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

Victor Falgas-Ravry
2013 Electronic Journal of Probability  
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. Abstract Let S n,k denote the random geometric graph obtained by placing points inside a square of area n according to a Poisson point process of intensity 1 and joining each such point to the k = k(n) points of the process nearest to it. In this paper we show that if P(S n,k connected) > n −γ 1 then the probability that S n,k contains a pair of 'small' components 'close' to each other is o(n −c 1 ) (in a precise sense of
more » ... recise sense of 'small' and 'close'), for some absolute constants γ1 > 0 and c1 > 0. This answers a question of Walters [13] . (A similar result was independently obtained by Balister.) As an application of our result, we show that the distribution of the connected components of S n,k below the connectivity threshold is asymptotically Poisson. Small components in the k-nearest neighbour random geometric graph points uniformly at random inside S n .) This gives us a random geometric vertex set P. The k-nearest neighbour graph on P, S n,k = S n,k (P), is obtained by setting an undirected edge between every vertex of P and the k vertices of P nearest to it. The Poisson law of the random pointset P coupled with our deterministic definitions of S n,k gives rise to a probability measure on the space of k-nearest neighbour graphs with vertices in S n , which we refer to as the k-nearest neighbour model S n,k . As this model has some inherent randomness, rather than proving deterministic statements about S n,k we are mostly interested in establishing that properties hold for 'typical' k-nearest neighbour graphs. Formally, given a property Q of k-nearest neighbour graphs (i.e a sequence of subsets Q n of the set of all geometric graphs on S n ) and a sequence k = k(n) of nonnegative integers, we say that S n,k(n) has property Q with high probability (whp) if lim n→∞ P(S n,k(n) ∈ Q n ) = 1. Previous work on the k-nearest neighbour model An important motivation for the study of the connectivity properties of the k-nearest neighbour model comes from the theory of ad-hoc wireless networks: suppose we have some radio transmitters (nodes) spread out over a large area and wishing to communicate using multiple hops, and that each transmitter can adjust its range so as to ensure two-way radio contact with the k nodes nearest to it. The connectivity of this radio network is modelled using S n,k in a natural way. As a result, the model has received a significant amount of attention, though many questions remain. (See e.g. the recent survey paper of Walters [12] .) Elementary arguments show that there exist constants c l ≤ c u such that for k(n) ≤ c l log n S n,k is whp not connected, while for k(n) ≥ c u log n S n,k is whp connected. A bound of c u ≤ 5.1774 was obtained by Xue and Kumar [14], using a substantial result of Penrose [10] for the related Gilbert disc model [8] . An earlier bound of c u ≤ 3.5897 could also be read out of earlier work of Gonzáles-Barrios and Quiroz [9]. 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. As part of their results, Balister, Bollobás, Sarkar and Walters [3] showed that when 0.3 log n < k < 0.6 log n whp all of the following hold: there is a unique 'giant' connected component, (iii) all other components have diameter O( √ log n). (Strictly speaking, they only showed that S n,k has at most one giant component; that such a component exists follows from the study of percolation in the k-nearest neighbour graph: see e.g. [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 EJP 18 (2013), paper 83. Page 2/22 ejp.ejpecp.org n −γ1 < P(S n,k connected) << 1 − o(n −c1 ). (For example, k = c log n for any c with c − δ < c < c will do.) Next we turn to the distribution of the small components of S n,k . Let X = X n,k denote the number of small connected components of S n,k . (Since there is whp a unique nonsmall connected component [3], X is whp the number of components of S n,k minus 1.) Also, given ν ≥ 0 and A ⊆ N ∪ {0}, let Po ν (A) denote the probability a Poisson random variable with parameter ν takes a value inside A. As an application of Theorem 1.3, we prove: Theorem 1.5. There exist absolute constants γ 2 and c 2 > 0 such that if k = k(n) is an integer sequence with P(S n,k connected) > n −γ2 for all n, then, writing ν = ν(n) for − log (P(S n,k connected)), we have sup A⊆N∪{0} |P(X ∈ A) − Po ν (A)| = o(n −c2 ).
doi:10.1214/ejp.v18-2465 fatcat:vfhj2nezx5gp3df3cn66pjxjcq