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. Bootstrap percolation with threshold K(d): Seed of active nodes, S. Deterministic dynamic: set if

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

Solving the RDE

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.

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

Site percolation Fountoulakis (07) Janson (09)

Coupling Type A if Janson-Luczak (07) A B

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

Percolated threshold model Bond percolation: immortal balls A A B

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

Death Processes for white balls For the white A and B balls: For the white A balls:

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

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)

Phase transition

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

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…