On the metric distortion of nearest-neighbour graphs on random point sets

TL;DR

This paper investigates the properties of nearest-neighbour graphs on random point sets, providing bounds on the critical number of connections needed for an infinite cluster and analyzing metric distortions relevant to wireless networks.

Contribution

It improves the upper bound on the critical value for percolation in 2D nearest-neighbour graphs and examines metric distortion properties of the infinite cluster.

Findings

Upper bound of 188 on critical k for percolation in 2D

Existence of an infinite subset with bounded metric distortion

Discussion of implications for wireless sensor networks

Abstract

We study the graph constructed on a Poisson point process in $d$ dimensions by connecting each point to the $k$ points nearest to it. This graph a.s. has an infinite cluster if $k > k_c(d)$ where $k_c(d)$, known as the critical value, depends only on the dimension $d$. This paper presents an improved upper bound of 188 on the value of $k_c(2)$. We also show that if $k \geq 188$ the infinite cluster of $\NN(2,k)$ has an infinite subset of points with the property that the distance along the edges of the graphs between these points is at most a constant multiplicative factor larger than their Euclidean distance. Finally we discuss in detail the relevance of our results to the study of multi-hop wireless sensor networks.