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Survival and extinction of epidemics on random graphs with general degrees
- Publication Year :
- 2019
-
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 $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution $\xi$ has an exponential tail, i.e., $\mathbb{E} e^{c\xi}<\infty$ for some $c>0$, settling a conjecture by Huang and Durrett [12]. On the random graph with degree distribution $\mu$, we show that if $\mu$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for $n^{1+o(1)}$-time w.h.p. (short survival), while for large enough $\lambda$ it runs over $e^{\Theta(n)}$-time w.h.p. (long survival). When $\mu$ is subexponential, we prove that the contact process w.h.p. displays long survival for any fixed $\lambda>0$.<br />Comment: 39 pages
- Subjects :
- Mathematics - Probability
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1902.03263
- Document Type :
- Working Paper