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

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

Let Sn,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(Sn,k connected) > n −γ1 then the probability that Sn,k contains a pair of ‘small’ components ‘close’ to each other is o(n −c1) (in a pre...

CONNECTIVITY OF RANDOM k-NEAREST-NEIGHBOUR GRAPHS

LetP be a Poisson process of intensity one in a squareSn of arean. 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 conject...

On the Connectivity of Random k-th Nearest Neighbour Graphs

Small components in k-nearest neighbour graphs

Let G = Gn,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 logn then the probability that G is connected tends to 0, whereas if k > 0.5139 logn then the probability that G is connected tends to 1. We prove that,...

Convergence of random k-nearest-neighbour imputation

Random k-nearest-neighbour (RKNN) imputation is an established algorithm for filling in missing values in data sets. Assume that data are missing in a random way, so that missingness is independent of unobserved values (MAR), and assume there is a minimum positive probability of a response vector being complete. Then RKNN, with k equal to the square root of the sample size, asymptotically produ...