
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 , consider Poisson() walkers performing independent lazy simple random walks on 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 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 is , 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 is regular the lower bound is improved to , with high probability. We determine up to a constant factor in the cases that is an expander and when it is the -cycle.
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