Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Entropy of Soft Random Geometric Graphs in General Geometries

Oliver Baker, Carl P. Dettmann

Abstract

We study the effect of the choice of embedding geometry on the entropy of random geometric graph ensembles with soft connection functions. First we show that when the connection range is small, the entropy is dependent only on the dimension of the geometry and not the shape, but for large connection ranges the boundaries of the domain matter. Next, we formulate the problem of estimating entropy as a problem of estimating the average degree of a graph with the binary entropy function as its connection function. We use this formulation to study the effect of boundaries on the entropy, and to estimate the entropy of soft random geometric graphs in complicated geometries where a closed form pair distance density is not available.

Entropy of Soft Random Geometric Graphs in General Geometries

Abstract

We study the effect of the choice of embedding geometry on the entropy of random geometric graph ensembles with soft connection functions. First we show that when the connection range is small, the entropy is dependent only on the dimension of the geometry and not the shape, but for large connection ranges the boundaries of the domain matter. Next, we formulate the problem of estimating entropy as a problem of estimating the average degree of a graph with the binary entropy function as its connection function. We use this formulation to study the effect of boundaries on the entropy, and to estimate the entropy of soft random geometric graphs in complicated geometries where a closed form pair distance density is not available.
Paper Structure (27 sections, 13 theorems, 192 equations, 7 figures)

This paper contains 27 sections, 13 theorems, 192 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Asymptotic notation
  4. Soft Random Geometric Graphs
  5. Entropy of Soft Random Geometric Graphs
  6. Entropy Scaling in the Connection Range
  7. Small Connection Range
  8. Large Connection Range
  9. Compressibility of SRGGs vs ER Graphs
  10. The Entropy Graph
  11. The Effect of Boundaries for Small Connection Ranges
  12. Entropy Estimation for General Geometries
  13. Wedge - Corner Effects by Connectivity-Style Analysis
  14. Cantor Set - Complicated Geometry with Unavailable $f(r)$
  15. Conclusion
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\Omega$ satisfy the conditions of Assumption assumption_omega, and $0 < \eta < \infty$. Then as $r_0 \rightarrow 0$, where $s_{d-1}$ is the surface area of the unit $d$-ball, $\Gamma$ is the gamma function and $\zeta$ is the Riemann-Zeta function.

Figures (7)

  • Figure 1: The conditional entropy-per-edge of an SRGG with Rayleigh fading connection function ($\eta=2$), $\overline{H}(\mathcal{G}(r_0)|\mathcal{R})$, numerically integrated, with the asymptotic from Theorem \ref{['small_r0_asymptotic']} and equation (\ref{['higher_order']}) for $\Omega = [0,1]^d$, $d=1,2,3$
  • Figure 2: Simulated conditional entropy against the theoretical prediction of Theorem \ref{['thm:large_r0']} for $\Omega = [0,1]^d$ and $\eta=2$ for $d=1,2,3$ (top left, middle and right respectively), and $\Omega$ as the unit $d$ ball with $\eta=4$ for $d=2, 3$ (bottom left and right respectively).
  • Figure 3: Entropy mass at each point of the unit square. We see that the main contribution to the entropy comes from the bulk, then the edges, then the corners as $r_0$ increases (connection function $p(r/r_0) = \exp(-(r/r_0)^2)$).
  • Figure 4: Left: Entropy curves with $\eta = 2$ in sectors of unit radius with angles $\frac{\pi}{2}, \frac{\pi}{4}$ and $\frac{\pi}{8}$, Right: The same curves rescaled by multiplying $r_0$ by $\sqrt{2\theta/\pi}$
  • Figure 5: The entropy mass of each point in a wedge of angle $\theta = \pi/4$ for $r_0 = 0.05, 0.5, 1$
  • ...and 2 more figures

Theorems & Definitions (32)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1
  • Theorem 3
  • proof
  • Definition 2: Entropy Mass
  • Lemma 1
  • proof
  • ...and 22 more