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…