Limit theorems for the number of crossings and stress in projections of the random geometric graph

TL;DR

This paper studies the asymptotic behavior of edge crossings and stress in projections of random geometric graphs, establishing convergence to Poisson processes and central limit theorems under certain conditions.

Contribution

It introduces new limit theorems for the number of crossings and stress in projected random geometric graphs, extending understanding of their probabilistic structure.

Findings

Number of crossings converges to a Poisson process under finite expected crossings.

A multivariate CLT relates crossings and stress when expected degree is finite.

Provides quantitative bounds in Kantorovich-Rubinstein distance.

Abstract

We consider the number of edge crossings in a random graph drawing generated by projecting a random geometric graph on some compact convex set $W\subset \mathbb{R}^d$, $d\geq 3$, onto a plane. The positions of these crossings form the support of a point process. We show that if the expected number of crossings converges to a positive but finite value, this point process converges to a Poisson point process in the Kantorovich-Rubinstein distance. We further show a multivariate central limit theorem between the number of crossings and a second variable called the stress that holds when the expected vertex degree in the random geometric graph converges to a positive finite value.