Connectivity of random k-nearest-neighbour graphs

Abstract

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 𝓅 to its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074 log n then the probability that Gn, k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774 log n, 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)) 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 Gn, 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 Sn tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover Sn tends to 1 as n → ∞.

A New Random Graph Model with Self-Optimizing Nodes: Connectivity and Diameter

“…connected) whp for sufficiently large n [6]. Similar bounds are derived for (strong) connectivity of directed kNNGs in [6]. Moreover, the existence of critical constant c ∈ [0.3043, 0.5139] is established for undirected kNNGs in [7].…”

“…k(n) ≥ c u ·log(n) with c u > 0.5139), an undirected kNNG is disconnected (resp. connected) whp for sufficiently large n [6]. Similar bounds are derived for (strong) connectivity of directed kNNGs in [6].…”

“…smaller) than log(n) [10,39]. Similarly, the smallest k needed for connectivity of kNNGs is shown to be (log(n)) for both directed and undirected graphs [5,6,7]. For example, if k(n) ≤ c l · log(n) with c l < 0.3043 (resp.…”

“…Note that k (n) cannot be smaller than the smallest k required for connectivity of the undirected kNNG. Because the lower bound for the smallest necessary k in undirected kNNGs is at least 0.3043· log(n) (whp) [6], together with the finding in [11], we have k (n) = (log(n)).…”

“…Another important random graph model, which is related to both the proposed model and random geometric graphs, is the k-nearest neighbor graph (kNNG) model [5,6,7,42]. The kNNGs are used to model the one-hop connectivity in wireless networks, in which each node initiates two-way communication with k nearest nodes; e.g., [42].…”