The contact process on random graphs and Galton Watson trees

Abstract

The key to our investigation is an improved (and in a sense sharp) understanding of the survival time of the contact process on star graphs. Using these results, we show that for the contact process on Galton-Watson trees, when the offspring distribution (i) is subexponential the critical value for local survival λ2 = 0 and (ii) when it is geometric(p) we have λ2 ≤ Cp, where the Cp are much smaller than previous estimates. We also study the critical value λc(n) for "prolonged persistence" on graphs with n vertices generated by the configuration model. In the case of power law and stretched exponential distributions where it is known λc(n) → 0 we give estimates on the rate of convergence. Physicists tell us that λc(n) ~ 1/Λ(n) where Λ(n) is the maximum eigenvalue of the adjacency matrix. Our results show that this is accurate for graphs with power-law degree distributions, but not for stretched exponentials.