Size of the giant component in a random geometric graph

Abstract

In this paper, we study the size of the giant component CG in the random geometric graph G = G(n, rn, f) of n nodes independently distributed each according to a certain density f(·) in [0, 1] 2 satisfying infx∈[0,1]2 f(x) > 0. If c1/n ≤ r2n ≤ c2 log n/n for some positive constants c1, c2 and nr2n → ∞ as n → ∞, we show that the giant component of G contains at least n - o(n) nodes with probability at least 1-e-βnr2n for all n and for some positive constant β. We also obtain estimates on the diameter and number of the non-giant components of G. © Association des Publications de l'Institut Henri Poincaré, 2013.