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Multipolar opinion evolution in biased networks

Luka Baković, David Ohlin, Giacomo Como, Emma Tegling

TL;DR

This work introduces a multidimensional nonlinear opinion dynamics model in which fixed agent biases shape evolving opinions that reside on the probability simplex. The dynamics are driven by a bias-modulated average of neighbors, with a normalization step that couples all dimensions, enabling multipolar outcomes beyond simple consensus. The authors characterize fixed points, provide conditions for dominance and suppression of alternatives, and prove a Lyapunov-based result guaranteeing consensus under aligned biases; they also illustrate polarization arising from spatially correlated biases in simulations on Watts–Strogatz networks. Overall, the model explains how bias heterogeneity and network structure can produce echo-chamber-like polarization or bias mediation, offering a framework for studying trust and opinion formation in biased information environments.

Abstract

Motivated by empirical research on bias and opinion formation, we formulate a multidimensional nonlinear opinion-dynamical model where agents have individual biases, which are fixed, as well as opinions, which evolve. The dimensions represent competing options, of which each agent has a relative opinion, and are coupled through normalization of the opinion vector. This can capture, for example, an individual's relative trust in different media. In special cases including where biases are uniform across agents our model achieves consensus, but in general, behaviors are richer and capture multipolar opinion distributions. We examine general fixed points of the system, as well as special cases such as zero biases toward certain options or partitioned decision sets. Lastly, we demonstrate that our model exhibits polarization when biases are spatially correlated across the network, while, as empirical research suggests, a mixed community can mediate biases.

Multipolar opinion evolution in biased networks

TL;DR

This work introduces a multidimensional nonlinear opinion dynamics model in which fixed agent biases shape evolving opinions that reside on the probability simplex. The dynamics are driven by a bias-modulated average of neighbors, with a normalization step that couples all dimensions, enabling multipolar outcomes beyond simple consensus. The authors characterize fixed points, provide conditions for dominance and suppression of alternatives, and prove a Lyapunov-based result guaranteeing consensus under aligned biases; they also illustrate polarization arising from spatially correlated biases in simulations on Watts–Strogatz networks. Overall, the model explains how bias heterogeneity and network structure can produce echo-chamber-like polarization or bias mediation, offering a framework for studying trust and opinion formation in biased information environments.

Abstract

Motivated by empirical research on bias and opinion formation, we formulate a multidimensional nonlinear opinion-dynamical model where agents have individual biases, which are fixed, as well as opinions, which evolve. The dimensions represent competing options, of which each agent has a relative opinion, and are coupled through normalization of the opinion vector. This can capture, for example, an individual's relative trust in different media. In special cases including where biases are uniform across agents our model achieves consensus, but in general, behaviors are richer and capture multipolar opinion distributions. We examine general fixed points of the system, as well as special cases such as zero biases toward certain options or partitioned decision sets. Lastly, we demonstrate that our model exhibits polarization when biases are spatially correlated across the network, while, as empirical research suggests, a mixed community can mediate biases.
Paper Structure (14 sections, 8 theorems, 19 equations, 6 figures)

This paper contains 14 sections, 8 theorems, 19 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions and paper outline
  4. The Model
  5. Notation and Preliminaries
  6. Model Setup
  7. Analysis
  8. General fixed points and bias with zero elements
  9. Rejection of non-dominant alternatives
  10. Fixed point stability analysis for $n=2$, $k=2$
  11. Numerical examples
  12. Two Agents
  13. Spatial Correlation and Polarization
  14. Conclusions

Key Result

Proposition 1

For any $k \in \mathbb N$, $n \in \mathbb N$ and $\mathcal{G}$ satisfying the definition, setting $R^i = \mathbb I_k \; (\forall i \in \mathcal{V})$ reduces the model to the DeGroot model degroot1974reaching.

Figures (6)

  • Figure 1: the maximal bias case
  • Figure 2: mediation
  • Figure 3: agent 2 influenced toward option 1
  • Figure 5: starting distribution
  • Figure 6: spatially correlated biases
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 1: k-simplex
  • Remark 1
  • Remark 2
  • Example 1
  • Example 2
  • Proposition 1
  • proof
  • Remark 3
  • Proposition 2
  • proof
  • ...and 14 more