A critical constant for the k nearest neighbour model

TL;DR

This paper establishes a critical constant for the connectivity of k-nearest neighbor graphs generated from a Poisson process, identifying the threshold between almost sure connectivity and disconnection as the domain size grows.

Contribution

The authors determine the precise critical constant for the phase transition in the connectivity of k-nearest neighbor graphs in a Poisson process setting.

Findings

Existence of a critical constant c for connectivity transition.

For c' < c, the graph is disconnected with high probability.

For c' > c, the graph is connected with high probability.

Abstract

Let P be a Poisson process of intensity one in a square S_n of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph G_{n,k}. We prove that there exists a critical constant c such that for c'<c, G_{n,c'log n} is disconnected with probability tending to 1 as n tends to infinity, and for c'>c G_{n,c'\log n} is connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors.