Random geometric graphs and the spherical Wishart matrix

TL;DR

This paper studies the properties of random geometric graphs on high-dimensional spheres, showing their similarity to Erdős–Rényi graphs in certain regimes and analyzing the spherical Wishart matrix to derive key statistical characteristics.

Contribution

It establishes the equivalence of high-dimensional spherical geometric graphs to Erdős–Rényi graphs in the sparse regime and derives exact statistics of the spherical Wishart matrix.

Findings

Graph laws are comparable in inclusion divergence for large dimensions.

Spherical Wishart matrix characteristic function is derived and approximated.

High-dimensional spherical geometric graphs can be distinguished from Erdős–Rényi graphs using inclusion divergence.

Abstract

We consider the random geometric graph on $n$ vertices drawn uniformly from a $d$--dimensional sphere. We focus on the sparse regime, when the expected degree is constant independent of $d$ and $n$. We show that, when $d$ is larger than $n$ by logarithmic factors, this graph is comparable to the Erd\H{o}s--R\'enyi random graph of the same edge density in the \emph{inclusion divergence} between the graph laws. This divergence functions in certain ways like a relaxation of the total variation distance, but is strong enough to distinguish Erd\H{o}s--R\'enyi graphs of different densities with a higher resolution than the total variation distance. To do the analysis, we derive some exact statistics of the \emph{spherical Wishart matrix}, the Gram matrix of $n$ independent uniformly random $d$--dimensional spherical vectors. In particular we give expressions for the characteristic function of the spherical Wishart matrix which are well--approximated using steepest descent.