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Connectivity of random k-nearest-neighbour graphs

Part of: Graph theory Equilibrium statistical mechanics

Published online by Cambridge University Press:  01 July 2016

Paul Balister ,
Béla Bollobás ,
Amites Sarkar  and
Mark Walters
Paul Balister
Affiliation:
University of Memphis
Béla Bollobás*
Affiliation:
University of Memphis and University of Cambridge
Amites Sarkar*
Affiliation:
University of Memphis
Mark Walters*
Affiliation:
University of Cambridge
*
∗ Postal address: Department of Mathematics, University of Memphis, Dunn Hall, 3725 Norriswood, Memphis, TN 38152, USA.
∗∗∗∗ Email address: [email protected]
∗∗∗∗∗∗ Postal address: Peterhouse, University of Cambridge, Cambridge CB2 1RD. UK.
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Abstract

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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 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 conjectured that the threshold for connectivity is k = (1 + o(1)) log n. In this paper we improve these lower and upper bounds to 0.3043 log n and 0.5139 log n, respectively, disproving this conjecture. We also establish lower and upper bounds of 0.7209 log n and 0.9967 log n 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 ≤ 0.7209 log n then the probability that these discs cover Sn tends to 0 as n → ∞ while, if k ≥ 0.9967 log n, then the probability that the discs cover Sn tends to 1 as n → ∞.

Keywords

Random geometric graphconnectivityPoisson process

MSC classification

Primary: 05C80: Random graphs
Secondary: 82B43: Percolation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 1 - 24
Copyright
Copyright © Applied Probability Trust 2005 

Footnotes

∗∗

Supported by NSF grant EIA-0130352.

∗∗∗

Supported by NSF grant DMS-9970404 and EIA-0130352 and DARPA grant F33615-01-C1900.

∗∗∗∗∗

Supported by NSF grant ITR-0225610.

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