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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 $\Gamma_{n}(r_{n},\xi)$ has $n$ vertices identified by $n$ independent uniformly distributed points $X_{1},\ldots,X_{n}$ in the unit square $[0,1]^{2}$. Each point $X_{i}$ selects $\xi_{i}$ neighbors at random, without replacement, among those points $X_{j}$ ($j\neq i$) for which $\Vert X_{i}-X_{j}\Vert <r_{n}$, and the selected vertices are connected to $X_{i}$ by an edge. The number $\xi_{i}$ of the neighbors is an integer-valued random variable, chosen independently with identical distribution for each $X_{i}$ such that $\xi_{i}$ satisfies $\xi_{i}\ge1$. We prove that when $r_{n}=\gamma_{n}\sqrt{\log n/n}$ for $\gamma_{n}\to\infty$ with $\gamma_{n}=o(n^{1/6}/\log^{5/6}n)$, the random geometric irrigation graph experiences explosive percolation in the sense that if ${\mathbf{E} \xi_{i}=1}$, then the largest connected component has $o(n)$ vertices but if $\mathbf{E} \xi_{i}>1$, then the number of vertices of the largest connected component is, with high probability, $n-o(n)$. This offers a natural noncentralized sparsification of a random geometric graph that is mostly connected.

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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

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Vol.26 • No. 5 • October 2016
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