AI 中文总结
研究在 assortative 稀疏随机块模型上的多数动态，表明加权优势决定达成一致速度，确定初始蓝色优势下蓝色达成一致的三种时间情况，分析依赖对单个顶点翻转概率的估计。
AI 中文摘要
多数动态是一个双观点过程，其中每个顶点反复更新为其邻居中的多数观点。我们在 assortative 状态下的重采样稀疏二元随机块模型上研究此过程。在每个时间步，从当前观点划分中采样一个图：具有相同观点的顶点以概率 $\alpha = a\log N / N$ 相连，而具有不同观点的顶点以概率 $\beta = b\log N / N$ 相连，其中 $a > b > 1$。令 $B_t$ 和 $R_t$ 表示时间 $t$ 时的蓝色阵营和红色阵营。我们表明，加权优势 $\widetilde{\Delta}_t = b|B_t| - a|R_t|$，而非仅未加权优势 $\Delta_t = |B_t| - |R_t|$，决定了达成一致的速度。我们的结果在 $N \to \infty$ 时以高概率成立，在初始蓝色优势（即 $\Delta_0 > 0$）下确定了蓝色达成一致的三种情况：恒定时间、次多项式时间和多项式时间。首先，当 $\widetilde{\Delta}_0 \gtrsim -N / \sqrt{\log N}$ 时，蓝色在三次更新内达成一致。其次，当 $\widetilde{\Delta}_0 < 0$ 且 $|\widetilde{\Delta}_0| = o(N)$ 时，蓝色在 $N^{o(1)}$ 次更新内达成一致。此外，当 $\widetilde{\Delta}_0 < 0$，$|\widetilde{\Delta}_0| = O(N)$ 且 $\Delta_0 \gg \sqrt{N / \log N}$ 时，蓝色仍在 $N^{I_0 + o(1)}$ 次更新内达成一致，其中 \[ I_0 = \left(\mathbf{ReLU}\Big(\sqrt{a\frac{|R_0|}{N}} - \sqrt{b\frac{|B_0|}{N}}\Big)\right)^2, \] 且 $\mathbf{ReLU}(x) = \max\{x, 0\}$。相反，远离加权阈值时，当 $|B_0| / |R_0| \leq a / b - \kappa$ 且 $\Delta_0 > 0$ 时，蓝色达成一致需要 $N^{I_0 - o(1)}$ 次更新。我们的分析依赖于对稀疏二项式差异中单个顶点翻转概率的详细估计，这可能具有独立的研究价值。
英文摘要
Majority dynamics is a two-opinion process in which each vertex repeatedly updates to the majority opinion among its neighbors. We study this process on a resampled sparse binary stochastic block model in the assortative regime. At each time step, a graph is sampled from the current opinion partition: vertices with the same opinion are joined with probability $α=a\log N/N$, while vertices with differing opinions are joined with probability $β=b\log N/N$, where $a>b>1$. Let $B_t$ and $R_t$ denote the blue and red camps at time $t$. We show that the weighted advantage $\widetildeΔ_t =b|B_t|-a|R_t|$, rather than the unweighted advantage $Δ_t=|B_t|-|R_t|$ alone, governs the pace to unanimity. Our results, which hold with high probability as \(N\to\infty\), identify three regimes for blue unanimity under the initial blue advantage, i.e., $Δ_0>0$: constant time, subpolynomial time, and polynomial time. First, when $\widetildeΔ_0 \gtrsim -N/\sqrt{\log N}$, blue unanimity occurs within three updates. Second, when $\widetildeΔ_0 < 0$ and $|\widetildeΔ_0| = o(N)$, blue unanimity occurs within $N^{o(1)}$ updates. Furthermore, when $\widetildeΔ_0 < 0$, $|\widetildeΔ_0| = O(N)$, and $Δ_0\gg\sqrt{N/\log N}$, blue unanimity still occurs within $N^{I_0+o(1)}$ updates, where \[ I_0= \left(\mathbf{ReLU}\Big(\sqrt{a\frac{|R_0|}{N}}-\sqrt{b\frac{|B_0|}{N}}\Big)\right)^2, \] and $\mathbf{ReLU}(x)=\max\{x,0\}$. Conversely, away from the weighted threshold, when $|B_0|/|R_0|\le a/b-κ$ and $Δ_0>0$, $N^{I_0 - o(1)}$ updates are necessary for blue unanimity. Our analysis relies on detailed estimates for one-vertex flip probabilities in sparse binomial differences, which could be of independent interest.
Comments56 pages, 6 figures
