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Diffusions et cascades dans les graphes aléatoires

Diffusions et cascades dans les graphes aléatoires. Marc Lelarge (INRIA-ENS) Journées MAS 2010, Bordeaux. Game-theoretic diffusion model…. Both receive payoff q . Both receive payoff 1-q>q . Both receive nothing. Morris (2000). …on a network. Everybody start with ICQ.

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Diffusions et cascades dans les graphes aléatoires

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  1. Diffusions et cascades dans les graphes aléatoires Marc Lelarge (INRIA-ENS) Journées MAS 2010, Bordeaux.

  2. Game-theoretic diffusion model… • Both receive payoff q. • Both receive payoff • 1-q>q. • Both receive nothing. • Morris (2000)

  3. …on a network. • Everybody start with ICQ. • Total payoff = sum of the payoffs with each neighbor. • A fraction of the population is forced to • If 2(1-q)>3q, i.e. 2>5q

  4. Threshold Model • State of agent i is represented by • Switch from to if:

  5. Morris, Contagion (2000) • Doesthereexist a finite groupe of playerssuchthattheir action underbest responsedynamicsspreadscontagiouslyeverywhere? • Contagion threshold: = largest q for whichcontagiousdynamics are possible. • Example: interaction on the line

  6. Anotherexample: d-regulartrees

  7. What happens for random graphs? • Random graphs with given degree sequence introduced by Molloy and Reed (1995). • Examples: • Random regular graphs. • Erdös-Réyni graphs, G(n,λ/n). • We are interested in large population asymptotics. • D is the asymptotic degree distribution.

  8. What happens for random graphs? • Random graphs are locallytree-like. • Random d-regular graph:

  9. Erdös-Réyni graphs G(n,λ/n) No cascade Global cascades

  10. Erdös-Réyni graphs G(n,λ/n) • Pivotalplayers: giant component of the subgraphwithdegrees

  11. Phase transition continued • What happens when q is bigger than the cascade capacity? • If the seed plays B forever, the model is monotone. • In a finite graph, there is only one possible final state for the epidemic

  12. Locally tree-like Independent computations on trees

  13. Branching Process Approximation • Local structure of G = random tree • Recursive Distributional Equation (RDE) or:

  14. Solving the RDE

  15. Phase transition in one picture

  16. Conclusion • The locally tree-like structure gives the intuition for the solution… • … but not the proof! • Configuration model + results of Janson and Luczak. • Generic epidemic model which allows to retrieve basic results for random graphs… • …. and new ones: contagion threshold, phase transitions.

  17. Merci! • Diffusion and Cascading Behavior in Random Networks. • Available at http://www.di.ens.fr/~lelarge

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