The Contact Process on the Complete Graph with Random Vertex-dependent Infection Rates
نویسنده
چکیده
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 λwiwj/n for some λ > 0, where wi are i.i.d. vertex weights. We show that when E[w 2 1] < ∞ there is a phase transition at λc > 0 so that for λ < λc the contact process dies out in logarithmic time, and for λ > λc the contact process lives for an exponential amount of time. Moreover, we give a formula for λc and when λ > λc we are able to give precise approximations for the probability 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 λc > 0 when in fact λc = 0.
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