Limiting shape for first-passage percolation models on random geometric graphs

Abstract

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape, and we show that the shape is a Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. In the latter case we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times.

Chase–escape in dynamic device-to-device networks

“…This is the setting of first-passage percolation (FPP) [1,30] and results often concern the speed of the data transmission over large (Euclidean) distances and the associated (asymptotic) shape of the set of vertices that have received the message up to some fixed time. In [12], such a shape theorem is presented for FPP on the supercritical Poisson-Gilbert graph. Conceiving the message as some infection, these growth models are often presented in the context of spatial probabilistic epidemiology, and then it is natural to generalize the pure growth process of FPP to processes in which infected devices can also spontaneously heal (or messages can be dropped), giving rise to contact processes on random graphs [32,33].…”