Open Access
October 2016 Almost optimal sparsification of random geometric graphs
Nicolas Broutin, Luc Devroye, Gábor Lugosi
Ann. Appl. Probab. 26(5): 3078-3109 (October 2016). DOI: 10.1214/15-AAP1170

Abstract

A random geometric irrigation graph Γn(rn,ξ) has n vertices identified by n independent uniformly distributed points X1,,Xn in the unit square [0,1]2. Each point Xi selects ξi neighbors at random, without replacement, among those points Xj (ji) for which XiXj<rn, and the selected vertices are connected to Xi by an edge. The number ξi of the neighbors is an integer-valued random variable, chosen independently with identical distribution for each Xi such that ξi satisfies ξi1. We prove that when rn=γnlogn/n for γn with γn=o(n1/6/log5/6n), the random geometric irrigation graph experiences explosive percolation in the sense that if Eξi=1, then the largest connected component has o(n) vertices but if Eξi>1, then the number of vertices of the largest connected component is, with high probability, no(n). This offers a natural noncentralized sparsification of a random geometric graph that is mostly connected.

Citation

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Nicolas Broutin. Luc Devroye. Gábor Lugosi. "Almost optimal sparsification of random geometric graphs." Ann. Appl. Probab. 26 (5) 3078 - 3109, October 2016. https://doi.org/10.1214/15-AAP1170

Information

Received: 1 November 2014; Revised: 1 September 2015; Published: October 2016
First available in Project Euclid: 19 October 2016

zbMATH: 1375.60031
MathSciNet: MR3563202
Digital Object Identifier: 10.1214/15-AAP1170

Subjects:
Primary: 05C80 , 60C05

Keywords: connectivity , irrigation graph , Random geometric graph

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.26 • No. 5 • October 2016
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