Asynchronous Majority Dynamics on Binomial Random Graphs

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 \emph{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 $\Gnp$ the process stabilizes in a \emph{correct} consensus in $\bigo(n\log^2 n/\log\log n)$ 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 \emph{incorrect} consensus with probability bounded away from zero.