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September 2012 Sharpness in the k-nearest-neighbours random geometric graph model
Victor Falgas-Ravry, Mark Walters
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Adv. in Appl. Probab. 44(3): 617-634 (September 2012). DOI: 10.1239/aap/1346955257


Let Sn,k denote the random graph obtained by placing points in a square box of area n according to a Poisson process of intensity 1 and joining each point to its k nearest neighbours. Balister, Bollobás, Sarkar and Walters (2005) conjectured that, for every 0 < ε < 1 and all sufficiently large n, there exists C = C(ε) such that, whenever the probability that Sn,k is connected is at least ε, then the probability that Sn,k+C is connected is at least 1 - ε. In this paper we prove this conjecture. As a corollary, we prove that there exists a constant C' such that, whenever k(n) is a sequence of integers such that the probability Sn,k(n) is connected tends to 1 as n → ∞, then, for any integer sequence s(n) with s(n) = o(logn), the probability Sn,k(n)+⌊C'slog logn is s-connected (i.e. remains connected after the deletion of any s - 1 vertices) tends to 1 as n → ∞. This proves another conjecture given in Balister, Bollobás, Sarkar and Walters (2009).


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Victor Falgas-Ravry. Mark Walters. "Sharpness in the k-nearest-neighbours random geometric graph model." Adv. in Appl. Probab. 44 (3) 617 - 634, September 2012.


Published: September 2012
First available in Project Euclid: 6 September 2012

zbMATH: 1278.60142
MathSciNet: MR3024602
Digital Object Identifier: 10.1239/aap/1346955257

Primary: 60K35
Secondary: 82B43

Keywords: connectivity , Random geometric graph , sharp transition

Rights: Copyright © 2012 Applied Probability Trust


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Vol.44 • No. 3 • September 2012
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