Polarization in Geometric Opinion Dynamics

TL;DR

This paper explores the nuanced emergence of polarization in geometric opinion dynamics models, demonstrating conditions under which strong or weak polarization occurs and extending the understanding of polarization's universality beyond specific cases.

Contribution

It advances the study of polarization by analyzing its nuanced forms in geometric models and broadening the conditions under which polarization phenomena are provable and robust.

Findings

Strong polarization holds in many variants of the HJMR model.

Weak polarization can occur even when strong polarization fails.

Polarization phenomena relate to Markov chain theory on general state spaces.

Abstract

In light of increasing recent attention to political polarization, understanding how polarization can arise poses an important theoretical question. While more classical models of opinion dynamics seem poorly equipped to study this phenomenon, a recent novel approach by H\k{a}z{\l}a, Jin, Mossel, and Ramnarayan (HJMR) proposes a simple geometric model of opinion evolution that provably exhibits strong polarization in specialized cases. Moreover, polarization arises quite organically in their model: in each time step, each agent updates opinions according to their correlation/response with an issue drawn at random. However, their techniques do not seem to extend beyond a set of special cases they identify, which benefit from fragile symmetry or contractiveness assumptions, leaving open how general this phenomenon really is. In this paper, we further the study of polarization in related geometric models. We show that the exact form of polarization in such models is quite nuanced: even when strong polarization does not hold, it is possible for weaker notions of polarization to nonetheless attain. We provide a concrete example where weak polarization holds, but strong polarization provably fails. However, we show that strong polarization provably holds in many variants of the HJMR model, which are also robust to a wider array of distributions of random issues -- this indicates that the form of polarization introduced by HJMR is more universal than suggested by their special cases. We also show that the weaker notions connect more readily to the theory of Markov chains on general state spaces.