AI 中文总结
该研究确认了稀疏随机图上多数动力学达到高概率共识的猜想，在pN≥N^ε条件下，通过揭示观点历史并利用图枚举工具，证明O(1/ε)天内达成完全一致。
AI 中文摘要
考虑一个在N个顶点上的简单无向图G，每个顶点持有两种选项之一的观点。从第1天的初始观点出发，我们运行多数动力学：在随后的每一天，所有顶点同时采纳其邻居中的多数观点，若出现平局则保留当前观点。Benjamini、Chan、O'Donnell、Tamuz和Tan的一个被广泛研究的猜想指出：若G来自随机二项模型G(N,p)，且初始观点均匀随机选取，则只要pN→∞，网络将以高概率收敛到99%共识。我们在pN≥N^ε（对任意固定ε>0）的条件下确认了这一猜想，表明在O(1/ε)天内以高概率达到完全一致。该结果改进了Fountoulakis、Kang和Makai（达到ε=1/2）、Chakraborti、Kim、Lee和Tran（达到ε=2/5）以及Jaffe（达到ε=1/3）的一系列结果。证明技术涉及迭代地揭示每个顶点随时间变化的“观点历史”，并跟踪每个顶点与第k天2^k个观点历史类别之间的度数。我们对这一过程的分析需要对度约束随机图模型进行精细估计，使用McKay和Wormald以及Canfield、Greenhill和McKay工作中的图枚举工具，以及Liebenau和Wormald的扩展。在此过程中，我们将离散动力学与确定性理想化过程联系起来，后者的主导阶行为由条件高斯概率和期望描述。
英文摘要
Consider a simple undirected graph $G$ on $N$ vertices, with each vertex holding an opinion from one of two options. Starting from the initial opinions on day 1, we run majority dynamics: on each subsequent day, all vertices simultaneously adopt the majority opinion among their neighbors, retaining their current opinion in case of a tie. A well-studied conjecture of Benjamini, Chan, O'Donnell, Tamuz, and Tan states that if $G$ is drawn from the random binomial model $\mathbb{G}(N,p)$, and if the starting opinions are chosen uniformly at random, then as long as $pN \to \infty$ the network will converge to 99% consensus with high probability. We confirm this conjecture as long as $pN \ge N^{\varepsilon}$ for any fixed $\varepsilon > 0$, showing that full unanimity is reached with high probability in $O(1/\varepsilon)$ days. This result improves on a line of results by Fountoulakis, Kang, and Makai, who achieve $\varepsilon = 1/2$; Chakraborti, Kim, Lee, and Tran, who achieve $\varepsilon = 2/5$; and Jaffe, who achieves $\varepsilon = 1/3$. The proof technique involves iteratively revealing the "opinion histories" for each vertex through time, keeping track of the degrees between every vertex and each of $2^k$ opinion history classes on day $k$. Our analysis of this process requires intricate estimates for degree-constrained random graph models using graph enumeration tools from the work of McKay and Wormald as well as Canfield, Greenhill, and McKay, and their extensions by Liebenau and Wormald. In doing so, we connect the discrete dynamics to a deterministic idealized process, whose leading-order behavior is described by conditional Gaussian probabilities and expectations.
Comments86 pages
