Infection spread in random geometric graphs

Abstract

In this paper we study the speed of infection spread and the survival of the contact process in the random geometric graph G = G(n, r_n, f) of n nodes independently distributed in S = [-½, ½]² according to a certain density f(·). In the first part of the paper we assume that infection spreads from one node to another at unit rate and that infected nodes stay in the same state forever. We provide an explicit lower bound on the speed of infection spread and prove that infection spreads in G with speed at least D₁nr_n². In the second part of the paper we consider the contact process ξ_t on G where infection spreads at rate λ > 0 from one node to another and each node independently recovers at unit rate. We prove that, for every λ > 0, with high probability, the contact process on G survives for an exponentially long time; there exist positive constants c₁ and c₂ such that, with probability at least 1 - c₁ / n⁴, the contact process ξ¹_t starting with all nodes infected survives up to time t_n = exp(c₂n/logn) for all n.