1. 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. Barrat LPT, Université Paris-Sud, France ISI Foundation, Turin, Italy

2. Introduction Statistical physics: study of the emergence of global complex properties from purely local rules “Sociophysics”: Simple (simplistic?) models which may allow to understand fundamental aspects of social phenomena =>Voter model, Axelrod model, Deffuant model….

3. Opinion formation models Simplified models of interaction between N agents Questions: ● Convergence to consensus without global external coordination? ● How? ● In how much time?

4. 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, heterogeneous structures, etc… Studies of agents on complex networks

5. 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?

6. 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

7. 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)

8. Minimal naming game: dynamical rules Speaker Hearer FAILURE Hearer SUCCESS ARBATI ZORGA GRA ARBATI ZORGA GRA ZORGA ARBATI ZORGA GRA ZORGA REFO TROG ZEBU REFO TROG ZEBU ZORGA TROG ZEBU

9. Naming game: other dynamical rules Speaker Hearer FAILURE Hearer SUCCESS 1.ARBATI 2.ZORGA 3.GRA 1.ARBATI 2.ZORGA 3.GRA 1.ZORGA 2.ARBATI 3.GRA 1.ARBATI 2.GRA 3.ZORGA 1.TROG 2.ZORGA 3.ZEBU 1.REFO 2.TROG 3.ZEBU 1.REFO 2.TROG 3.ZEBU 4.ZORGA 1.TROG 2.ZEBU 3.ZORGA Possibility of giving weights to words, etc... => more complicate rules

10. 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)

11. Simplest case: complete graph interactions among individuals create complex networks: a population can be represented as a graph on which interactions agentsnodes edges a node interacts equally with all the others, prototype of mean-field behavior Naming game: example of social dynamics

12. Baronchelli et al. JSTAT 2006 Complete graph Total number of words=total memory used N=1024 agents Number of different words Success rate Memory peak Building of correlations Convergence

13. Complete graph: Dependence on system size ● Memory peak: t max / N 1.5 ; N max w / N 1.5 average maximum memory per agent / N 0.5 ● Convergence time: t conv / N 1.5 Baronchelli et al. JSTAT 2006 diverges as N 1

14. Another extreme case: agents on a regular lattice N=1000 agents MF=complete graph 1d, 2d: agents on a regular lattice N w =total number of words; N d =number of distinct words; R=success rate Baronchelli et al., PRE 73 (2006) 015102(R)

15. 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)

16. 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 t conv » N 3

17. Another extreme case: agents on a regular lattice d=1 t max / N t conv / N 3 d=2 t max / N t conv / N 2

18. Regular lattice: Dependence on system size ● Memory peak: t max / N ; N max w / N average maximum memory per agent: finite! ● Convergence by coarsening: power-law decrease of N w /N towards 1 ● Convergence time: t conv / N 3 =>Slow process! (in d dimensions / N 1+2/d )

19. Two extreme cases Complete graphdimension 1 maximum memory / N 1.5 / N/ N convergence time / N 1.5 / N3/ N3

20. 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

21. 1DRandom topology p: 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

22. increasing p p=0 p=0: linear chain p À 1/N : small-world -slower at intermediate times (partial “pinning”) -faster convergence

23. Naming Game on a small-world convergence time: / N 1.4 maximum memory: / N

24. Complete graph dimension 1small-world maximum memory / N 1.5 / N/ N / N/ N convergence time / N 1.5 / N3/ N3 What about other types of networks ? Better not to have all-to-all communication, nor a too regular network structure

25. 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)

26. NG on heterogeneous networks Different behaviours shows the importance of understanding the role of the hubs! Example: agents on a BA network:

27. NG on heterogeneous networks Speaker first: hubs accumulate more words Hearer first: hubs have less words and “polarize” the system, hence a faster dynamics

28. NG on homogeneous and heterogeneous networks -Long reorganization phase with creation of correlations, at almost constant N w and decreasing N d -similar behaviour for BA and ER networks (except for single node dynamics), as also observed for Voter model

29. NG on complex networks: dependence on system size ● Memory peak: t max / N ; N max w / N average maximum memory per agent: finite! ● Convergence time: t conv / N 1.5

30. Effects of average degree larger ● larger memory, ● faster convergence

31. larger clustering ● smaller memory, ● slower convergence Effects of enhanced clustering (more triangles, at constant number of edges) C increases

32. 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 => convergence  0 : no elimination of names => no convergence Expect a transition at some  c A. Baronchelli et al, cond-mat/0611717

33. Mean-field case Stability of the consensus state ? consider a state with only 2 words A, B Evolution equations for the densities: n A, n B, n AB  > 1/3 : states (n A =n AB =0, n B =1), (n B =n AB =0, n A =1)  0, n A =n B > 0

34. Mean-field case At  c = 1/3, Consensus to Polarization transition t conv / (  -  c ) -1 The polarized state is active (  Axelrod model, in which the polarized state is frozen)

35. Mean-field case: numerics Usual NG NG with at most m different words =>At least 2 different universality classes

36. Series of transitions t m =time to reach a state with m different words Transitions to more and more disordered active states

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

38. On networks, as in MF At  c, Consensus to Polarization transition (  c depends on strategy+network heterogeneity) The polarized state is active

39. 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…

40. Alain.Barrat@u-psud.fr http://www.th.u-psud.fr/ http://cxnets.googlepages.com

41. On networks Possible to write evolution equations =>  c (  )