We study information aggregation in networks when agents interact to learn a binary state of the world. Initially each agent privately observes an independent signal which is "correct" with probability $\frac{1}{2}+\delta$ for some $\delta > 0$. At each round, a node is selected uniformly at random to update their public opinion to match the majority of their neighbours (breaking ties in favour of their initial private signal). Our main result shows that for sparse and connected binomial random graphs $\mathcal G(n,p)$ the process stabilizes in a "correct" consensus in $\mathcal O(n\log^2 n/\log\log n)$ steps with high probability. In fact, when $\log n/n \ll p = o(1)$ the process terminates at time $\hat T = (1+o(1))n\log n$, where $\hat T$ is the first time when all nodes have been selected at least once. However, in dense binomial random graphs with $p=\Omega(1)$, there is an information cascade where the process terminates in the "incorrect" consensus with probability bounded away from zero.

翻译：我们研究网络中当智能体通过交互学习世界的二元状态时的信息聚合问题。初始时，每个智能体独立观测到一个信号，该信号以概率 $\frac{1}{2}+\delta$（其中 $\delta > 0$）为“正确”。每一轮中，均匀随机选择一个节点，将其公开意见更新为其邻居中的多数意见（平局时倾向于初始私人信号）。我们的主要结果表明，对于稀疏且连通的二项随机图 $\mathcal G(n,p)$，该过程以高概率在 $\mathcal O(n\log^2 n/\log\log n)$ 步内稳定于一个“正确”共识。实际上，当 $\log n/n \ll p = o(1)$ 时，过程在时间 $\hat T = (1+o(1))n\log n$ 终止，其中 $\hat T$ 是所有节点至少被选择一次的首个时刻。然而，在 $p=\Omega(1)$ 的稠密二项随机图中，存在信息级联现象，使得该过程以有界非零概率终止于“错误”共识。