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arXiv · 0708.4007

A critical constant for the k nearest neighbour model

Paul Balister·Bela Bollobas·Amites Sarkar·Mark Walters
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Abstract

Let P be a Poisson process of intensity one in a square S_n of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph G_{n,k}. We prove that there exists a critical constant c such that for c' c G_{n,c'\log n} is connected with probability tending to 1 as n tends to infinity. This answers a question previously posed by the authors.

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Paul Balister, Bela Bollobas, Amites Sarkar, Mark Walters. 2007-08-29. A critical constant for the k nearest neighbour model. https://arxiv.org/abs/0708.4007

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