Sharpness in the k-nearest-neighbours random geometric graph model
2012 (English)In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 44, no 3, 617-634 p.Article in journal (Refereed) Published
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+Cis 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 sequences(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).
Place, publisher, year, edition, pages
Applied Probability Trust , 2012. Vol. 44, no 3, 617-634 p.
Random geometric graph, connectivity, sharp transition
Probability Theory and Statistics
IdentifiersURN: urn:nbn:se:umu:diva-80522DOI: 10.1239/aap/1346955257OAI: oai:DiVA.org:umu-80522DiVA: diva2:650004