On the Entropy of a Random Geometric Graph

2026-01-15

AI Summary

This study investigates the asymptotic behavior of the structural entropy of unlabeled random geometric graphs (RGGs) on the unit cube and the d-dimensional unit torus 𝕋ᵈ as a function of the number of nodes m and the connection radius r. Employing tools from probabilistic graph theory, information-theoretic entropy analysis, and geometric probability—along with careful upper and lower bound constructions—the work establishes, for the first time, a tight asymptotic characterization of structural entropy on high-dimensional tori: when 0 < r ≤ 1/4, H(Gₘ) ∼ d m log₂ m; on the one-dimensional interval [0,1], H(Gₘ) ∼ m log m for any 0 < r < 1. Furthermore, the paper derives a general lower bound of Ω((d−1)m log₂ m) for the structural entropy of unlabeled RGGs, substantially advancing the theoretical understanding of information complexity in high-dimensional random geometric graphs.

Technology Category

• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• random geometric graph
• entropy
• spatial networks
• asymptotic behavior
• structural entropy

Innovation

• Random Geometric Graph
• Entropy
• Asymptotic Analysis
• Structural Entropy
• Spatial Networks