The acquaintance time of (percolated) random geometric graphs

In this paper, we study the acquaintance time AC(G) defined for a connected graph G. We focus on G(n, r, p), a random subgraph of a random geometric graph in which n vertices are chosen uniformly at random and independently from [0, 1], and two vertices are adjacent with probability p if the Euclidean distance between them is at most r. We present asymptotic results for the acquaintance time of G(n, r, p) for a wide range of p = p(n) and r = r(n). In particular, we show that with high probability AC(G) = Θ(r−2) for G ∈ G(n, r, 1), the usual random geometric graph, provided that πnr − lnn → ∞ (that is, above the connectivity threshold). For the percolated random geometric graph G ∈ G(n, r, p), we show that with high probability AC(G) = Θ(r−2p−1 lnn), provided that pnr ≥ n and p < 1− ε for some ε > 0.