Connectivity of random k-nearest-neighbour graphs

Abstract

Let 𝓅 be a Poisson process of intensity one in a square S n of area n. We construct a random geometric graph G n,k by joining each point of 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that G n, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, then the probability that G n, k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n for the directed version of this problem. A related question concerns coverage. With G n, k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209 log n then the probability that these discs cover S n tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover S n tends to 1 as n → ∞.

Highly connected random geometric graphs

“…However, the value of this constant c remains unknown. Numerical results [1] indicate that it is close to the above lower bound, namely 0.3043.…”

“…Section 3 contains our results on connectivity. We need the main theorem of [1] and three technical lemmas before we can begin. These are followed by Lemma 7, which embodies the main idea relating s-connectivity to connectivity, and it together with Lemma 8 enables us to establish Theorem 1.…”

“…These are followed by Lemma 7, which embodies the main idea relating s-connectivity to connectivity, and it together with Lemma 8 enables us to establish Theorem 1. For Theorem 2, we require a strengthened version of a sharpness result ("sharpness in n") from [1], and for Theorem 10 we need to prove a new sharpness result ("sharpness in k") which we believe is of considerable interest in its own right. Section 4 contains analogous results for coverage, and we conclude with some open problems in Section 5.…”

“…In [1] we prove that if k ≤ 0.3043 log n then G n,k is not connected whp, while if k ≥ 0.5139 log n then G n,k is connected whp. This greatly improved the earlier bounds due to Xue and Kumar [11] and Gonzáles-Barrios and Quiroz [4].…”

“…Here, it is important that every point in the region falls within the range of some transmitter. In [1] we investigated the question of whether a network of n transmitters in a square, each choosing its range so as to reach at least k other transmitters, would in fact cover the entire region in the above sense. More precisely, with G n,k as above, surround each vertex by the smallest disc containing its k nearest neighbours: we studied the values of k for which these discs cover the square S n whp.…”