Entropy of Soft Random Geometric Graphs in General Geometries

TL;DR

This paper investigates how the shape and size of the embedding space influence the entropy of soft random geometric graphs, revealing boundary effects and providing estimation methods for complex geometries.

Contribution

It introduces a formulation linking entropy estimation to average degree and boundary effects in diverse geometries, advancing understanding of geometric graph entropy.

Findings

Entropy depends only on dimension for small connection ranges.

Boundary effects become significant for large connection ranges.

A new estimation approach for complex geometries without closed-form densities.

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.