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Agreement dynamics on interaction networks: the Naming game

Agreement dynamics on interaction networks: the Naming game. A. Barrat LPT, Université Paris-Sud, France ISI Foundation, Turin, Italy. A. Baronchelli (La Sapienza, Rome, Italy) L. Dall’Asta (LPT, Orsay, France) V. Loreto (La Sapienza, Rome, Italy). http://www.th.u-psud.fr/.

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Agreement dynamics on interaction networks: the Naming game

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  1. Agreement dynamics on interaction networks: the Naming game A. BarratLPT, Université Paris-Sud, FranceISI Foundation, Turin, Italy 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

  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 (Talking Heads experiment,Steels ’98) 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 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 addsto its memory the word given by the speaker (Baronchelli et al, JSTAT 2006)

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

  9. FAILURE Speaker Hearer Speaker Hearer 1.ARBATI 2.ZORGA 3.GRA 1.REFO 2.TROG 3.ZEBU 1.ARBATI 2.GRA 3.ZORGA 1.REFO 2.TROG 3.ZEBU 4.ZORGA SUCCESS Speaker Speaker Hearer Hearer 1.ZORGA 2.ARBATI 3.GRA 1.TROG 2.ZORGA 3.ZEBU 1.ARBATI 2.ZORGA 3.GRA 1.TROG 2.ZEBU 3.ZORGA Naming game: other dynamical rules 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. Naming game:example of social dynamics interactions among individuals create complex networks: a population can be represented as a graph on which agents nodes interactions edges Simplest case: complete graph a node interacts equally with all the others, prototype of mean-field behavior

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

  13. 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 diverges as N 1 Baronchelli et al. JSTAT 2006

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

  15. Another extreme case:agents on a regular lattice 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. Few neighbors: coarsening phenomena (slow!)

  16. Another extreme case:agents on a regular lattice 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 tconv» N3

  17. Another extreme case:agents on a regular lattice d=1 tmax/ N tconv/ N3 d=2 tmax/ N tconv/ N2

  18. 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)

  19. Two extreme cases

  20. Naming Game on a small-world N nodes forms a regular lattice. With probability p, each edge is rewired randomly =>Shortcuts N = 1000 • Large clustering coeff. • Short typical path Watts & Strogatz, Nature393, 440 (1998)

  21. Naming Game on a small-world Dall'Asta et al., EPL 73 (2006) 969 1D Random 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.

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

  23. Naming Game on a small-world maximum memory: /N convergence time: /N1.4

  24. Better not to have all-to-all communication, nor a too regular network structure What about other types of networks ?

  25. Definition of the Naming Game on heterogeneous networks Dall’Asta et al., PRE 74 (2006) 036105 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 (cf also Suchecki et al, 2005 and Castellano, 2005)

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

  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 Nw and decreasing Nd -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: tmax/ N ; Nmaxw/ N average maximum memory per agent: finite! • Convergence time: tconv/ N1.5

  30. Effects of average degree larger <k> • larger memory, • faster convergence

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

  32. Bad transmissions/errors? A. Baronchelli et al, cond-mat/0611717 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

  33. 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

  34. 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)

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

  36. Series of transitions tm=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 ()

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