发表机构
University of California San Diego; University of Pennsylvania; University of Washington（加州大学圣地亚哥分校; 宾夕法尼亚大学; 华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析重采样稀疏Erdős–Rényi图上的双观点多数动态，确定由初始优势决定的三个区域，给出各区域达成一致的时间界与概率，解决了Tran和Vu2025年提出的相关猜想。
AI 中文摘要
我们研究双观点多数动态过程：每一时间步，每个顶点采用邻居中的多数观点，平局时保留自身当前观点。每一步独立地，交互图从稀疏Erdős–Rényi模型$\boldsymbol{\text{G}}(N,p)$中重采样，其中$p = b\boldsymbol{\text{log}} N/N$，且固定$b>1$。我们的结果确定了由初始优势$\boldsymbol{\text{Δ}}_0 = |B_0| - |R_0|$决定的三个区域，其中$|B_0|$和$|R_0|$分别表示初始蓝方和红方阵营的规模。第一，若初始蓝方优势超过$N/\boldsymbol{\text{log}} N$的某个显式常数倍，则高概率下蓝方将在两次更新内达成一致。第二，在中间区域$\boldsymbol{\text{√}}(N/\boldsymbol{\text{log}} N) \boldsymbol{\text{ll}} \boldsymbol{\text{Δ}}_0 \boldsymbol{\text{lesssim}} N/\boldsymbol{\text{log}} N$中，我们得到了蓝方达成一致时间的显式高概率上下界。最后，在临界窗口$\boldsymbol{\text{Δ}}_0 \boldsymbol{\text{√}} p = O(1)$中，蓝方和红方达成一致的概率分别等于$\boldsymbol{\text{Φ}}(\boldsymbol{\text{√}}(2/\boldsymbol{\text{π}}) \boldsymbol{\text{Δ}}_0 \boldsymbol{\text{√}} p) + o(1)$和$\boldsymbol{\text{Φ}}(-\boldsymbol{\text{√}}(2/\boldsymbol{\text{π}}) \boldsymbol{\text{Δ}}_0 \boldsymbol{\text{√}} p) + o(1)$，且高概率下达成一致所需的更新次数为$(1+o(1))\boldsymbol{\text{log}} N/\boldsymbol{\text{log}} \boldsymbol{\text{log}} N$。这解决了Tran和Vu（2025）提出的“最优少数力量”猜想的重采样版本。
英文摘要
Collective decisions in agent networks emerge from repeated local interactions, as agents update their opinions (colors) based on their neighbors, potentially leading to consensus. In sparse communication graphs, repeated interactions can amplify both the initial majority preference and the random fluctuations caused by individual agents. Understanding when the initial majority determines the final preference and how quickly an agreement emerges is, therefore, a basic question in collective decision-making. We study binary majority dynamics as an idealized model of this process. At each round, the interaction graph is independently resampled from the Erdős--Rényi model $\mathbb{G}(N,p)$, with $p=b\log N/N$ and fixed $b>1$. Each agent adopts the majority opinion among its neighbors, retaining its current opinion if there is a tie. For a fixed initial configuration, let $Δ_0$ denote the initial difference between the number of blue agents and the number of red agents. We identify three regimes for the consensus time, determined by $|Δ_0|$. First, when $Δ_0\sqrt p=O(1)$, the probability of blue unanimity is $$ Φ\left(Δ_0\sqrt{2p/π}\right) + o(1), $$ where $Φ$ is the cumulative distribution function of a standard Gaussian. Moreover, consensus is reached after $(1+o(1))\log N/\log\log N$ rounds with high probability. Second, in the intermediate regime $1\llΔ_0 \sqrt{p}\lesssim \sqrt{N}$, we establish explicit high probability upper and lower bounds on the time to blue-unanimity. Lastly, when the initial blue advantage is above an explicit constant multiple of $N/\sqrt{\log N}$, blue unanimity is guaranteed within two rounds with high probability. Our findings provide a complete description of how the initial advantage influences winner selection and the speed of consensus in this type of majority dynamics with sparse interactions.
Comments42 pages, 4 figures
