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.