Document Type

Article

Publication Date

2005

Keywords

Random geometric graph, connectivity, Poisson process

Abstract

Let P be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph Gn,k by joining each point of Pto its k ≡ k(n) nearest neighbours. Recently, Xue and Kumar proved that if k ≤ 0.074logn then the probability that Gn,k is connected tends to 0 as n → ∞ while, if k ≥ 5.1774logn, then the probability that Gn,k is connected tends to 1 as n → ∞. They conjectured that the threshold for connectivity is k = (1 + o(1))logn. In this paper we improve these lower and upper bounds to 0.3043logn and 0.5139logn, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209logn and 0.9967logn for the directed version of this problem. A related question concerns coverage. With Gn,k as above, we surround each vertex by the smallest (closed) disc containing its k nearest neighbours. We prove that if k ≤ 0.7209logn then the probability that these discs cover Sn tends to 0 as n → ∞ while, if k ≥ 0.9967logn, then the probability that the discs cover Sn tends to 1 as n → ∞.

Publication Title

Advances in Applied Probability

Volume

37

Issue

1

First Page

1

Last Page

24

Required Publisher's Statement

Published by Project Euclid

doi:10.1239/aap/1113402397

Comments

This is the authors' post refereed version of the article. Here is a link to the publisher's version: http://projecteuclid.org/euclid.aap/1113402397

Language

English

Format

application/pdf

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Mathematics Commons

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