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

We study the contact process on the complete graph on nn vertices where the rate at which the infection travels along the edge connecting vertices ii and jj is equal to λwiwj/nλwiwj/n for some λ>0λ>0, where wiwi are i.i.d. vertex weights. We show that when E[w12]<∞ there is a phase transition at λc>0λc>0 such that for λ<λcλ<λc the contact process dies out in logarithmic time, and for λ>λcλ>λc the contact process lives for an exponential amount of time. Moreover, we give a formula for λcλc and when λ>λcλ>λc 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 λc>0λc>0 when in fact λc=0λc=0.