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Geometric opinion exchange polarizes in every dimension

Abdou Majeed Alidou, Júlia Baligács, Jan Hązła

TL;DR

This work studies geometric opinion exchange where $n$ agents hold $d$-dimensional unit vector opinions updated by a biased assimilation rule. The authors develop a potential-based, probabilistic analysis using inter- and intra-cluster correlations, encoded as $\delta_0$, $\delta_1$ and their logs $Q_0=-\log\delta_0$, $Q_1=-\log\delta_1$, and prove a block-wise drift that drives $Q_0$ downward while keeping $Q_1$ controlled. They introduce the notions of inactive configurations and cluster-consistency, show that inactive states become consistent in a bounded number of steps, and establish concentration over blocks of updates to guarantee eventual polarization. As a result, they resolve the general $d\ge 3$ case by proving that the process polarization occurs almost surely unless the initial configuration is separable, and they connect the analysis to a two-cluster collapse mechanism with high probability. Overall, the paper advances the understanding of polarized dynamics in high-dimensional opinion-exchange models and provides a rigorous framework for tracking cross-cluster biases via potential functions and martingale concentration.

Abstract

A recent line of work studies models of opinion exchange where agent opinions about $d$ topics are tracked simultaneously. The opinions are represented as vectors on the unit $(d-1)$-sphere, and the update rule is based on the overall correlation between the relevant vectors. The update rule reflects the assumption of biased assimilation, i.e., a pair of opinions is brought closer together if their correlation is positive and further apart if the correlation is negative. This model seems to induce the polarization of opinions into two antipodal groups. This is in contrast to many other known models which tend to achieve consensus. The polarization property has been recently proved for $d=2$, but the general case of $d \ge 3$ remained open. In this work, we settle the general case, using a more detailed understanding of the model dynamics and tools from the theory of random processes.

Geometric opinion exchange polarizes in every dimension

TL;DR

This work studies geometric opinion exchange where agents hold -dimensional unit vector opinions updated by a biased assimilation rule. The authors develop a potential-based, probabilistic analysis using inter- and intra-cluster correlations, encoded as , and their logs , , and prove a block-wise drift that drives downward while keeping controlled. They introduce the notions of inactive configurations and cluster-consistency, show that inactive states become consistent in a bounded number of steps, and establish concentration over blocks of updates to guarantee eventual polarization. As a result, they resolve the general case by proving that the process polarization occurs almost surely unless the initial configuration is separable, and they connect the analysis to a two-cluster collapse mechanism with high probability. Overall, the paper advances the understanding of polarized dynamics in high-dimensional opinion-exchange models and provides a rigorous framework for tracking cross-cluster biases via potential functions and martingale concentration.

Abstract

A recent line of work studies models of opinion exchange where agent opinions about topics are tracked simultaneously. The opinions are represented as vectors on the unit -sphere, and the update rule is based on the overall correlation between the relevant vectors. The update rule reflects the assumption of biased assimilation, i.e., a pair of opinions is brought closer together if their correlation is positive and further apart if the correlation is negative. This model seems to induce the polarization of opinions into two antipodal groups. This is in contrast to many other known models which tend to achieve consensus. The polarization property has been recently proved for , but the general case of remained open. In this work, we settle the general case, using a more detailed understanding of the model dynamics and tools from the theory of random processes.

Paper Structure

This paper contains 12 sections, 22 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Inactive configurations
  3. Our contribution and previous work
  4. Outline of the proof
  5. Plan of the proof of \ref{['thm:technical-informal']}
  6. \ref{['thm:technical-informal']} implies \ref{['thm:main-polarization']}
  7. Properties of $Q_0$ and $Q_1$
  8. Consistent configurations and expectation of $Q_0$
  9. Taking $T$ steps at once and martingale concentration
  10. Proof of \ref{['prop:two-chains']}
  11. Proof of \ref{['cor:bounded_step']}
  12. Proof that \ref{['thm:technical-informal']} implies \ref{['thm:main-polarization']}

Key Result

Theorem 1.3

Let $\mathcal{U}^0$ be an initial configuration which is not separable. Then, almost surely, $(\mathcal{U}^t)_t$ polarizes.

Theorems & Definitions (51)

  • Definition 1.1
  • Definition 1.2: Separable configuration
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6: Cluster
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1: Lemma 2.20 in ABH24
  • ...and 41 more