Inevitability of Polarization in Geometric Opinion Exchange

TL;DR

The paper analyzes a geometric, multi-topic opinion-exchange model where agents hold unit vectors on a sphere and update via pairwise interactions governed by a biased-assimilation function $f$. It establishes that polarization to antipodal clusters is ubiquitous in the two-dimensional case under stable/update rules with full-support interaction and extends partial polarization results to higher dimensions using notions like inactive/separable/strict convex configurations and a $(d,n)$-stable framework. The results show a clear dimension-dependent behavior: $d=2$ admits a robust path to polarization, while $d\ge 3$ requires additional structure or assumptions, with counterexamples illustrating limitations of generalization. The work introduces technical tools including convexity-based progress measures, transitivity lemmas, and cluster-concentration arguments, contributing a rigorous treatment of polarization phenomena in discrete-time, pairwise-geometric opinion dynamics. Overall, the findings advance understanding of how modest bias and geometric coupling among topics drive robust polarization in social systems.

Abstract

Polarization and unexpected correlations between opinions on diverse topics (including in politics, culture and consumer choices) are an object of sustained attention. However, numerous theoretical models do not seem to convincingly explain these phenomena. This paper is motivated by a recent line of work, studying models where polarization can be explained in terms of biased assimilation and geometric interplay between opinions on various topics. The agent opinions are represented as unit vectors on a multidimensional sphere and updated according to geometric rules. In contrast to previous work, we focus on the classical opinion exchange setting, where the agents update their opinions in discrete time steps, with a pair of agents interacting randomly at every step. The opinions are updated according to an update rule belonging to a general class. Our findings are twofold. First, polarization appears to be ubiquitous in the class of models we study, requiring only relatively modest assumptions reflecting biased assimilation. Second, there is a qualitative difference between two-dimensional dynamics on the one hand, and three or more dimensions on the other. Accordingly, we prove almost sure polarization for a large class of update rules in two dimensions. Then, we prove polarization in three and more dimensions in more limited cases and try to shed light on central difficulties that are absent in two dimensions.

Figures (4)

• Figure 1: Issue radicalization with an asymmetric update rule: $n = 100, d = 3, \eta_+ = 0.9, \eta_- = 0.1$. The initial configuration is obtained by choosing $n$ opinions uniformly at random on the unit sphere. The update rule used is \ref{['eq:asymmetric']}. Each of the first three images shows the configuration at time $t = 2500$, for three independent runs of the experiments (i.e., three different initial configurations). Furthermore, over $100$ independent runs of the experiments, the histogram of the size of the cluster containing $\boldsymbol{u}_1$ is plotted. The colors represent the $z$ dimension and are added for visual clarity.
• Figure 2: Almost orthogonal configurations polarize: $n = 100, d = 3$ and the update rule is $f(A) = 0.1 A$. Initially, for every pair of opinions, their dot product $A$ satisfies $\left|{A}\right| \leq 0.1$ or $\left|{A}\right| \geq 0.9$. The process escapes the "almost orthogonal" configuration and polarizes.
• Figure 3: Illustration of the example from \ref{['sec:2d-inactive']}.
• Figure 4: All the cases from the proof of \ref{['lem:opinions_in_a_quadrant']}. For Case 3 applying $b_1\boldsymbol{u}_1$ and $b_2\boldsymbol{u}_2$ to all other opinions in their respective clusters will make the configuration strictly convex.

Theorems & Definitions (86)

• Definition 1.1
• Definition 1.2: Stable function
• Definition 1.3: Active function
• Definition 1.4: Separable configuration
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Claim 2.2