We introduce two models of consensus following a majority rule on time-evolving stochastic block models (SBM), in which the network evolution is Markovian or non-Markovian. Under the majority rule, in each round, each agent simultaneously updates their opinion according to the majority of their neighbors. Our network has a community structure and randomly evolves with time. In contrast to the classic setting, the dynamics is not purely deterministic, and reflects the structure of SBM by resampling the connections at each step, making agents with the same opinion more likely to connect than those with different opinions. In the Markovian model, connections between agents are resampled at each step according to the SBM law and each agent updates their opinion via the majority rule. We prove a power-of-one type result, i.e., any initial bias leads to a non-trivial advantage of winning in the end, uniformly in the size of the network. In the non-Markovian model, a connection between two agents is resampled according to the SBM law only when at least one of them changes opinion and is otherwise kept the same. We identify the phase-transition threshold, up to the second-order leading term, between halting and fast convergence to consensus. We also give sufficient initial-lead conditions for consensus to occur within one, two, or three rounds.

翻译：我们引入了两个基于多数规则的时间演化随机分块模型（SBM）上的共识模型，其中网络演化具有马尔可夫性或非马尔可夫性。在多数规则下，每轮每个智能体根据其邻居的多数意见同时更新自己的观点。我们的网络具有社区结构，并随时间随机演化。与经典设定不同，该动力学过程并非纯确定性，而是通过每步重新采样连接来反映SBM结构，使得持有相同意见的智能体比持不同意见者更可能相互连接。在马尔可夫模型中，智能体之间的连接根据SBM法则每步重新采样，每个智能体通过多数规则更新其观点。我们证明了一种"一票之力"型结果，即任何初始偏向都会导致最终获胜的非平凡优势，且该优势对网络规模一致成立。在非马尔可夫模型中，仅当至少一个智能体改变观点时，两者之间的连接才根据SBM法则重新采样，否则保持不变。我们确定了从停滞到快速达成共识的相变阈值，精确到二阶主导项。此外，我们还给出了在一轮、两轮或三轮内达成共识的充分初始领先条件。