## Annals of Applied Probability

### Almost optimal sparsification of random geometric graphs

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 5 (2016), 3078-3109.

Dates
Revised: September 2015
First available in Project Euclid: 19 October 2016

https://projecteuclid.org/euclid.aoap/1476884312

Digital Object Identifier
doi:10.1214/15-AAP1170

Mathematical Reviews number (MathSciNet)
MR3563202

Zentralblatt MATH identifier
1375.60031

#### Citation

Broutin, Nicolas; Devroye, Luc; Lugosi, Gábor. Almost optimal sparsification of random geometric graphs. Ann. Appl. Probab. 26 (2016), no. 5, 3078--3109. doi:10.1214/15-AAP1170. https://projecteuclid.org/euclid.aoap/1476884312

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