Agreement dynamics on interaction networks: the Naming game

A. Baronchelli (La Sapienza, Rome, Italy)L. Dall’Asta (LPT, Orsay, France)V. Loreto (La Sapienza, Rome, Italy)

http://www.th.u-psud.fr/Phys. Rev. E 73 (2006) 015102(R)Europhys. Lett. 73 (2006) 969 Phys. Rev. E 74 (2006) 036105

http://cxnets.googlepages.com

A. BarratLPT, Université Paris-Sud, France

ISI Foundation, Turin, Italy

Introduction

Statistical physics: study of the emergence of global complex properties from purely local rules

“Sociophysics”: Simple (simplistic?) models which mayallow to understand fundamental aspectsof social phenomena

=>Voter model, Axelrod model, Deffuant model….

Opinion formation models

Simplified models of interaction between N agents

Questions:● Convergence to consensus without global

external coordination?● How?● In how much time?

Opinion formation models

Most initial studies:● “mean-field”: each agent can interact with all the others● regular lattices

Recent progresses in network science:social networks: complex networks

small-world, large clustering, heterogeneousstructures, etc…

Studies of agents on complex networks

Naming game

Interactions of N agents who communicate on how to associate a name to a given object

=> Emergence of a communication system?Agents: -can keep in memory different words/names-can communicate with each other

Example of social dynamics/agreement dynamics

(Talking Heads experiment, Steels ’98)

Convergence? Convergence mechanism?Dependence on N of memory/time requirements?Dependence on the topology of interactions?

Naming game: dynamical rules

At each time step:-2 agents, a speaker and a hearer, are randomly selected-the speaker communicates a name to the hearer(if the speaker has nothing in memory –at the beginning- it invents a name)

-if the hearer already has the name in its memory: success-else: failure

Minimal naming game: dynamical rules

success => speaker and hearer retain the uttered word as the correct one and cancel all other words from their memory

failure => the hearer adds to its memory the word given by the speaker

(Baronchelli et al, JSTAT 2006)

Minimal naming game: dynamical rules

Speaker

Speaker Speaker

SpeakerHearerFAILURE

HearerHearer

Hearer

SUCCESS

ARBATIZORGAGRA

ARBATIZORGAGRA

ZORGA

ARBATIZORGAGRA

ZORGA

REFOTROGZEBU

REFOTROGZEBUZORGA

ZORGATROGZEBU

Naming game: other dynamical rules

Speaker

Speaker Speaker

SpeakerHearerFAILURE

HearerHearer

Hearer

SUCCESS

1.ARBATI2.ZORGA3.GRA

1.ARBATI2.ZORGA3.GRA

1.ZORGA2.ARBATI3.GRA

1.ARBATI2.GRA3.ZORGA

1.TROG2.ZORGA3.ZEBU

1.REFO2.TROG3.ZEBU

1.REFO2.TROG3.ZEBU4.ZORGA

1.TROG2.ZEBU3.ZORGA

Possibility of giving weights to words, etc...=> more complicate rules

Naming game:example of social dynamics

-no bounded confidence( Axelrod model, Deffuant model)

-possibility of memory/intermediate states( Voter model, cf also Castello et al 2006)

-no limit on the number of possible states(no parameter)

Simplest case: complete graph

interactions among individuals create complex networks: a population can be represented as a graph on which

interactions

agents nodes

edges

a node interacts equally with all the others, prototype of mean-field behavior

Naming game:example of social dynamics

Baronchelli et al. JSTAT 2006

Complete graph

Total number of words=totalmemory used

N=1024 agents

Number of different words

Success rate

Memory peak

Building of correlations

Convergence

Complete graph:Dependence on system size

● Memory peak: tmax / N1.5 ; Nmaxw / N1.5

average maximum memory per agent / N0.5

● Convergence time: tconv / N1.5

Baronchelli et al. JSTAT 2006

diverges asN 1

Another extreme case:agents on a regular lattice

N=1000 agents

MF=complete graph

1d, 2d: agents on a regularlattice

Nw=total number of words; Nd=number of distinct words; R=success rate

Baronchelli et al., PRE 73 (2006) 015102(R)

Local consensus is reached very quickly through repeated interactions. Then:-clusters of agents with the same unique word start to grow, -at the interfaces series of successful and unsuccessful interactions take place.

coarsening phenomena (slow!)

Few neighbors:

Another extreme case:agents on a regular lattice

Baronchelli et al., PRE 73 (2006) 015102(R)

The evolution of clusters is described as the diffusion of interfaces which remain localized i.e. of finite width

Diffusion equation for the probability P(x,t) that an interface is at the position x at time t:

Each interface diffuses with a diffusion coefficient D(N)» 0.2/N

The average cluster size grows as

Another extreme case:agents on a regular lattice

tconv » N3

Another extreme case:agents on a regular lattice

d=1tmax/ Ntconv/ N3

d=2tmax/ Ntconv/ N2

Regular lattice:Dependence on system size

● Memory peak: tmax / N ; Nmaxw / N

average maximum memory per agent: finite!

● Convergence by coarsening: power-law decrease of Nw/N towards 1

● Convergence time: tconv / N3 =>Slow process!(in d dimensions / N1+2/d)

Two extreme cases

Complete graph dimension 1maximummemory

/ N1.5 / N

convergence

time

/ N1.5 / N3

Naming Game on a small-world

Watts & Strogatz, Nature 393, 440 (1998)

N = 1000

•Large clustering coeff. •Short typical path

N nodes forms a regular lattice. With probability p, each edge is rewired randomly

=>Shortcuts

1D Random topologyp: shortcuts

(rewiring prob.)

(dynamical) crossover expected:

● short times: local 1D topology implies (slow) coarsening

● distance between two shortcuts is O(1/p), thus when a cluster is of order 1/p the mean-field behavior emerges.

Dall'Asta et al., EPL 73 (2006) 969

Naming Game on a small-world

Naming Game on a small-world

increasing p

p=0

p=0: linear chainp À 1/N : small-world

-slower at intermediate times (partial “pinning”)-faster convergence

Naming Game on a small-world

convergence time:/ N1.4

maximum memory:/ N

Complete graph

dimension 1 small-world

maximummemory

/ N1.5 / N / N

convergence time

/ N1.5 / N3 / N1.5

What about other types of networks ?

Better not to haveall-to-all communication,nor a too regular network structure

Definition of the Naming Game on heterogeneous networks

recall original definition of the model:

select a speaker and a hearer at random among all nodes

=>various interpretations once on a network:

-select first a speaker i and then a hearer among i’s neighbours

-select first a hearer i and then a speaker among i’s neighbours

-select a link at random and its 2 extremities at random as hearer and speaker

can be important in heterogeneous networks because: -a randomly chosen node has typically small degree-the neighbour of a randomly chosen node has typically large degree

Dall’Asta et al., PRE 74 (2006) 036105

(cf also Suchecki et al, 2005 and Castellano, 2005)

NG on heterogeneous networks

Different behaviours

shows the importanceof understanding the roleof the hubs!

Example: agents on a BA network:

NG on heterogeneous networks

Speaker first: hubs accumulate more words

Hearer first: hubs have less words and “polarize” the system,hence a faster dynamics

NG on homogeneous and heterogeneous networks

-Long reorganization phasewith creation of correlations, at almost constant Nw and decreasing Nd

-similar behaviour for BAand ER networks(except for single node dynamics),as also observed for Voter model

NG on complex networks:dependence on system size

● Memory peak: tmax / N ; Nmaxw / N

average maximum memory per agent: finite!

● Convergence time: tconv / N1.5

Effects of average degree

larger <k>

● larger memory, ● faster convergence

larger clustering

● smaller memory, ● slower convergence

Effects of enhanced clustering(more triangles, at constant number of edges)

C increases

Bad transmissions/errors?

Modified dynamical rules:in case of potential successful communication:

● With probability : success● With probability 1-: nothing happens (irresolute attitude)

1 : usual Naming Game => convergence0 : no elimination of names => no convergence

Expect a transition at some c

A. Baronchelli et al, cond-mat/0611717

Mean-field case

Stability of the consensus state ? consider a state with only 2 words A, B

Evolution equations for the densities: nA, nB, nAB

> 1/3 : states (nA=nAB=0, nB=1), (nB=nAB=0, nA=1)< 1/3 : state with nAB > 0 , nA=nB > 0

Mean-field case

At c = 1/3, •Consensus to Polarization transition•tconv / (-c)-1

The polarized state is active( Axelrod model, in which the polarized state is frozen)

Mean-field case:numerics

Usual NG NG with at most m different words

=>At least 2 different universality classes

Series of transitions

tm=time to reach a state with m different words

Transitions to more and more disordered active states

On networks

-Influence of strategy-Transition preserved on het. networks( Axelrod model)

On networks, as in MF

At c , Consensus to Polarization transition(c depends on strategy+network heterogeneity)

The polarized state is active

Other issues

● Community structures (slow down/stop convergence)(cf also Castello et al, arXiv:0705.2560)

● Other (more efficient) strategies (dynamical rules) (A. Baronchelli et al., physics/0511201; Q. Lu et al., cs.MA/0604075)

● Activity of single nodes(L. Dall’Asta and A. Baronchelli, J. Phys A 2006)

● Coupling the dynamics of the network with the dynamics on the network: transitions between consensus and polarized states, effect of intermediate states…

Alain.Barrat@u-psud.fr

http://www.th.u-psud.fr/

http://cxnets.googlepages.com

On networks

Possible to write evolution equations=> c ()