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Une analyse simple d’épidémies sur les graphes aléatoires

Une analyse simple d’épidémies sur les graphes aléatoires. Marc Lelarge (INRIA-ENS) ALEA 2009. Diluted Random Graphs. Molloy-Reed (95). Percolated Threshold Model. Bond percolation: randomly delete each edge with probability 1- π .

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Une analyse simple d’épidémies sur les graphes aléatoires

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  1. Une analyse simple d’épidémies sur les graphes aléatoires Marc Lelarge (INRIA-ENS) ALEA 2009.

  2. Diluted Random Graphs Molloy-Reed (95)

  3. Percolated Threshold Model • Bond percolation: randomly delete each edge with probability 1-π. • Bootstrap percolation with threshold K(d): Seed of active nodes, S. Deterministic dynamic: set if

  4. Branching Process Approximation • Local structure of G = random tree • Recursive Distributional Equation:

  5. Solving the RDE

  6. Algorithm • Remove vertices S from graph G • Recursively remove vertices i such that: • All removed vertices are active and all vertices left are inactive. • Variations: • remove edges instead of vertices. • remove half-edges of type B.

  7. Configuration Model • Vertices = bins and half-edges = balls Bollobás (80)

  8. Site percolation Fountoulakis (07) Janson (09)

  9. Coupling • Type A if Janson-Luczak (07) A B

  10. Deletion in continuous time • Each white ball has an exponential life time. A B

  11. Percolated threshold model • Bond percolation: immortal balls A A B

  12. Death processes • Rate 1 death process (Glivenko-Cantelli): • Death process with immortal balls:

  13. Death Processes for white balls • For the white A and B balls: • For the white A balls:

  14. Epidemic Spread • largest solution in [0,1] of: • If , then final outbreak: • If , and not local minimum, outbreak:

  15. Applications • K=0, π>0, α>0: site + bond percolation. • If α->0, giant component: • Bootstrap percolation, π=1, K(d)=k. Regular graphs (Balogh Pittel 07) K-core for Erdӧs-Rényi (Pittel, Spencer, Wormald 96)

  16. Phase transition

  17. Phase transition • Cascade condition: • Contagion threshold K(d)=qd (Watts 02)

  18. Conclusion • Percolated threshold model (bond-bootstrap percolation) • Analysis on a diluted random graph (coupling) • Recover results: giant component, sudden emergence of the k-core… • New results: cascade condition, vaccination strategies…

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