A new density limit for unanimity in majority dynamics on random graphs

TL;DR

The paper addresses unanimity in majority dynamics on random graphs $G(n,p)$, establishing a new density-advantage threshold that enables Red (or Blue) to win with high probability. It develops a two-day analytic framework that first guarantees day-2 expansion by creating and propagating $D$-almost Red vertices, then leverages known landslide results to finish the process in $O(\log_{pn} n)$ days; the approach relies on sharp binomial concentration and Berry–Esseen-type bounds. In the moderately sparse regime, the results yield a new lower bound $Δ \ge C_ε p^{-3/2} n^{-1/2} \log n$ for unanimity, and for the random $1/2$ coloring model, a near-optimal density $p \gg n^{-2/3}\log^{2/3} n$, improving previous work. The methods clarify the interaction between connectivity, initial advantage, and dynamic updates, and contribute to the broader understanding of the “power of few” in threshold phenomena for stochastic processes on graphs.

Abstract

Majority dynamics is a process on a simple, undirected graph $G$ with an initial Red/Blue color for every vertex of $G$. Each day, each vertex updates its color following the majority among its neighbors, using its previous color for tie-breaking. The dynamics achieves \textit{unanimity} if every vertex has the same color after finitely many days, and such color is said to \textit{win}. When $G$ is a $G(n,p)$ random graph, L. Tran and Vu (2019) found a codition in terms of $p$ and the initial difference $2Δ$ beteween the sizes of the Red and Blue camps, such that unanimity is achieved with probability arbitrarily close to 1. They showed that if $pΔ^2 \gg1 $, $pΔ\geq 100$, and $p\geq (1+\varepsilon) n^{-1}\log n$ for a positive constant $\varepsilon$, then unanimity occurs with probability $1 - o(1)$. If $p$ is not extremely small, namely $p > \log^{-1/16} n $, then Sah and Sawhney (2022) showed that the condition $pΔ^2 \gg 1$ is sufficient. If $n^{-1}\log^2 n \ll p \ll n^{-1/2}\log^{1/4} n$, we show that $p^{3/2}Δ\gg n^{-1/2}\log n$ is enough. Since this condition holds if $pΔ\geq 100$ for $p$ in this range, this is an improvement of Tran's and Vu's result. For the closely related problem of finding the optimal condition for $p$ to achieve unanimity when the initial coloring is chosen uniformly at random among all possible Red/Blue assignments, our result implies a new lower bound $p \gg n^{-2/3}\log^{2/3} n$, which improves upon the previous bound of $n^{-3/5}\log n$ by Chakraborti, Kim, Lee and T. Tran (2021).

Theorems & Definitions (24)

• Definition 1.1: Majority Dynamics
• Theorem 1.4
• Corollary 1.5
• proof
• Lemma 2.1
• Definition 2.2
• Lemma 2.3: Stronger Lemma 1.3 in jhk-etal2021
• Lemma 2.4
• proof : Proof of Lemma \ref{['lem:day2']}
• proof : Proof of Lemma \ref{['lem:unst-neighs-with-swing-neighs']}