Small components in k-nearest neighbour graphs

TL;DR

This paper investigates the connectivity threshold of k-nearest neighbour graphs formed in a square, showing boundary vertices are part of the giant component near the threshold and improving the known bounds for connectivity.

Contribution

It proves boundary vertices are included in the giant component near the connectivity threshold and refines the upper bound for the threshold to 0.4125 log n.

Findings

Vertices near the boundary are part of the giant component near the threshold.

Improved upper bound for connectivity threshold to 0.4125 log n.

Connectivity probability transitions are clarified around the threshold.

Abstract

Let $G=G_{n,k}$ denote the graph formed by placing points in a square of area $n$ according to a Poisson process of density 1 and joining each point to its $k$ nearest neighbours. Balister, Bollob\'as, Sarkar and Walters proved that if $k<0.3043\log n$ then the probability that $G$ is connected tends to 0, whereas if $k>0.5139\log n$ then the probability that $G$ is connected tends to 1. We prove that, around the threshold for connectivity, all vertices near the boundary of the square are part of the (unique) giant component. This shows that arguments about the connectivity of $G$ do not need to consider `boundary' effects. We also improve the upper bound for the threshold for connectivity of $G$ to $k=0.4125\log n$.