Une analyse simple dpidmies sur les graphes alatoires

1
Une analyse simple dépidémies 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-p.
• 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
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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)
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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

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Death Processes for white balls

• For the white A and B balls
• For the white A balls

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Epidemic Spread

• largest solution in 0,1 of
• If , then final outbreak
• If , and not local minimum, outbreak

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Applications

• K0, pgt0, agt0 site bond percolation.
• If a-gt0, giant component
• Bootstrap percolation, p1, K(d)k.
• Regular graphs (Balogh Pittel 07)
• K-core for Erd?s-Rényi (Pittel, Spencer, Wormald
  96)

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Phase transition
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Phase transition

• Cascade condition
• Contagion threshold
• K(d)qd (Watts 02)

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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