On the Entropy of a Random Geometric Graph

TL;DR

This paper investigates the entropy of random geometric graphs (RGGs), deriving bounds and asymptotic behaviors for different domains and connection ranges, revealing how the structural complexity scales with the number of nodes.

Contribution

The paper provides the first asymptotic characterizations and bounds for the entropy of RGGs on various domains, advancing understanding of their structural complexity.

Findings

Entropy of RGGs on the torus scales as dm log m for small r

One-dimensional RGG entropy behaves like m log m for all r

Asymptotic structural entropy is at least proportional to (d-1)m log m

Abstract

In this paper, we study the entropy of a hard random geometric graph (RGG), a commonly used model for spatial networks, where the connectivity is governed by the distances between the nodes. Formally, given a connection range $r$, a hard RGG $G_m$ on $m$ vertices is formed by drawing $m$ random points from a spatial domain, and then connecting any two points with an edge when they are within a distance $r$ from each other. The two domains we consider are the $d$-dimensional unit cube $[0,1]^d$ and the $d$-dimensional unit torus $\mathbb{T}^d$. We derive upper bounds on the entropy $H(G_m)$ for both these domains and for all possible values of $r$. In a few cases, we obtain an exact asymptotic characterization of the entropy by proving a tight lower bound. Our main results are that $H(G_m) \sim dm \log_2m$ for $0 < r \leq 1/4$ in the case of $\mathbb{T}^d$ and that the entropy of a one-dimensional RGG on $[0,1]$ behaves like $m\log m$ for all $0<r<1$. As a consequence, we can infer that the asymptotic structural entropy of an RGG on $\mathbb{T}^d$, which is the entropy of an unlabelled RGG, is $\Omega((d-1)m \log_2m)$ for $0 < r \leq 1/4$. For the rest of the cases, we conjecture that the entropy behaves asymptotically as the leading order terms of our derived upper bounds.