Almost optimal sparsification of random geometric graphs

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]. Each point Xi selects ξi neighbors at random, without replacement, among those points Xj (j 6= i) for which ‖Xi − Xj‖ < 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 1 ≤ ξi ≤ κ for a constant κ > 1. We prove that when rn = γn √ log n/n for γn → ∞ with γn = o(n/ log n), the random geometric irrigation graph experiences explosive percolation in the sense that when 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, n − o(n). This offers a natural non-centralized sparsification of a random geometric graph that is mostly connected.