Distribution of components in the k-nearest neighbour random geometric graph for k below the connectivity threshold

TL;DR

This paper analyzes the structure of k-nearest neighbor random geometric graphs near the connectivity threshold, showing that small components are unlikely to be close together and that component distribution is asymptotically Poisson.

Contribution

It proves that the probability of close small components is negligible and establishes the asymptotic Poisson distribution of components below the connectivity threshold.

Findings

Small components rarely occur close together near the threshold

Component distribution converges to a Poisson process

Answers a question posed by Walters

Abstract

Let S_{n,k} denote the random geometric graph obtained by placing points inside a square of area n according to a Poisson point process of intensity 1 and joining each such point to the k=k(n) points of the process nearest to it. In this paper we show that if Pr(S_{n,k} connected) > n^{-\gamma_1} then the probability that S_{n,k} contains a pair of `small' components `close' to each other is o(n^{-c_1}) (in a precise sense of `small' and 'close'), for some absolute constants \gamma_1>0 and c_1 >0. This answers a question of Walters. (A similar result was independently obtained by Balister.) As an application of our result, we show that the distribution of the connected components of S_{n,k} below the connectivity threshold is asymptotically Poisson.