The contact process on the complete graph with random vertex-dependent infection rates

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:121

Article

Peterson, Jonathon

关键词: Contact process;    Random environment;    Phase transition;   

DOI  :  10.1016/j.spa.2010.11.003

来源: Elsevier

【 摘 要 】

We study the contact process on the complete graph on n vertices where the rate at which the infection travels along the edge connecting vertices i and j is equal to lambda omega i omega j/n for some lambda > 0, where omega i are i.i.d. vertex weights. We show that when E[omega(2)(1)] < infinity there is a phase transition at A, > 0 such that for lambda < lambda(c) the contact process dies out in logarithmic time, and for lambda > lambda(c) the contact process lives for an exponential amount of time. Moreover, we give a formula for lambda(c) and when lambda > lambda(c) we are able to give precise approximations for the probability that a given vertex is infected in the quasi-stationary distribution. Our results are consistent with a non-rigorous mean field analysis of the model. This is in contrast to some recent results for the contact process on power law random graphs where the mean field calculations suggested that lambda(c) > 0 when in fact lambda(c) = 0. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free