On the Distribution of Random Geometric Graphs

TL;DR

This paper derives the joint distribution of distances among three nodes in a disk, enabling analysis of random geometric graphs' entropy and connectivity, with implications for network modeling and protocol design.

Contribution

It provides the first joint distribution of three nodes in a disk, improving entropy bounds for arbitrary node counts in random geometric graphs.

Findings

Derived the joint distribution of three nodes in a disk.

Established tighter entropy bounds for random geometric graphs.

Numerical analysis of graph connectedness and entropy bounds.

Abstract

Random geometric graphs (RGGs) are commonly used to model networked systems that depend on the underlying spatial embedding. We concern ourselves with the probability distribution of an RGG, which is crucial for studying its random topology, properties (e.g., connectedness), or Shannon entropy as a measure of the graph's topological uncertainty (or information content). Moreover, the distribution is also relevant for determining average network performance or designing protocols. However, a major impediment in deducing the graph distribution is that it requires the joint probability distribution of the $n(n-1)/2$ distances between $n$ nodes randomly distributed in a bounded domain. As no such result exists in the literature, we make progress by obtaining the joint distribution of the distances between three nodes confined in a disk in $\mathbb{R}^2$. This enables the calculation of the probability distribution and entropy of a three-node graph. For arbitrary $n$, we derive a series of upper bounds on the graph entropy; in particular, the bound involving the entropy of a three-node graph is tighter than the existing bound which assumes distances are independent. Finally, we provide numerical results on graph connectedness and the tightness of the derived entropy bounds.