We study binary opinion dynamics in a fully connected network of interacting agents. The agents are assumed to interact according to one of the following rules: (1) Voter rule: An updating agent simply copies the opinion of another randomly sampled agent; (2) Majority rule: An updating agent samples multiple agents and adopts the majority opinion in the selected group. We focus on the scenario where the agents are biased towards one of the opinions called the {\em preferred opinion}. Using suitably constructed branching processes, we show that under both rules the mean time to reach consensus is $\Theta(\log N)$, where $N$ is the number of agents in the network. Furthermore, under the majority rule model, we show that consensus can be achieved on the preferred opinion with high probability even if it is initially the opinion of the minority. We also study the majority rule model when stubborn agents with fixed opinions are present. We find that the stationary distribution of opinions in the network in the large system limit using mean field techniques.

中文翻译：

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带有偏见和顽固代理的选民和多数派动态

我们在一个完全连接的交互代理网络中研究二元意见动态。假定代理根据以下规则之一进行交互： (1) 投票规则：更新代理简单地复制另一个随机采样的代理的意见；(2) 多数规则：更新代理对多个代理进行抽样，并在所选组中采用多数意见。我们专注于代理偏向于一种称为 {\em 首选意见} 的意见的情况。使用适当构建的分支过程，我们表明在这两个规则下达成共识的平均时间是 $\Theta(\log N)$，其中 $N$ 是网络中的代理数量。此外，在多数规则模型下，我们表明即使最初是少数人的意见，也可以高概率就首选意见达成共识。当存在具有固定意见的顽固代理时，我们还研究了多数规则模型。我们发现使用平均场技术限制了大型系统中网络中意见的平稳分布。