Fluctuations of the connectivity threshold and largest nearest-neighbour link

TL;DR

This paper analyzes the asymptotic distribution of the connectivity threshold in random geometric graphs, revealing different limiting behaviors depending on dimension and connectivity level, with implications for boundary effects and nearest-neighbor links.

Contribution

It provides new asymptotic distribution results for the connectivity threshold and nearest-neighbor links in random geometric graphs, including boundary effect considerations.

Findings

In 2D, the scaled threshold converges to a Gumbel distribution.

For other (d,k) cases, the limit involves a Gumbel or extreme value distribution.

Results extend to Poisson and non-uniform samples.

Abstract

Consider a random uniform sample of $n$ points in a compact region $A$ of Euclidean $d$-space, $d \geq 2$, with a smooth or (when $d=2$) polygonal boundary. Fix $k \in {\bf N}$. Let $T_{n,k}$ be the threshold $r$ at which the geometric graph on these $n$ vertices with distance parameter $r$ becomes $k$-connected. We show that if $d=2$ then $n (\pi/|A|) T_{n,1}^2 - \log n$ is asymptotically standard Gumbel. For $(d,k) \neq (2,1)$, it is $n (\theta_d/|A|) T_{n,k}^d - (2-2/d) \log n - (4-2k-2/d) \log \log n$ that converges in distribution to a nondegenerate limit, where $\theta_d$ is the volume of the unit ball. The limit is Gumbel with scale parameter 2 except when $(d,k)=(2,2)$ where the limit is two component extreme value distributed. The different cases reflect the fact that boundary effects are more more important in some cases than others. We also give similar results for the largest $k$-nearest neighbour link $U_{n,k}$ in the sample, and show $T_{n,k}=U_{n,k}$ with high probability. We provide estimates on rates of convergence and give similar results for Poisson samples in $A$. Finally, we give similar results even for non-uniform samples, with a less explicit sequence of centring constants.