Connectivity of soft random geometric graphs

Abstract

Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the inter-point distance is at most $r$. We show that as $n\to\infty$ the probability of full connectivity is governed by that of having no isolated vertices, itself governed by a Poisson approximation for the number of isolated vertices, uniformly over all choices of $p,r$. We determine the asymptotic probability of connectivity for all $(p_{n},r_{n})$ subject to $r_{n}=O(n^{-\varepsilon })$, some $\varepsilon >0$. We generalize the first result to higher dimensions and to a larger class of connection probability functions.