Survival and extinction of epidemics on random graphs with general\n degrees

2019/02/08 by Shankar Bhamidi, Bhamidi, Shankar, Danny Nam +6 · 1 citation
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1902.03263

openalex publication_date 2019/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish the necessary and sufficient criterion for the\ncontact process on Galton-Watson trees (resp. random graphs) to exhibit the\nphase of extinction (resp. short survival). We prove that the survival\nthreshold \λ₁ for a Galton-Watson tree is strictly positive if and only\nif its offspring distribution \ξ has an exponential tail, i.e., \𝔼≠^c\ξ<\∞ for some c>0, settling a conjecture by Huang and Durrett\n[12]. On the random graph with degree distribution \μ, we show that if \μ\nhas an exponential tail, then for small enough \λ the contact process\nwith the all-infected initial condition survives for n^1+o(1)-time w.h.p.\n(short survival), while for large enough \λ it runs over\ne^\Θ(n)-time w.h.p. (long survival). When \μ is subexponential, we\nprove that the contact process w.h.p. displays long survival for any fixed\n\λ>0.\n

Cited by

• The effects of individual versus community-influenced isolation on SIS epidemic persistence on finite random graphs

Related

• Critical value asymptotics for the contact process on random graphs
• The Contact Process on Random Graphs and Galton-Watson Trees
• Branching random walks and contact processes on Galton-Watson trees
• Exponential rate for the contact process extinction time
• Degree-penalized contact processes