# Asynchronous Majority Dynamics in Preferential Attachment Trees

## TL;DR

This paper analyzes how information spreads and stabilizes in large preferential attachment trees where agents asynchronously adopt majority decisions, showing that correct consensus is reached efficiently with high probability.

## Contribution

It proves that in preferential attachment trees, the majority dynamics stabilize correctly within near-linear rounds, extending results to other tree structures.

## Key findings

- Correct majority stabilizes in $O(n ext{ log } n / ext{ log log } n)$ rounds with high probability
- Results extend to balanced M-ary trees and similar structures
- Asynchronous, non-Bayesian decision dynamics effectively aggregate information

## Abstract

We study information aggregation in networks where agents make binary decisions (labeled incorrect or correct). Agents initially form independent private beliefs about the better decision, which is correct with probability $1/2+\delta$. The dynamics we consider are asynchronous (each round, a single agent updates their announced decision) and non-Bayesian (agents simply copy the majority announcements among their neighbors, tie-breaking in favor of their private signal). Our main result proves that when the network is a tree formed according to the preferential attachment model \cite{BarabasiA99}, with high probability, the process stabilizes in a correct majority within $O(n \log n/ \log\log n)$ rounds. We extend our results to other tree structures, including balanced $M$-ary trees for any $M$.