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We propose a distribution-free approach to the study of random geometricgraphs. The distribution of vertices follows a Poisson point process withintensity function $nf(\cdot)$, where $n\in \mathbb{N}$, and $f$ is aprobability density function on $\mathbb{R}^d$. A vertex located at $x$connects via directed edges to other vertices that are within a cut-offdistance $r_n(x)$. We prove strong law results for (i) the critical cut-offfunction so that almost surely, the graph does not contain any node without-degree zero for sufficiently large $n$ and (ii) the maximum and minimumvertex degrees. We also provide a characterization of the cut-off function forwhich the number of nodes with out-degree zero converges in distribution to aPoisson random variable. We illustrate this result for a class of densitieswith compact support that have at most polynomial rates of decay to zero.Finally, we state a sufficient condition for an enhanced version of the abovegraph to be almost surely connected eventually.

Nonuniform random geometric graphs with location-dependent radii