Abstract: We propose a \italic {distribution free} approach to the study of random geometric graphs. The distribution of vertices follows a Poisson point process with intensity function $nf(\cdot)$, where $n \in \mathcal{N}$, and $f$ is a probability density function on $\mathcal{R}^d$. A vertex located at $x$ connects via directed edges to other vertices that are within a \italic{cut-off} distance $r_n(x)$. We prove strong law results for, (i) the critical cut-off function so that almost surely, the graph does not contain any node with out-degree zero for sufficiently large $n$, (ii) the maximum and minimum vertex degrees. We also provide a characterization of the cut-off function for which the number of nodes with out-degree zero converges in distribution to a Poisson random variable. We illustrate this result for a class of densities with compact support that have at most polynomial rates of decay to zero. Finally, we state a sufficient condition for an enhanced version of the above graph to be almost surely connected eventually.

This is a joint work with Srikanth K. Iyer.