Survival and extinction of epidemics on random graphs with general degrees

Abstract

In this paper, we establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold λ1 for a Galton-Watson tree is strictly positive if and only if its offspring distribution ξ has an exponential tail, i.e., Ee cξ < ∞ for some c > 0, settling a conjecture by Huang and Durrett [12]. On the random graph with degree distribution µ, we show that if µ has an exponential tail, then for small enough λ the contact process with the all-infected initial condition survives for n 1+o(1) -time whp (short survival), while for large enough λ it runs over e Θ(n) -time whp (long survival). When µ is subexponential, we prove that the contact process whp displays long survival for any fixed λ > 0.Contents 24 Acknowledgment 31 References 31 8. Appendix 33

A discontinuous phase transition in the threshold-$θ\geq 2$ contact process on random graphs

“…Kim [15] used this method to give a sharp estimate on the k-core threshold of Erdős-Rényi random graphs. It turns out that the method can also be very useful in the study of G(n, µ) as shown in [2]. In this subsection, we introduce the algorithm and derive some properties that are used in Sections 2.2 and 2.3.…”

“…Our first goal is to derive estimates on H T . For our purpose, it is enough to have some cheap estimates as in [2], nevertheless sharper ones can be found in [15]. We first make the following simple observation.…”

Exponential growth and continuous phase transitions for the contact process on trees

“…The critical value is positive on Galton-Watson trees with tails thinner than the subexponential distribution. Recently Shankar Bhamidi, Danny Nam, Oanh Nguyen and Allan Sly [2] proved that Theorem 2. Consider the contact process on the Galton-Watson tree with offspring distribution D. If E(exp(cD)) < ∞ for some c > 0, then λ 1 > 0.…”

Critical value asymptotics for the contact process on random graphs

“…By contrast, if the offspring distribution ξ has an exponential tail, i.e., Ee cξ < ∞ for some c > 0, Bhamidi and the authors [2] showed that there is an extinction phase: λ gw 1 (ξ) ≥ λ 0 (ξ) for some constant λ 0 (ξ) > 0. Our first main result derives the first-order asymptotics on λ gw 1 (ξ) for ξ concentrated around its mean, which turns out to have the same form as (1.1).…”

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Recent progress in the study of the contact process [2] has verified that the extinctionsurvival threshold λ1 on a Galton-Watson tree is strictly positive if and only if the offspring distribution ξ has an exponential tail. In this paper, we derive the first-order asymptotics of λ1 for the contact process on Galton-Watson trees and its corresponding analog for random graphs.

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