Contact Process in Random Environment
The Contact Process on the Complete Graph with
Random, Vertex-dependent, Infection Rates
Jonathon Peterson
Purdue University
Department of Mathematics
December 2, 2011
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Contact Process in Random Environment
Introduction
Contact Process
Graph G = (V , E).
Markov Process ηt on {0, 1}V .
ηt (x) = 0 healthy
ηt (x) = 1 infected .
Contact Process dynamics:
Transition
Infected → Healthy
Healty → Infected
rate
1
λ (# infected neighbors).
Classical examples: Zd or the homogeneous tree Td .
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Contact Process in Random Environment
Introduction
Phase Transition
Q: How does the infection rate λ affect the long term behavior of ηt ?
τ = inf{t ≥ 0 : ηt ≡ 0} (lifetime of infection).
λc (G) = sup{λ ≥ 0 : P {x} (τ < ∞) = 1}
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Contact Process in Random Environment
Introduction
Phase Transition
Q: How does the infection rate λ affect the long term behavior of ηt ?
τ = inf{t ≥ 0 : ηt ≡ 0} (lifetime of infection).
λc (G) = sup{λ ≥ 0 : P {x} (τ < ∞) = 1}
Finite Graphs: Contact process on {1, 2, . . . , n}d = Gn
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12/2/2011
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Contact Process in Random Environment
Introduction
Phase Transition
Q: How does the infection rate λ affect the long term behavior of ηt ?
τ = inf{t ≥ 0 : ηt ≡ 0} (lifetime of infection).
λc (G) = sup{λ ≥ 0 : P {x} (τ < ∞) = 1}
Finite Graphs: Contact process on {1, 2, . . . , n}d = Gn
λ < λc (Zd )
=⇒
P Gn (τ < C log n) → 1
λ > λc (Zd )
=⇒
P Gn (τ > ecn ) → 1.
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d
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Contact Process in Random Environment
Introduction
Power-law Random Graphs
Limiting degree distribution P(D0 ≥ k ) ∼ Ck −(α−1) .
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Contact Process in Random Environment
Introduction
Power-law Random Graphs
Limiting degree distribution P(D0 ≥ k ) ∼ Ck −(α−1) .
I
Barbasi-Albert: Preferential attachment
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Contact Process in Random Environment
Introduction
Power-law Random Graphs
Limiting degree distribution P(D0 ≥ k ) ∼ Ck −(α−1) .
I
Barbasi-Albert: Preferential attachment
I
Newman-Strogatz-Watts: Configuration model
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Contact Process in Random Environment
Introduction
Power-law Random Graphs
Limiting degree distribution P(D0 ≥ k ) ∼ Ck −(α−1) .
I
Barbasi-Albert: Preferential attachment
I
Newman-Strogatz-Watts: Configuration model
I
Chung-Lu: Vertex weights wi (expected degree).
P(i ↔ j) = ρwi wj
(ρ = 1/
X
wi )
i
≈
wi wj
.
nE[w1 ]
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Contact Process in Random Environment
Introduction
Contact Process on Power-law Random Graphs
Gn a power-law random graph (index α).
Contact process on Gn with infection rate λ.
Physics: Mean field treatment (non-rigorous)
I
λc = 0 if α ≤ 3
I
λc > 0 if α > 3
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Contact Process in Random Environment
Introduction
Contact Process on Power-law Random Graphs
Gn a power-law random graph (index α).
Contact process on Gn with infection rate λ.
Physics: Mean field treatment (non-rigorous)
I
λc = 0 if α ≤ 3
I
λc > 0 if α > 3
Mathematics: λc = 0 for any α ≥ 3.
I
Berger, Borgs, J. Chayes, & Saberi - Preferential attachment
I
Shirshendu Chatterjee & Durrett - Configuration model
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Contact Process in Random Environment
Introduction
Contact Process with Vertex-dependent Edge Weights
Motivation:
I
Sex in Sweden: Power law random graph
I
Spread of STD: Contact process
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Contact Process in Random Environment
Introduction
Contact Process with Vertex-dependent Edge Weights
Motivation:
I
Sex in Sweden: Power law random graph
I
Spread of STD: Contact process
Jonathon Peterson
(past contacts)
(future contacts)
12/2/2011
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Contact Process in Random Environment
Introduction
Contact Process with Vertex-dependent Edge Weights
Motivation:
I
Sex in Sweden: Power law random graph
I
Spread of STD: Contact process
(past contacts)
(future contacts)
Model:
I
Assign vertices i.i.d. weights wi with distribution µ.
I
Infection rate between i and j:
λwi wj
n .
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12/2/2011
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Contact Process in Random Environment
Introduction
Contact Process with Vertex-dependent Edge Weights
Motivation:
I
Sex in Sweden: Power law random graph
I
Spread of STD: Contact process
(past contacts)
(future contacts)
Model:
I
Assign vertices i.i.d. weights wi with distribution µ.
I
Infection rate between i and j:
λwi wj
n .
If µ(wi > x) ∼ Cx −α , graph of potential transmissions by time t is a
PLRG.
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Contact Process in Random Environment
Introduction
Results - Phase Transition
Mean field computation: λc = 1/Eµ [w12 ]
Theorem (P. ’11)
I
If λ < 1/Eµ [w12 ], there exists C > 0 such that
[n]
Pw (τ ≤ C log n) → 1.
I
If λ > 1/Eµ [w12 ], there exists c > 0 such that
[n]
Pw (τ ≥ ecn ) → 1.
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Contact Process in Random Environment
Introduction
Results - Quasi-stationary distribution
Mean field computation: Quasi-stationary distribution
w12
σ(λ)wi
[n]
Pw (ηt (i) = 1) ≈
, where 1 = λEµ
1 + σ(λ)wi
1 + σ(λ)w1
Theorem (P. ’11)
If λ > λc , then for any ε > 0 there exists c, C > 0 such that
[n]
σ(λ)w
i
≤ ε.
lim
sup
sup Pw (ηt (i) = 1) −
n→∞ C log n≤t≤ecn i≤n
1 + σ(λ)wi Jonathon Peterson
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