Local majority dynamics on preferential attachment graphs

TL;DR

This paper analyzes how local majority voting dynamics lead to consensus on the initial majority in preferential attachment graphs with power-law degree distribution, showing rapid convergence under certain bias conditions.

Contribution

It provides the first analysis of local majority dynamics on preferential attachment graphs, demonstrating fast consensus formation when initial bias exceeds a threshold.

Findings

Consensus is reached quickly (O(log_d log_d t) steps) under sufficient initial bias.

The results apply to graphs with power-law degree distribution with exponent around 3.

Analysis extends to graphs with larger degree exponents, including regular graphs.

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 the preferential attachment model of Albert and Barab\'asi, which gives rise to a degree distribution that has roughly power-law $P(x) \sim \frac{1}{x^{3}}$, as well as generalisations which give exponents larger than $3$. The setting is as follows: 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 away from $\frac{1}{2}$, then with high probability, consensus is reached on the initial global majority within $O(\log_d \log_d t)$ steps. Here $t$ is the number of vertices and $d \geq 5$ is the minimum of $k$ and $m$ (or $m-1$ if $m$ is even), $m$ being the number of edges each new vertex adds in the preferential attachment generative process. Additionally, our analysis reduces the required bias of $\alpha$ for graphs of a given degree sequence studied by the first author (which includes, e.g., random regular graphs).