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
•
• If
• If
largest solution in [0,1] of:
, then final outbreak:
, 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…