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Naming game: dynamics on complex networks A. Barrat, LPT, Université Paris-Sud, France

Naming game: dynamics on complex networks A. Barrat, LPT, Université Paris-Sud, France. A. Baronchelli (La Sapienza, Rome, Italy) L. Dall’Asta (LPT, Orsay, France) V. Loreto (La Sapienza, Rome, Italy). -Phys. Rev. E 73 (2006) 015102(R) -Europhys. Lett. 73 (2006) 969 -Preprint (2006).

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Naming game: dynamics on complex networks A. Barrat, LPT, Université Paris-Sud, France

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  1. Naming game: dynamics on complex networksA. Barrat, LPT, Université Paris-Sud, France A. Baronchelli (La Sapienza, Rome, Italy) L. Dall’Asta (LPT, Orsay, France) V. Loreto (La Sapienza, Rome, Italy) -Phys. Rev. E 73 (2006) 015102(R) -Europhys. Lett. 73 (2006) 969 -Preprint (2006) http://www.th.u-psud.fr/

  2. Naming game Interactions of N agents who communicate on how to associate a name to a given object Agents: -can keep in memory different words -can communicate with each other Example of social dynamics or agreement dynamics

  3. Minimal 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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. 2005 (physics/0509075)

  9. 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. 2005 (physics/0509075)

  10. 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!)

  11. Another extreme case:agents on a regular lattice N=1000 agents MF=complete graph 1d, 2d: agents on a regular lattice Nw=total number of words; Nd=number of distinct words; R=sucess rate

  12. 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)

  13. Two extreme cases

  14. 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)

  15. 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.

  16. Naming Game on a small-world p=0: linear chain p À 1/N : small-world p=0 increasing p

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

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

  19. Networks:Homogeneous and heterogeneous 1.Usual random graphs: Erdös-Renyi model (1960) N points, links with proba p: static random graphs Poisson distribution (p=O(1/N))

  20. P(k) ~k-3 Networks:Homogeneous and heterogeneous 2.Scale-free graphs: Barabasi-Albert (BA) model (1)GROWTH: At every timestep we add a new node with m edges (connected to the nodes already present in the system). (2)PREFERENTIAL ATTACHMENT :The probability Π that a new node will be connected to node i depends on the connectivity ki of that node / ki A.-L.Barabási, R. Albert, Science 286, 509 (1999)

  21. 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

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

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

  24. 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

  25. 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

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

  27. Effects of enhanced clustering larger clustering C increases • smaller memory, • slower convergence

  28. Other issues • Hierarchical structures • Community structures • Other (more efficient?) strategies (i.e. dynamical rules) • ... Slow down/stop the dynamics

  29. Conclusions and (Some) Perspectives • Importance of the topological properties for the • processes taking place on the network • Weighted networks • Dynamical networks (e.g. peer to peer) • Coupling (evolving) topology and dynamics • on the network

  30. Alain.Barrat@th.u-psud.fr http://www.th.u-psud.fr/

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