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Naming game:

ZEBU. ZORGA. ZORGA. TROG. ZEBU. Naming game: other dynamical rules. Speaker ... 3.ZEBU. 4.ZORGA. 1.TROG. 2.ZEBU. 3.ZORGA. Possibility of giving weights to ...

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Naming game:

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

    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 adds to its memory the word given by the speaker

    5. Minimal naming game: dynamical rules

    Speaker Speaker Speaker Speaker Hearer FAILURE Hearer Hearer Hearer SUCCESS ARBATI ZORGA GRA ARBATI ZORGA GRA ZORGA ARBATI ZORGA GRA ZORGA REFO TROG ZEBU REFO TROG ZEBU ZORGA ZORGA TROG ZEBU

    6. Naming game: other dynamical rules

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

    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. 2005 (physics/0509075) Complete graph Total number of words=total memory used N=1024 agents Number of different words Success rate Building of correlations Convergence

    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 Baronchelli et al. 2005 (physics/0509075) diverges as N 1

    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)

    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

    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 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

    16. Naming Game on a small-world

    increasing p p=0 p=0: linear chain p Ŕ 1/N : small-world

    17. Naming Game on a small-world

    convergence time: / N1.4 maximum memory: / N

    What about other types of networks ? Better not to have all-to-all communication, nor a too regular network structure 1.Usual random graphs: Erdös-Renyi model (1960) N points, links with proba p: static random graphs Poisson distribution (p=O(1/N)) Networks: Homogeneous and heterogeneous (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 ? that a new node will be connected to node i depends on the connectivity ki of that node A.-L.Barabási, R. Albert, Science 286, 509 (1999) Networks: Homogeneous and heterogeneous 2.Scale-free graphs: Barabasi-Albert (BA) model ? / ki

    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

    Different behaviours shows the importance of understanding the role of the hubs! Example: agents on a BA network:

    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

    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

    Effects of average degree larger <k> larger memory, faster convergence larger clustering smaller memory, slower convergence Effects of enhanced clustering C increases

    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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