Small components in -nearest neighbour graphs

Abstract

Let G = G n,k denote the graph formed by placing points in a square of area n according to a Poisson process of density 1 and joining each point to its k nearest neighbours. In [2] Balister, Bollobás, Sarkar and Walters proved that if k < 0.3043 log n then the probability that G is connected tends to 0, whereas if k > 0.5139 log n then the probability that G is connected tends to 1.We prove that, around the threshold for connectivity, all vertices near the boundary of the square are part of the (unique) giant component. This shows that arguments about the connectivity of G do not need to consider 'boundary' effects.We also improve the upper bound for the threshold for connectivity of G to k = 0.4125 log n.

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

“…This answers Question 1 in Walters [13]. Balister has independently obtained a similar result (personal communication).…”

“…In this paper we show that if P(S n,k connected) > n −γ1 then the probability that S n,k contains a pair of 'small' components 'close' to each other is o(n −c1 ) (in a precise sense of 'small' and 'close'), for some absolute constants γ 1 > 0 and c 1 > 0. This answers a question of Walters [13]. (A similar result was independently obtained by Balister.…”

“…[2].) Refining their techniques, Walters [13] showed that around the connectivity threshold, there are no 'small' components (of diameter O( √ log n)) lying 'close' to the boundary of S n (within distance O(log n)), and used this to improve the upper bound on c ⋆ to 0.4125. Towards the end of his paper [13], Walters asked a number of questions about the properties of small components of S n,k below the connectivity threshold, with the aim of completing the picture we currently have and perhaps improving the upper bound on c ⋆ .…”

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

“…The nice feature of T n is that it is not very close to any of the boundary of S n . The following lemma is a minor restatement of Theorem 1 of [9].…”

The Influence of Communication Range on Connectivity for Resilient Wireless Sensor Networks Using a Probabilistic Approach

“…But for the directed network the upper and lower bounds are 0.7209 log and 0.9967 log , respectively. Reference [25] improves the upper bound to be 0.4125 log . For disk model [26] states that 6 to 10 average numbers of neighbors almost make sure that network will be fully connected no matter how many nodes there are totally in the network.…”