A new bound in Majority Dynamics on Random Graphs

TL;DR

This paper analyzes majority dynamics on random graphs $G(n,p)$, establishing a new bound on the initial majority gap $\Delta$ that guarantees the initial majority colour wins with high probability for $p$ in the range $\frac{(1+\lambda)\log n}{n} \le p \le \tfrac{1}{4}$. The bound is $\Delta \ge \max\left\{\dfrac{1}{\sqrt{p}} \exp\left[A\sqrt{\log(1/p)}\right],\ Bp^{-3/2} n^{-1/2}\right\}$, improving previous $\Delta$ thresholds and yielding $\mathcal{O}(\log_{pn} n)$ convergence time. The authors also extend the results to random initial colourings, showing unanimity in the random $1/2$ scheme for $p \ge \lambda n^{-2/3}$ and strengthening bias-conditions for general $(q_1,q_2)$ schemes. The proof combines Berry–Esseen-based Gaussian approximations with a detailed day-by-day, multi-step analysis, including a novel Step 4 that uses sets $S^{(1)}_{uv}$, $S^{(2)}_{uv}$, and $S^{*}_{uv}$ to control dependencies and variance, enabling sharp concentration and progression to unanimity. These results advance understanding of consensus and information diffusion in sparse random networks and sharpen thresholds for rapid consensus in two-colour majority dynamics.

Abstract

We study the evolution of majority dynamics on Erdős-Rényi $G(n,p)$ random graphs. In this process, each vertex of a graph is assigned one of two initial states. Subsequently, on every day, each vertex simultaneously updates its state to the most common state in its neighbourhood. If the difference in the numbers of vertices in each state on day $0$ is larger than $ \max \left\{\frac{1}{\sqrt{p}} \exp\left[A\sqrt{\log \left(\frac{1}{p}\right)}\right] , Bp^{-3/2} n^{-1/2} \right\}$ for constants $A$ and $B$, we demonstrate that the state with the initial majority wins with overwhelmingly high probability. This extends work by Linh Tran and Van Vu (2023), who previously considered this phenomenon. We also study majority dynamics with a random initial assignment of vertex states. When each vertex is assigned to a state with equal probability, we show that unanimity occurs with high probability for every $p \geq λn^{-2/3}$, for some constant $λ$. This improves work by Fountoulakis, Kang and Makai (2020). Furthermore, we also consider a random initial assignment of vertex states where a vertex is slightly more likely to be in the first state than the second state. Previous work by Zehmakan (2018) and Tran and Vu (2023) provided conditions on how big this bias needs to be for the first colour to achieve unanimity with high probability. We strengthen these results by providing a weaker sufficient condition.

Theorems & Definitions (93)

• Definition 1.1
• Theorem 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• Definition 5.1
• Lemma 6.1
• proof