Fixation to Consensus on Tree-related Graphs

TL;DR

This paper analyzes a Markov process on tree-related graphs where vertices update their states based on neighbors, demonstrating that with a high initial density of +1 spins, the system eventually reaches a consensus state.

Contribution

It introduces a novel geometric percolation-based approach to study consensus formation on complex tree-like graphs under majority dynamics.

Findings

System reaches fixation to consensus when initial +1 density is high.

Percolation arguments are effective for analyzing dynamics on tree-related graphs.

Long-term behavior depends on initial configuration density.

Abstract

We study a continuous time Markov process whose state space consists of an assignment of +1 or -1 to each vertex of a graph G. The graphs that we treat are related to homogeneous trees of degree K $\geq$ 3, such as finite or infinite stacks of such trees. The initial spin configuration is chosen from a Bernoulli product measure with density $\theta$ of +1 spins. The system evolves according to an agreement inducing dynamics: each vertex, at rate 1, changes its spin value to agree with the majority of its neighbors. We study the long time behavior of this system and prove that, if $\theta$ is close enough to 1, the system reaches fixation to consensus. The geometric percolation-type arguments introduced here may be of independent interest.