Nonuniform random geometric graphs with location-dependent radii

TL;DR

This paper introduces a distribution-free framework for analyzing nonuniform random geometric graphs with location-dependent radii, providing laws of large numbers, degree distribution characterizations, and connectivity conditions.

Contribution

It develops a novel approach to study nonuniform random geometric graphs with variable radii, including laws of large numbers and degree distribution results.

Findings

Strong law results for critical cut-off functions

Characterization of degree zero node distribution as Poisson

Sufficient conditions for eventual connectivity

Abstract

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.