Majority Dynamics on Resampled Sparse Erdős--Rényi Graphs: Gaussian Winner Selection and Pace to Unanimity

2026-08-06

AI Summary

This study analyzes the convergence behavior of two-opinion majority dynamics on sparse Erdős–Rényi graphs under independent edge resampling at each step. Depending on the initial opinion imbalance Δ₀, three distinct regimes emerge: when Δ₀ ≫ N/√log N, consensus is achieved with high probability within two steps; in an intermediate regime, tight upper and lower bounds on the consensus time are established; and within the critical window where Δ₀√p = O(1), the winning probability converges to a Gaussian limit, and consensus occurs with high probability in (1+o(1))log N / log log N steps. Combining tools from random graph theory, concentration inequalities, and asymptotic analysis, this work precisely characterizes the interplay between dynamic graph structure and opinion evolution, resolving a resampled-graph version of Tran and Vu’s conjecture on the “optimal power of minorities.”

Abstract

We study the two-opinion majority dynamics process: at each time step, every vertex adopts the majority opinion among its neighbors, retaining its current opinion if there is a tie. Independently at each step, the interaction graph is resampled from the sparse Erdős--Rényi model $\mathbb G(N,p)$ with $p=b\log N/N$ and fixed $b>1$. Our results identify three regimes governed by the initial advantage $Δ_0=|B_0|-|R_0|$, where $|B_0|$ and $|R_0|$ denote the initial blue and red camps, respectively. First, an initial blue advantage above an explicit constant multiple of $N/\sqrt{\log N}$ leads to blue unanimity within two updates with high probability. Second, throughout the intermediate regime $\sqrt{N/\log N}\llΔ_0\lesssim N/\sqrt{\log N}$, we obtain explicit high-probability upper and lower bounds on the blue-unanimity time. Finally, uniformly in the critical window $Δ_0\sqrt p=O(1)$, the blue- and red-unanimity probabilities equal $Φ(\sqrt{2/π}\,Δ_0\sqrt p)+o(1)$ and $Φ(-\sqrt{2/π}\,Δ_0\sqrt p)+o(1)$, respectively, and unanimity is reached within $(1+o(1))\log N/\log\log N$ many updates with high probability. This resolves the resampled version of the \emph{optimal power-of-few} conjecture raised by Tran and Vu (2025).

Problem

• Majority Dynamics
• Resampled Graphs
• Sparse Erdős–Rényi Graphs
• Unanimity Time
• Initial Advantage

Innovation

• majority dynamics
• resampled Erdős–Rényi graphs
• Gaussian limit
• unanimity time
• phase transition