The Contact Process on Random Graphs and Galton-Watson Trees
arXiv:1810.06040
The paper analyzes the contact process on random graphs and Galton‑Watson trees, providing sharper estimates for survival times on star graphs and determining critical infection rates for local survival under various offspring distributions.
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 $λ_2=0$ and (ii) when it is geometric($p$) we have $λ_2 \le C_p$, where the $C_p$ are much smaller than previous estimates. We also study the critical value $λ_c(n)$ 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 $λ_c(n) \to 0$ we give estimates on the rate of convergence. Physicists tell us that $λ_c(n) \sim 1/Î(n)$ where $Î(n)$ is the maximum eigenvalue of the adjacency matrix. Our results show that this is not correct.
25 pages, 2 figures