Reaching consensus on a connected graph

TL;DR

This paper analyzes a simple random process on connected graphs where vertices reach consensus through pairwise interactions, providing probabilities, expected times, and bounds, with implications for graph structure and delay effects.

Contribution

It introduces a model for consensus reaching on graphs, computes outcome probabilities independent of structure, and establishes bounds and delay effects on the process.

Findings

Outcome probabilities are independent of graph structure.

Expected consensus time is minimized by the complete graph $K_n$.

Delays increase mean absorption time monotonically for certain parameters.

Abstract

We study a simple random process in which vertices of a connected graph reach consensus through pairwise interactions. We compute outcome probabilities, which do not depend on the graph structure, and consider the expected time until a consensus is reached. In some cases we are able to show that this is minimised by $K_n$. We prove an upper bound for the case $p=0$ and give a family of graphs which asymptotically achieve this bound. In order to obtain the mean of the waiting time we also study a gambler's ruin process with delays. We give the mean absorption time and prove that it monotonically increases with $p\in[0,1/2]$ for symmetric delays.