Sharpness in the k-Nearest-Neighbours Random Geometric Graph Model

Abstract

Let Sn, k denote the random graph obtained by placing points in a square box of area n according to a Poisson process of intensity 1 and joining each point to its k nearest neighbours. Balister, Bollobás, Sarkar and Walters (2005) conjectured that, for every 0 < ε < 1 and all sufficiently large n, there exists C = C(ε) such that, whenever the probability that Sn, k is connected is at least ε, then the probability that Sn, k+C is connected is at least 1 - ε. In this paper we prove this conjecture. As a corollary, we prove that there exists a constant C' such that, whenever k(n) is a sequence of integers such that the probability Sn,k(n) is connected tends to 1 as n → ∞, then, for any integer sequence s(n) with s(n) = o(logn), the probability Sn,k(n)+⌊C'slog logn⌋ is s-connected (i.e. remains connected after the deletion of any s − 1 vertices) tends to 1 as n → ∞. This proves another conjecture given in Balister, Bollobás, Sarkar and Walters (2009).

Small components in -nearest neighbour graphs

“…Since the first draft of this paper Falgas-Ravry [4] has answered this question in the affirmative provided that the probability that G is connected is not too small: more precisely he proves it whenever P(G is connected) = Ω(n γ ) (where γ is an absolute constant).…”

Distribution of components in the $k$-nearest neighbour random geometric graph for $k$ below the connectivity threshold

“…Remark 1. Theorem 1 of [7] showed that, for n large enough, we need to increase k by at most a constant times log(1/ε) for S n,k to go from having an ε chance of being connected to having a 1 − ε chance of being connected. Assuming there is a 'bluntness' converse to this result, i.e.…”

“…These results were substantially improved by Balister, Bollobás, Sarkar and Walters in a series of papers [3,5,4] in which they established inter alia the existence of a critical constant c ⋆ : 0.3043 < c ⋆ < 0.5139 such that for c < c ⋆ and k ≤ c log n, S n,k is whp not connected while for c > c ⋆ and k ≥ c log n, S n,k is whp connected. Building on their work, Walters and the author [7] recently proved that the transition from whp not connected to whp connected is sharp in k: there is an absolute constant C > 0 such that if S n,k is connected with probability at least ε > 0 and n is sufficiently large, then for k ′ ≥ k + C log(1/ε), S n,k ′ is connected with probability at least 1 − ε.…”

“…We follow the strategy introduced by Balister, Bollobás, Sarkar and Walters in [4] and developed in [7]: we prove Theorem 2 by looking at local events. In Section 2, we prove a local result, Theorem 5, which can be thought of as an analogue of the local sharpness result Lemma 12 in [7].…”

Speed and concentration of the covering time for structured coupon collectors

“…For any D as above, it is easy to show (see e.g. Lemma 8 of [19]) that the boundary of D meets at most 18πr 1)) n log n πr 2 . Since by construction of the random variable X − and X + we have that T (X − ) ≤ T (X) ≤ T (X + ), we deduce that whp the covering time for the torus V satisfies T (X) = (1 + o(1)) n log n πr 2 .…”