Hamiltonian given by

The critical temperature is given by

The magnetization is non-homogeneous€

Tc = Jθ 2

θ= J

k(k −1)

k

€

si = tanh β θ iJS + hi( )[ ]

€

H = −J

2 θ Nsiθ iθ js j − hisi

i

∑i≠ j

∑

G. Bianconi 2002,S.N. Dorogovtsev et al. 2002, Leone et al. 2002, Goltsev et al. 2003,Lee et al. 2009

Critical exponents of the Ising model on complex topologies

M C(T<Tc)

>5 |Tc-T|1/2 Jump at Tc |Tc-T|-1

=5 |T-Tc|1/2/(|ln|TcT||)1/2 1/ln|Tc-T| |Tc-T|-1

3<<5 |Tc-T|1/( |Tc-T| |Tc-T|-1

=3 e-2T/ T2e-4T/ T-1

2<<3 T T T-1

But the critical fluctuations still remain mean-field !

€

P(k) ∝ k −γ

Ensembles of spatial complex networks

The function J(d) can be measured in real spatial

networks

€

pij =θ iθ jJ(

r r i,

r r j )

1+θ iθ jJ(r r i,

r r j )

≅θ iθ jJ(r r i,

r r j )

QuickTime™ and a decompressor

are needed to see this picture.

The maximally entropic network with spatial structure has link probability given by

Airport Network Bianconi et al. PNAS 2009

J(d)

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

€

pij = θ iθ jJ(r r i,

r r j )

€

H( si{ }) = −1

2siθ iJijθ js j − H isi

i

∑i≠ j

∑

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 €

mi0 = tanh β(H i + θ iJijθ jm j

0

j

∑ ) ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥€

Z = e−βH si{ }( )

si{ }

∑

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 deff =2S +2

€

ρ(λ ) ∝ (λ c − λ )δ S

Δ = Λ − λ c

ρ

c

(S. Bradde, F. Caccioli, L. Dall’Asta and G. Bianconi PRL 2010)

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

Human social interaction Human social interaction

are characterized by are characterized by networks networks

at different level of at different level of organizationorganization

Friendships Cities Political parties

Human social interactionsare 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

Bursty human activities

Einstein and Darwin correspondence

Olivera and Barabasi Nature (2005)

Human dynamics is not described by Poisson processes

Inter-event time of human activities

Vazquez et al. PRE (2006)

€

N(τ ) ∝ τ −α

α ≈1

Queuing model for bursty human activities

Priorities and random activities

Queuing model(Barabasi Nature 2005,Vazquez PRL 2005)

€

P(τ ) ∝1

τe−τ /τ 0

€

P(τ ) ∝1

τ

Only priorities

New data: face-to-face interactions

Bluetooth sensors (Infocom 2005 conference)IMOTE data set

-Temporal resolution 120sMIT 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

Distribution of contact lifetimes and intercontact duration

Infocom 2005 conference41 sensor for 3 daysSampling period 120s

100 MIT students for 9 monthsSampling period 300s

Chaintreaux et al. 2005

Eagle and Pentland Reality Mining 2006

αττ −∝> )( tP Contact Intercontact

IMOTE

MIT

Duration of contacts

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

Weighted social network

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

Cognitive Hebbian mechanismsReinforcement 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

Stochastic processes with reinforcement

Polya urnsReinforced random walkHebbian LearningReplicator DynamicsChinese restaurant processesPreferential attachment in networks

The ingredients of the dynamical pairwise model

The individual i is associated to a state ni=0,1 indicating if he/she is isolated or interacting to a time ti which is the last time it has changed his state

Reinforcement dynamicsThe 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

The dynamical paiwise model

Choose one random agent

If n=0 with probability p0(t,ti)he connects to another isolatedagent chosen with probability p0(t,ti)

If n=1, with probability p1(t,ti)there is a transitionand he/she disconnects from his/her group

Choice of pn(t,ti)

Absence of reinforcement

Presence of reinforcement

€

p0,1(t, ti) =b0,1

τ +1( )

τ =t − ti

N

€

p0,1(t, ti) = b0,1

The dynamical equationswith reinforcement

The dynamical equations for

the number of individuals N0,1(t,ti) that at time t

are in state 0,1 since time ti are given by

€

€

∂N0(t, ti)

∂t= −2

b0

(τ +1)N0(t, ti) + π10(ti)δ t ,ti

∂N1(t, ti)

∂τ= −2

b1

(τ +1)N1(t, ti) + π 01(ti)δ t ,ti

τ =t − ti

N

Structure of the dynamical solution

€

N0(t, t ') = π10(t ') 1+τ( )−2b0

N1(t, t ') = π 01(t ') 1+τ( )−2b1

€

π10(t) =2

Np1(t, t ')N1(t, t')

t '

∑

π 01(t) =2

Np0(t, t ')N0(t, t')

t '

∑

Where the transition rates are given in terms of N0,1(t,t’)

Self-consistent assumptionand phase diagram

€

πm,n (t) = ˜ π m,n

t

N

⎛

⎝ ⎜

⎞

⎠ ⎟−α

€

α =0

α = max(1 − 2b0,1− 2b1)

Stationary phase (white)

Non-stationary phase

Transition rates:simulation vs. analytical

results

Green

Stationary region

Red

b0<0.5,b1>0.5

Blue

b0<0.5, b1<0.5

Contact and inter-contact time distributions

Stationary region Non-stationary region (b1<0.5 b0<0.5)

The dynamical model with groups of any size

The individual i is associated to a state ni=0,1,2… indicating the number of other individual

in his/her group to a time ti which is the last time it has changed his state

Reinforcement dynamicsThe 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-

The dynamical model

Choose one random agentIf n=0 with probability p0(t,ti)he connects to another isolatedagent chosen with probability p0(t,ti)

If n>0, with probability pn(t,ti)there is a transition-with probability he/she connects to an insolated agent chosen with probability p0(t,ti)-with probability 1- he/she disconnects from his/her group

Choice of pn(t,ti)

In presence of reinforcement

€

pn (t, ti) =bn

τ +1( )

bn = b1 for n ≥1

τ =t − ti

N

The dynamical equationswith reinforcement

The dynamical equations for the number of individuals Nn(t,ti) that at time t are in state n since time ti are given by

with (t) given by

€

€

∂N0(t, ti)

∂t= −

b0

(τ +1)2 + (1− λ )β[ ]N0(t, ti) + π 0(ti)δ t,t i

∂Nn (t, ti)

∂τ= −(n +1)

b1

(τ +1)Nn (t, ti) + π n (ti)δ t ,ti

n ≥1

€

=Nn (ti,τ )

b1

(τ +1)dτ∫

n≥1

∑

N0(ti,τ )b0

(τ +1)dτ∫

Phase diagram of the model

(I) Stationary region

(II) Non-stationary region

(III) Self-consistent assumption breaks-down

€

πm (t) = ˜ π mt

N

⎛

⎝ ⎜

⎞

⎠ ⎟−α

€

α =0

€

α =max 1 − 2b1,1 − b0

3λ −1

2λ −1

⎛

⎝ ⎜

⎞

⎠ ⎟

Lifetime of a group of size n+1

€

N0(τ ) ∝ (τ +1)−b0 (2+(1−λ )α )

N1(τ ) ∝ (τ +1)−2b1

N i(τ ) ∝ (τ +1)−b1 ( i+1)

Langer groups are more unstable

(J. Stehle, A. Barrat and G. Bianconi PRE 2010)

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 )

Instability for the formation of a large group in region I

€

n ∝ (λ − λ c )−1

λ c = 0.5

The model present an instability for the formation of a large group of the order of magnitude of N

<n>

Strong finite size effects in region III

Features of the nodesFeatures 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

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

€

p0(η i, t, ti) =η i

τ +1( )

pn (η i, t, ti) =1 −η i

τ +1( )for n ≥1

τ =t − ti

N

Pairwise heterogeneous model:

The duration of contacts of agents with sociability

€

N0(η, t, ti) = π10η (t ') 1+

t − t '

N

⎛

⎝ ⎜

⎞

⎠ ⎟−2η

N1(η,η ', t, ti) = π 01η ,η '(t') 1+

t − t'

N

⎛

⎝ ⎜

⎞

⎠ ⎟−2+η +η '

Self-consistent solution

€

π10η (t) = ˜ π 10

η t

N

⎛

⎝ ⎜

⎞

⎠ ⎟−α (η )

π 01η ,η '(t) = ˜ π 01

η ,η ' t

N

⎛

⎝ ⎜

⎞

⎠ ⎟−α (η ,η ' )

€

α(η) = max(1− 2η,η −1/2)

α (η,η ') = α (η) + α (η ')

Aggregated data for the pairwise heterogeneous model:

simulations versus analytical results

Heterogeneous model with groups of any size

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

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)