On the components of random geometric graphs in the dense limit

TL;DR

This paper analyzes the asymptotic behavior of components in dense random geometric graphs, providing formulas for their expected counts, variances, and distributional limits in various dimensions and settings.

Contribution

It introduces new asymptotic formulas for component counts in dense regimes, extending results to non-uniform distributions and different dimensions.

Findings

Asymptotic formulas for the number of components, giant component, and isolated vertices.

Variance asymptotics and CLTs for component counts in high dimensions.

Extension of results to non-uniform point distributions.

Abstract

Consider the geometric graph on $n$ independent uniform random points in a connected compact region $A$ of ${\bf R}^d, d \geq 2$, with $C^2$ boundary, or in the unit square, with distance parameter $r_n$. Let $K_n$ be the number of components of this graph, and $R_n$ the number of vertices not in the giant component. Let $S_n$ be the number of isolated vertices. We show that if $r_n$ is chosen so that $nr_n^d$ tends to infinity but slowly enough that ${\bf E}[S_n]$ also tends to infinity, then $K_n$, $R_n$ and $S_n$ are all asymptotic to $\mu_n$ in probability as $n \to \infty$ where (with $|A|$, $\theta_d$ and $|\partial A|$ denoting the volume of $A$, of the unit $d$-ball, and the perimeter of $A$ respectively) $\mu_n := ne^{-\pi n (r_n)^d/|A|}$ if $d=2$ and $\mu_n := ne^{-\theta_d n (r_n)^d/|A|} + \theta_{d-1}^{-1} |\partial A| (r_n)^{1-d} e^{- \theta_d n (r_n)^d/(2|A|)}$ if $d\geq 3$. We also give variance asymptotics and central limit theorems for $K_n$ and $R_n$ in this limiting regime when $d \geq 3$, and for Poisson input with $d \geq 2$. We extend these results (substituting ${\bf E}[S_n]$ for $\mu_n$) to a class of non-uniform distributions on $A$.