Random Geometric Graph Diameter in the Unit Ball

Abstract

The unit ball random geometric graph $G=G_p^d(\lambda ,n)$ has as its vertices $n$ points distributed independently and uniformly in the $d$-dimensional unit ball, with two vertices adjacent if and only if their $l_p$-distance is at most $\lambda$. Like its cousin the Erdos-Renyi random graph, $G$ has a connectivity threshold: an asymptotic value for $\lambda$ in terms of $n$, above which $G$ is connected and below which $G$ is disconnected (and in fact has isolated vertices in most cases). In the connected zone, we determine upper and lower bounds for the graph diameter of $G$. Specifically, almost always, $\diam_p(\mathbf{B})(1-o(1))/λ\leq \diam(G) \leq \diam_p(\mathbf{B})(1+O((\ln \ln n/\ln n)^{1/d}))/λ$, where $\diam_p(\mathbf{B})$ is the ${\ell }_p$-diameter of the unit ball $\mathbf{B}$. We employ a combination of methods from probabilistic combinatorics and stochastic geometry.

17 pages, 4 figures; exposition revised substantially, particularly in Sections 3 and 5