Rapid social connectivity

Given a graph G = (V, E), consider Poisson(vertical bar V vertical bar) walkers performing independent lazy simple random walks on G simultaneously, where the initial position of each walker is chosen independently with probability proportional to the degrees. When two walkers visit the same vertex at the same time they are declared to be acquainted. The social connectivity time SC(G) is defined as the first time in which there is a path of acquaintances between every pair of walkers. It is shown that when the average degree of G is d, with high probability c log vertical bar V vertical bar <= SC(G) <= Cd1+5.1G is not regular log(3 )vertical bar V vertical bar. When G is regular the lower bound is improved to SC(G) >= log vertical bar V vertical bar - 6 log log vertical bar V vertical bar, with high probability. We determine SC(G) up to a constant factor in the cases that G is an expander and when it is the n-cycle.