Majority dynamics on finite trees

TL;DR

This paper determines the worst-case stabilization time of majority dynamics on finite trees and shows that for random initial opinions on perfect rooted cubic trees, the process stabilizes quickly within a specific time interval.

Contribution

It provides the exact worst-case stabilization time for majority dynamics on any finite tree and establishes probabilistic bounds for stabilization time on perfect rooted cubic trees.

Findings

Worst-case stabilization time for any finite tree is precisely characterized.

On perfect rooted cubic trees, stabilization occurs in a narrow time window with high probability.

Stability time scales proportionally with the diameter of the tree.

Abstract

For an arbitrary finite tree $T$, we find the exact value of the wort-case stabilisation time of majority dynamics on $T$. We also prove that for a perfect rooted cubic tree $T$ with diameter $D$ and uniformly random initial opinions, the dynamics stabilises in time $\tau\in(D/4,D/3)$ with high probability.