Global Majority Consensus by Local Majority Polling on Graphs of a Given Degree Sequence

TL;DR

This paper analyzes a local majority voting protocol on graphs with a given degree sequence, showing rapid consensus on the initial global majority under certain conditions, and establishing lower bounds for local protocols.

Contribution

It introduces bounds on the speed of consensus in majority polling on graphs with specified degree sequences and extends the analysis to Erdős-Rényi random graphs.

Findings

Consensus is reached in O(log_k log_k n) steps for biased initial majority.

Any local protocol without colour change in uniform neighbourhoods takes at least Ω(log_d log_d n) steps.

Technique applies to Erdős-Rényi graphs in the connected regime.

Abstract

Suppose in a graph $G$ vertices can be either red or blue. Let $k$ be odd. At each time step, each vertex $v$ in $G$ polls $k$ random neighbours and takes the majority colour. If it doesn't have $k$ neighbours, it simply polls all of them, or all less one if the degree of $v$ is even. We study this protocol on graphs of a given degree sequence, in the following setting: initially each vertex of $G$ is red independently with probability $\alpha < \frac{1}{2}$, and is otherwise blue. We show that if $\alpha$ is sufficiently biased, then with high probability consensus is reached on the initial global majority within $O(\log_k \log_k n)$ steps if $5 \leq k \leq d$, and $O(\log_d \log_d n)$ steps if $k > d$. Here, $d\geq 5$ is the effective minimum degree, the smallest integer which occurs $\Theta(n)$ times in the degree sequence. We further show that on such graphs, any local protocol in which a vertex does not change colour if all its neighbours have that same colour, takes time at least $\Omega(\log_d \log_d n)$, with high probability. Additionally, we demonstrate how the technique for the above sparse graphs can be applied in a straightforward manner to get bounds for the Erd\H{o}s-R\'enyi random graphs in the connected regime.