Power of the Crowd

TL;DR

This paper analyzes how opinions propagate in a Galton-Watson tree with multiple opinions, focusing on the long-term distribution of the root's opinion as the tree height grows infinitely.

Contribution

It introduces a model for opinion spread in a branching process and studies the asymptotic distribution of the root's opinion under majority rule.

Findings

Distribution converges as tree height increases

Certain opinions dominate in the limit

Behavior depends on initial opinion probabilities

Abstract

Consider an Galton Watson tree of height $m$: each leaf has one of $k$ opinions or not. In other words, for $i \in \{1, . . . , k\}$, $x$ at generation $m$ thinks $i$ with probability $p_i$ and nothing with probability $p_0$. Moreover the opinions are independently distributed for each leaf. Opinions spread along the tree based on a specific rule: the majority wins! In this paper, we study the asymptotic behavior of the distribution of the opinion of the root when $m \to +\infty$.