A critical constant for the k nearest-neighbour model

Abstract

Let 𝒫 be a Poisson process of intensity 1 in a square Sn of area n. For a fixed integer k, join every point of 𝒫 to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that, for c < ccrit, Gn,⌊c log n⌋ is disconnected with probability tending to 1 as n → ∞ and, for c > ccrit, Gn,⌊c log n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed in Balister et al. (2005).

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

“…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].…”

“…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.…”

“…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].…”

Percolation, Connectivity, Coverage and Colouring of Random Geometric Graphs

“…The natural conjecture that c l = c u = c was made in [7] and proved in [10]. More precisely, we have the following theorem.…”

“…although in a paper to be published, Balister, Bollobás and Walters [15] used a certain oriented 1-independent percolation model to prove that…”