Abstract
We consider the contact process on a random graph with fixed degree distribution given by a power law. We follow the work of Chatterjee and Durrett (2009), who showed that for arbitrarily small infection parameter $\lambda$, the survival time of the process is larger than a stretched exponential function of the number of vertices, $n$. We obtain sharp bounds for the typical density of infected sites in the graph, as $\lambda$ is kept fixed and $n$ tends to infinity. We exhibit three different regimes for this density, depending on the tail of the degree law.
Highlights
In this paper we study the contact process on a random graph with a fixed degree distribution equal to a power law
The contact process is an interacting particle system that is commonly taken as a model for the spread of an infection in a population
We study the contact process on the Galton-Watson tree started with the root infected,t≥0
Summary
In this paper we study the contact process on a random graph with a fixed degree distribution equal to a power law. Vin,k , with 1 ≤ in,1 < · · · < in,k ≤ n) With this convergence at hand, in [2], the right-hand side of (1.5) (and by a second moment argument, the density of infected sites) is shown to be related to the probability of survival of the contact process on the random tree given by the measure Qp,q. Since we concentrate our efforts in proving Proposition 1.4, in all the remaining sections of the paper (except the Appendix) we do not consider the random graph Gn. Rather, we study the contact process on the Galton-Watson tree started with the root infected, (ξto)t≥0.
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