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 zero as n → ∞, while if k ≥ 5.1774 log n then the probability that G n,k is connected tends to one 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, 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 zero as n → ∞, while if k ≥ 0.9967 log n then the probability that the discs cover S n tends to one as n → ∞.
Study REF finds that if k ≤ 0.3043 log n the network is not connected with high probability; if k ≥ 0.5139 log n, then the network is connected with high probability.
Connectivity of random k-nearest-neighbour graphs
Advances in Applied Probability
Mathematics