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

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.