Majority dynamics on trees and the dynamic cavity method

TL;DR

This paper studies how opinions evolve on infinite trees under majority dynamics, providing bounds on initial bias needed for consensus and introducing a novel cavity method inspired by spin-glass theory.

Contribution

It introduces a new analytical approach using the dynamic cavity method to analyze majority dynamics on trees, with bounds on convergence thresholds.

Findings

Bounded the initial bias threshold for consensus.

Characterized the process in the large degree limit.

Derived bounds applicable to small, odd degrees.

Abstract

A voter sits on each vertex of an infinite tree of degree $k$, and has to decide between two alternative opinions. At each time step, each voter switches to the opinion of the majority of her neighbors. We analyze this majority process when opinions are initialized to independent and identically distributed random variables. In particular, we bound the threshold value of the initial bias such that the process converges to consensus. In order to prove an upper bound, we characterize the process of a single node in the large $k$-limit. This approach is inspired by the theory of mean field spin-glass and can potentially be generalized to a wider class of models. We also derive a lower bound that is nontrivial for small, odd values of $k$.