Rapid social connectivity

Abstract

Given a graph $G=(V,E)$, consider Poisson($ |V|$) 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 $\mathrm{SC}(G)$ 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 \[ c\log |V| \le \mathrm{SC}(G) \le C d^{1+5 \cdot 1_{G \text{ is not regular}} } \log^3 |V|.\] When $G$ is regular the lower bound is improved to $\mathrm{SC}(G) \ge \log |V| -6 \log \log |V| $, with high probability. We determine $\mathrm{SC}(G)$ up to a constant factor in the cases that $G$ is an expander and when it is the $n$-cycle.