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Entropy of Random Geometric Graphs in High and Low Dimensions

Oliver Baker, Carl P. Dettmann

TL;DR

This work analyzes the Shannon entropy of labelled random geometric graph ensembles in high and low dimensions on the cube $[0,1]^d$ and the torus $\mathbb{T}^d$, using a multivariate central limit theorem and an Edgeworth expansion. It shows that, for uniform-node distributions, hard RGGs on the torus converge to the Erdős–Rényi ensemble as $d\to\infty$, while nonuniform-node distributions and any kurtosis$>1$ in the cube prevent ER convergence and yield lower entropy; soft RGGs converge to ER in both geometries. The authors provide exact entropy calculations for 3-node hard RGGs in 1D, numerical simulations in low dimensions, and a detailed Edgeworth correction that captures the leading $O(d^{-1/2})$ scaling of entropy. These results have implications for high-dimensional geometry testing and the design of proximity networks, highlighting how node distributions shape asymptotic randomness. The work also demonstrates that, while ER is a universal limit for soft RGGs, hard RGGs retain memory of the underlying spatial distribution in high dimensions.

Abstract

We use a multivariate central limit theorem (CLT) to study the distribution of random geometric graphs (RGGs) on the cube and torus in the high-dimensional limit with general node distributions. We find that the distribution of RGGs on the torus converges to the Erd\H os-Rényi (ER) ensemble when the nodes are uniformly distributed, but that the distribution for RGGs with non-uniformly distributed nodes on the torus, and for RGGs with any distribution of nodes with kurtosis greater than 1 on the cube is different. In these cases, the distribution has a lower maximum entropy than the ER ensemble, but is still symmetric. Soft RGGs in either geometry converge to the ER ensemble. An Edgeworth correction to the CLT is then developed to derive the $\mathcal{O}\left(d^{-\frac{1}{2}}\right)$ sub-leading term of the Shannon entropy of RGGs in dimension for both geometries. We also provide numerical approximations of maximum entropy in low-dimensional hard and soft RGGs, and calculate exactly the entropy of hard RGGs with 3 nodes in the one-dimensional cube and torus.

Entropy of Random Geometric Graphs in High and Low Dimensions

TL;DR

This work analyzes the Shannon entropy of labelled random geometric graph ensembles in high and low dimensions on the cube and the torus , using a multivariate central limit theorem and an Edgeworth expansion. It shows that, for uniform-node distributions, hard RGGs on the torus converge to the Erdős–Rényi ensemble as , while nonuniform-node distributions and any kurtosis in the cube prevent ER convergence and yield lower entropy; soft RGGs converge to ER in both geometries. The authors provide exact entropy calculations for 3-node hard RGGs in 1D, numerical simulations in low dimensions, and a detailed Edgeworth correction that captures the leading scaling of entropy. These results have implications for high-dimensional geometry testing and the design of proximity networks, highlighting how node distributions shape asymptotic randomness. The work also demonstrates that, while ER is a universal limit for soft RGGs, hard RGGs retain memory of the underlying spatial distribution in high dimensions.

Abstract

We use a multivariate central limit theorem (CLT) to study the distribution of random geometric graphs (RGGs) on the cube and torus in the high-dimensional limit with general node distributions. We find that the distribution of RGGs on the torus converges to the Erd\H os-Rényi (ER) ensemble when the nodes are uniformly distributed, but that the distribution for RGGs with non-uniformly distributed nodes on the torus, and for RGGs with any distribution of nodes with kurtosis greater than 1 on the cube is different. In these cases, the distribution has a lower maximum entropy than the ER ensemble, but is still symmetric. Soft RGGs in either geometry converge to the ER ensemble. An Edgeworth correction to the CLT is then developed to derive the sub-leading term of the Shannon entropy of RGGs in dimension for both geometries. We also provide numerical approximations of maximum entropy in low-dimensional hard and soft RGGs, and calculate exactly the entropy of hard RGGs with 3 nodes in the one-dimensional cube and torus.

Paper Structure

This paper contains 25 sections, 10 theorems, 179 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. RGG Entropy for Small $n$ and $d$
  4. Exact Entropy Calculations
  5. Exact Entropy Calculation for $\Omega = \mathbb{T}$
  6. Exact Entropy Calculation for $\Omega = [0,1]$
  7. Simulations for $d \in \{1,2,3\},$ $n=3$
  8. RGG Entropy in High Dimensions
  9. Central Limit Theorem for High Dimensional Cubes
  10. Central Limit Theorem for High-Dimensional Tori
  11. Entropy of Hard and Soft RGG Ensembles in High Dimensional Tori
  12. Entropy of Hard RGG Ensembles in High Dimensional Tori
  13. Non-Convergence to $G(n,p)$ for Non-Uniform Node Distributions on $\mathbb{T}^d$
  14. Entropy of Soft RGG Ensembles in High Dimensional Tori
  15. Entropy of Hard RGG Ensembles in the High Dimensional Cube
  16. ...and 10 more sections

Key Result

Lemma 1

Let $X_1,...,X_n$ be i.i.d. points distributed according to $\nu$ of the form (eq:general_distribution) in $[0,1]^d$. Then in the limit $d\rightarrow\infty$, their normalised squared distances satisfy in distribution, where $Z \sim N(0_{\binom{n}{2}}, \Sigma_c)$ is a Gaussian random vector, and $\Sigma_c = \{\mathbb{E}[|q_{ij}-\mu_c||q_{kl}-\mu_c|]\}_{1\leq i<j\leq n, 1\leq k<l \leq n}$ is the co

Figures (6)

  • Figure 1: Exact entropy (left) and edge probability (right) curves in $r_0$ for 3-node hard RGGs with uniform node distributions in $\mathbb{T}^1$ (top) and $[0,1]$ (bottom). The dotted line shows the $r_0$ value where $p_0$ becomes 0, and where $p_1=p_2$.
  • Figure 2: Top: Normalised distribution of edge count in the $d\rightarrow\infty$ limit for the hard RGG in the cube and torus, with uniformly distributed nodes for $n=3$ (top left), and $n=7$ (top right). Bottom: Comparison of the unnormalised distribution of edge counts for $n=7$ for hard RGGS with uniformly distributed nodes in the cube and torus.
  • Figure 3: Top: Normalised distribution of edge count in the $d\rightarrow\infty$ limit for the hard RGG in the cube (red) and torus (blue), with Gaussian distributed nodes for $n=3$ (top left), and $n=7$ with a log scale for the probabilities (top right). Bottom: Comparison of the unnormalised distribution of edge counts for $n=7$ for hard RGGS with Gaussian distributed nodes in the cube and torus.
  • Figure 4: Entropy curves in the theoretical $d\rightarrow\infty$ limit for hard RGGs in the cube, with $n=3$ and $11$ nodes and uniformly (left) and Gaussian (right) distributed nodes. All 4 curves show evidence for a unique maximum at $t=0$.
  • Figure 5: The entropy estimate for an ensemble of $4$-node hard RGGs with uniform distributed nodes in the $d$-dimensional hypercube (left) and hypertorus (right) given by the Gaussian and Edgeworth estimates against the simulated entropy. The dotted lines show a fit of the curve $H(\mathcal{G}) = a-b(d^{-\frac{1}{2}}+c)$. The solid line is the entropy of the uniform distribution on $\binom{4}{2}=6$ variables, which is the theoretical maximum entropy.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6: Normalised Euclidean Distance
  • Lemma 1: erba2020random
  • proof
  • Remark 1
  • Definition 7
  • ...and 24 more