Discrete Incremental Voting: New Bounds for General Graphs and Expanders

Abstract

We analyze the discrete incremental voting process (DIV) introduced by Cooper, Radzik, and Shiraga [OPODIS '23]. In this process, we consider a set $V$ of $n$ nodes connected in an undirected graph $G=(V,E)$ where each node has an integer opinion. In one step a randomly selected node interacts with its randomly selected neighbor and changes its opinion by $1$ in the direction of the neighbour's opinion. The process converges to a unique opinion that, in expectation, is the degree-weighted average of the initial opinions. We show that if the graph has conductance $\hat{I}¦(G)$, the ratio of the average to smallest degree is ${\hat{I}}^3(G)$, and the maximal difference between initial opinions is $K$, then the expected convergence time is $O\left(n\left(K\log (Kn)+{\hat{I}}^3(G)n\right)/\hat{I}¦(G{)}^2\right)$. This bound is essentially optimal for a large class of graphs of bounded expansion. We also show that for regular graphs, if the second largest eigenvalue is $o(1/{\log }^{2}n)$ and $K$ is $o\left(n/{\log }^{2}n\right)$, then w.h.p.\ DIV converges to the initial average opinion (rounded up or down).