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On the Entropy of a Random Geometric Graph

Praneeth Kumar Vippathalla, Justin P. Coon, Mihai-Alin Badiu

TL;DR

It is inferred 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$.

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 $Ω((d-1)m \log_2m)$ 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.

On the Entropy of a Random Geometric Graph

TL;DR

It is inferred that the asymptotic structural entropy of an RGG on , which is the entropy of an unlabelled RGG, is for 0<r \leq 1/4$.

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 , a hard RGG on vertices is formed by drawing random points from a spatial domain, and then connecting any two points with an edge when they are within a distance from each other. The two domains we consider are the -dimensional unit cube and the -dimensional unit torus . We derive upper bounds on the entropy for both these domains and for all possible values of . In a few cases, we obtain an exact asymptotic characterization of the entropy by proving a tight lower bound. Our main results are that for in the case of and that the entropy of a one-dimensional RGG on behaves like for all . As a consequence, we can infer that the asymptotic structural entropy of an RGG on , which is the entropy of an unlabelled RGG, is for . For the rest of the cases, we conjecture that the entropy behaves asymptotically as the leading order terms of our derived upper bounds.
Paper Structure (17 sections, 12 theorems, 79 equations, 2 figures)

This paper contains 17 sections, 12 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Entropy of $G_m$
  4. Entropy of an RGG
  5. $d$-dimensional unit cube $[0,1]^d$
  6. $d$-dimensional unit torus $\mathbb{T}^d$
  7. Structural Entropy of an RGG
  8. Proofs
  9. Proof of Theorem \ref{['thm:ent_rgg_upper_euclidean']}: Upper bound in the case of $[0,1]^d$
  10. Proof of Theorem \ref{['thm:ent_rgg_upper_torus']}: Upper bound in the case of $\mathbb{T}^d$
  11. Proof of Theorem \ref{['thm:ent_rgg_lower_torus']}: Lower Bound in the case of $\mathbb{T}^d$
  12. Conclusion
  13. A general approach for proving a lower bound on the entropy of an RGG
  14. Proof of Theorem \ref{['thm:ent_rgg_lower_one_d']}: Lower bound on the entropy of the one dimensional RGG $(d=1)$ on $[0,1]$
  15. $0< r\leq \frac{1}{2}$
  16. ...and 2 more sections

Key Result

Theorem 1

An upper bound on the entropy of an RGG $G_m$ on the $d$-dimensional unit cube $[0,1]^d$ is given by where $\beta(r)$ is the volume of the ball $B\left((1/2,1/2,\ldots,1/2); r-\sqrt{d}/2\right) \cap [0,1]^d$.

Figures (2)

  • Figure 1: Limit of $H(G_m)/m\log m$ of a one-dimensional RGG
  • Figure 2: The volume of the two crescents $C_{y,y'}$ is twice the volume of the unshaded region of the sphere centered at origin shown on the right side. The unshaded region is lower bounded by the volume enclosed by the two cones whose bases are in the hyperplane $y_1=0$ with radii $R$ and the vertices are at points $\left(-\frac{||y-y'||}{2}, 0,\ldots, 0\right)$ and $\left(\frac{||y-y'||}{2}, 0,\ldots, 0\right)$.

Theorems & Definitions (21)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 11 more