Rapid social connectivity

TL;DR

This paper analyzes the time until a network of random walkers becomes socially connected, providing bounds based on graph properties and identifying exact times for specific graph types.

Contribution

It establishes bounds on social connectivity time for various graph classes, including regular graphs, expanders, and cycles, advancing understanding of social mixing processes.

Findings

Social connectivity time scales with graph size and degree.

Bounds are tight for regular graphs and expanders.

Exact times are determined for cycles.

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.