  We propose a 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 $n\in \mathbb{N}$, and $f$ is a
probability density function on $\mathbb{R}^d$. A vertex located at $x$
connects via directed edges to other vertices that are within a 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 $n$ and (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.

Probability