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 ≤ r 2 n ≤ c2 logn n for some positive constants c1, c2 and nr 2 n −→ ∞, we show that the giant component of G contains at least n − o(n) nodes with probability at least 1 − o(1) as n → ∞. We also obtain estimates on the diameter and number of the non-giant components of G.