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

作者：Falgas Ravry Victor^*

来源：Electronic Journal of Probability, 2013, 18: 1-22.

DOI：10.1214/EJP.v18-2465

• Random geometric graphs

Let S-n,S-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,S-k connected) %26gt; n(-gamma 1) then the probability that S-n,S-k contains a pair of %26apos;small%26apos; components %26apos;close%26apos; to each other is o(n (c1)) (in a precise sense of %26apos;small%26apos; and %26apos;close%26apos;), for some absolute constants gamma(1) %26gt; 0 and c(1) %26gt; 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,S-k below the connectivity threshold is asymptotically Poisson.

• 出版日期2013-9-18