Majority dynamics on sparse random graphs

TL;DR

This paper proves that majority dynamics on sparse Erdős–Rényi random graphs converges to near consensus with high probability for a broader range of edge probabilities than previously known, specifically for sparser graphs.

Contribution

It extends the validity of the conjecture to sparser graphs where the edge probability p is between  n^{-3/5} log n and  n^{-1/2}, breaking the previous  n^{-1/2} barrier.

Findings

Convergence to near consensus in sparser graphs is proven.

The conjecture holds for p down to  n^{-3/5} log n.

The result broadens understanding of majority dynamics in sparse networks.

Abstract

Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $\pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnel, Tamuz and Tan conjectured that, in the Erd\H{o}s--R\'enyi random graph $G(n,p)$, the random initial $\pm 1$-assignment converges to a $99\%$-agreement with high probability whenever $p=\omega(1/n)$. This conjecture was first confirmed for $p\geq\lambda n^{-1/2}$ for a large constant $\lambda$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for $p< \lambda n^{-1/2}$. We break this $\Omega(n^{-1/2})$-barrier by proving the conjecture for sparser random graphs $G(n,p)$, where $\lambda' n^{-3/5}\log n \leq p \leq \lambda n^{-1/2}$ with a large constant $\lambda'>0$.