Highly connected random geometric graphs

Abstract

Let $\mathcal{P}$ be a Poisson process of intensity 1 in a square $S_n$ of area $n$. We construct a random geometric graph $G_{n,k}$ by joining each point of $\mathcal{P}$ to its $k$ nearest neighbours. For many applications it is desirable that $G_{n,k}$ is highly connected, that is, it remains connected even after the removal of a small number of its vertices. In this paper we relate the study of the $s$-connectivity of $G_{n,k}$ to our previous work on the connectivity of $G_{n,k}$. Roughly speaking, we show that for $s=o(logn)$, the threshold (in $k$) for $s$-connectivity is asymptotically the same as that for connectivity, so that, as we increase $k$, $G_{n,k}$ becomes $s$-connected very shortly after it becomes connected.