Let P be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph Gn,k by joining each point of Pto its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074logn then the probability that Gn,k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774logn, then the probability that Gn,k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1))logn. In this paper we improve these lower and upper bounds to 0.3043logn and 0.5139logn, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209logn and 0.9967logn for the directed version of this problem. A related question concerns coverage. With Gn,k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209logn then the probability that these discs cover Sn tends to 0 as n → ∞ while, if k ≥ 0.9967logn, then the probability that the discs cover Sn tends to 1 as n → ∞.
Advances in Applied Probability
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Balister, Paul; Bollobás, Béla; Sarkar, Amites; and Walters, Mark, "Connectivity of Random k-Nearest-Neighbor Graphs" (2005). Mathematics. Paper 83.