🤖 AI Summary
This paper studies information aggregation via asynchronous majority dynamics in agent networks: each agent initially receives a biased binary signal (accuracy $1/2+delta$) and iteratively updates its public opinion based on neighbors’ opinions, aiming to reach global correct consensus. Focusing on Erdős–Rényi random graphs $G(n,p)$, the work precisely characterizes convergence behavior across the sparse-to-dense transition. When $log n/n ll p = o(1)$, the process converges to the correct consensus within $(1+o(1))nlog n$ steps with high probability; in contrast, when $p = Omega(1)$, a phase transition occurs where incorrect consensus emerges with non-vanishing probability. The key contributions are: (i) establishing a tight time scale of $O(n log^2 n / log log n)$ for convergence, and (ii) revealing a critical interplay between graph sparsity and consensus robustness—quantifying how diminishing edge density fundamentally limits resilience to initial bias.
📝 Abstract
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 $mathcal O(nlog^2 n/loglog n)$ steps with high probability. In fact, when $log n/n ll p = o(1)$ the process terminates at time $hat T = (1+o(1))nlog n$, 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.
