Super-Exponential Extinction Time of the Contact Process on Random Geometric Graphs

Abstract

In this paper we prove lower and upper bounds for the extinction time of the contact process on random geometric graphs with connection radius tending to infinity. We obtain that for any infection rate λ > 0, the contact process on these graphs survives a time super-exponential in the number of vertices.

Exponential Extinction Time of the Contact Process on Rank-One Inhomogeneous Random Graphs

“…Similarly to Lemma 3.2 in [9], we can prove (18) by using a comparison between the contact process and an oriented percolation on 1, n with density close to 1. Note that here, we use a mechanism of infection between star graphs instead of complete graphs as it was the case in [9].…”

“…I, Theorem 6.6]. Finally, the convergence in law of τ n /E(τ n ) can be proved similarly as in the proof of Theorem 1.1 in [9] by using Propositions 3.1, 3.3 and the following:…”

“…Then by comparing with an oriented percolation with density close to 1, we get the lower bound. The convergence in law can be proved similarly as in [9].…”

“…The upper bound on τ n follows from a general result in [9]. To prove the lower bound, we will show that G n contains a sequence of disjoint star graphs with large degree, whose total size is of order n. Moreover, the distance between two consecutive star graphs is not too large, so that the virus starting from a star graph can infect the other one with high probability.…”

“…Observe that Proposition 3.1 and Lemma 3.2 imply the lower bound on τ n . On the other hand, the upper bound follows from Lemma 3.4 in [9] and the fact that |E n | ≍ n w.h.p., see [11, Vol. I, Theorem 6.6].…”