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Diffusion and Cascading in Random Graphs

Explore the game-theoretic diffusion model, contagion threshold, and phase transitions in random networks. Learn about cascading behavior and the impact of contagion threshold on large population dynamics. Delve into the concept of pivotal players and the recursive distributional equation in understanding cascades. Discover the interplay between individual behaviors and network structures in the spread of contagion in random graphs.

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Diffusion and Cascading in Random Graphs

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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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