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Exponential extinction time of the contact process on finite graphs Qiang YAO Exponential extinction time of the Contents contact process on finite graphs Qiang YAO (Joint work with Thomas MOUNTFORD, Jean-Christophe MOURRAT and Daniel
Exponential extinction time of the contact process on finite graphs Qiang YAO Contents
(Joint work with Thomas MOUNTFORD, Jean-Christophe MOURRAT and Daniel VALESIN) School of Finance and Statistics, East China Normal University December 7th, 2013
Exponential extinction time of the contact process on finite graphs Qiang YAO Contents
1
Introduction Basic definitions Contact process on infinite graphs Contact process on finite graphs
2
Main results
3
Idea of proof
4
Application
5
Further work
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
First introduced by T. E. Harris (1974). A model to describe the spread of diseases. Two classical books:
Systems: Contact, Voter and Exclusion Processes.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
G = (V , E): a connected undirected graph. The process (ξt : t ≥ 0): a continuous-time Markov process. State space: {A : A ⊆ V }. At each t, each vertex is either healthy or infected. ξt is the collection of infected vertices at time t. Transition rates: ξt → ξt \ {x} for x ∈ ξt at rate 1, ξt → ξt ∪ {x} for x / ∈ ξt at rate λ · |{y ∈ ξt : x ∼ y}|. (ξA
t : t ≥ 0): the process with initial state A.
Absorbing state: ∅.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
We say that the process survives if the infection will persist with positive probability; otherwise we say that it dies out. survival strong survival weak survival Strong survival: every site will be infected infinitely many times with positive probability. Weak survival: the infection will persist with positive probability, but every site will be infected only finite times with probability 1. (Infection moves away to infinity.)
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
λ1 := inf{λ : the process survives}, λ2 := inf{λ : the process survives strongly}. Three phases: ξt dies out if λ < λ1, ξt survives weakly if λ1 < λ < λ2, ξt survives strongly if λ > λ2.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
On integer lattices Zd (d ≥ 1), λ1 = λ2 λc. On homogeneous trees Td (d ≥ 3), λ1 < λ2.
References:
dies out, Ann. Probab. 18 1462-1482 (1990).
contact process on trees, Ann. Probab. 24 1711-1726 (1996).
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
If G is a finite graph, then the contact process on G must die out. We are interested in the extinction time from full
τ = τ(G) = inf{t ≥ 0 : ξV
t = ∅}.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Consider the graph {0, . . . , n}d (viewed as a subgraph of Zd) and the distribution of τ for this graph, as n goes to infinity. If λ < λc, then τ/ log n converges in probability to a constant.
References:
dimensions, Chinese J. Contemp. Math. 15 13-20 (1994).
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
If λ > λc, then log E[τ]/nd converges to a positive constant, and τ/E[τ] converges in distribution to the unit exponential distribution.
References:
set II, Ann. Probab. 16 1570-1583 (1988).
contact process, Canad. Math. Bull. 36 216-226 (1993).
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
In the supercritical case, the order of magnitude of the extinction time is exponential in the number of vertices of the graph; the process is said to exhibit metastability, meaning that it persists for a long time in a state that resembles an equilibrium and then quickly moves to its true equilibrium (∅ in this case).
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
If d = 1 and λ = λc, then τ/n → ∞ and τ/n4 → 0 in probability.
Reference:
(1989).
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Fix d ≥ 2, let Th
d be the finite subgraph of Td
defined by considering up to h generations from the root and again take the contact process started from full occupancy on this graph, with associated extinction time τ. If λ < λ2, then there exist constants c, C > 0 such that P(ch ≤ τ ≤ Ch) → 1 as h → ∞.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
If λ > λ2, then for any σ < 1 there exist c1, c2 > 0 such that P
→ 1 as h → ∞. This implies that τ is at least as large as a stretched exponential function of the number of vertices, (d + 1)h.
Reference:
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
As far as we know, no rigorous results are available concerning finite graphs which are not regular.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
For n ∈ N and d > 0, let Λ(n, d) be the set of all trees with n vertices and degree bounded by d, and let G(n, d) be the set of graphs having a spanning tree in Λ(n, d). Theorem 1. For any d ≥ 2 and λ > λc(Z), there exists c > 0 such that lim
n→∞
inf
T∈Λ(n,d) P [τT ≥ ecn] = 1.
In particular, lim inf
n→
inf
T∈Λ(n,d)
log E[τT] n ≥ c.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Theorem 2. Let d ≥ 2, λ > λc(Z), and (Gn)n∈N be a sequence of graphs with Gn ∈ G(n, d). Then the distribution of τGn/E[τGn] converges to the unitary exponential distribution as n tends to infinity.
Reference:
extinction time of the contact process on finite graphs, preprint. Available at http://arxiv.org/abs/1203.2972.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
whose removal separates T into two subtrees T1 and T2 both of size at least ⌊n/d⌋.
into two subtrees T1, T2 as described by the above
E[τT] ≥ n−9 E[τT1]E[τT2].
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
The graph: G n := (V n, E n). Vertex set: V n := {1, 2, · · · , n}. Degree of vertex i is denoted by di (i = 1, 2, · · · , n), which follows (1) d1, d2, · · · , dn i.i.d. (2) pk(:= P(d1 = k)) ∼ Ck−α (α > 2, C > 0) if k is large.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
(2001): each vertex i was issued with di half edges and these half edges were matched up in a uniformly chosen manner. If α > 3, then P(no loops or multiple edges) → 1 as n → ∞. If 2 < α ≤ 3, not so. We treat both cases in our work.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Assumptions: (1) p0 = p1 = p2 = 0; (2) Conditioned on En := {d1 + · · · + dn is even}.
n→∞ P(En) = 1
2
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Theorem 3. For any λ > 0, there exists c > 0 such that P [τG n ≥ ecn] → 1 as n → ∞. As a result, the critical infection parameter for these graphs is 0.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
δ > 0, P
→ 1 as n → ∞.
Reference:
with power law degree distributions have critical value 0, Ann.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Idea of proof. Treat the “stars” as a single point and use the results of Theorems 1 and 2.
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
The study of metastable densities for the contact processes on NSW random graphs.
Reference:
contact processes on random graphs, Electron. J. Probab. 18 Article 103 (2013).
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction
Basic definitions Contact process on infinite graphs Contact process on finite graphs
Main results Idea of proof Application Further work
E-mail: qyao@sfs.ecnu.edu.cn