Advances in Applied Probability

Sharpness in the k-nearest-neighbours random geometric graph model

Victor Falgas-Ravry and Mark Walters

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Abstract

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).

Article information

Source
Adv. in Appl. Probab., Volume 44, Number 3 (2012), 617-634.

Dates
First available in Project Euclid: 6 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.aap/1346955257

Digital Object Identifier
doi:10.1239/aap/1346955257

Mathematical Reviews number (MathSciNet)
MR3024602

Zentralblatt MATH identifier
1278.60142

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Keywords
Random geometric graph connectivity sharp transition

Citation

Falgas-Ravry, Victor; Walters, Mark. Sharpness in the k -nearest-neighbours random geometric graph model. Adv. in Appl. Probab. 44 (2012), no. 3, 617--634. doi:10.1239/aap/1346955257. https://projecteuclid.org/euclid.aap/1346955257


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References

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