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Do Stubborn Users Always Cause More Polarization and Disagreement? A Mathematical Study

Mohammad Shirzadi, Ahad N. Zehmakan

Abstract

We study how the stubbornness of social network users influences opinion polarization and disagreement. Our work is in the context of the popular Friedkin-Johnson opinion formation model, where users update their opinion as a function of the opinion of their connections and their own innate opinion. Stubbornness then is formulated in terms of the stress a user puts on its innate opinion. We examine two scenarios: one where all nodes have uniform stubbornness levels (homogeneous) and another where stubbornness varies among nodes (inhomogeneous). In the homogeneous scenario, we prove that as the network's stubbornness factor increases, the polarization and disagreement index grows. In the more general inhomogeneous scenario, our findings surprisingly demonstrate that increasing the stubbornness of some users (particularly, neutral/unbiased users) can reduce the polarization and disagreement. We characterize specific conditions under which this phenomenon occurs. Finally, we conduct an extensive set of experiments on real-world network data to corroborate and complement our theoretical findings.

Do Stubborn Users Always Cause More Polarization and Disagreement? A Mathematical Study

Abstract

We study how the stubbornness of social network users influences opinion polarization and disagreement. Our work is in the context of the popular Friedkin-Johnson opinion formation model, where users update their opinion as a function of the opinion of their connections and their own innate opinion. Stubbornness then is formulated in terms of the stress a user puts on its innate opinion. We examine two scenarios: one where all nodes have uniform stubbornness levels (homogeneous) and another where stubbornness varies among nodes (inhomogeneous). In the homogeneous scenario, we prove that as the network's stubbornness factor increases, the polarization and disagreement index grows. In the more general inhomogeneous scenario, our findings surprisingly demonstrate that increasing the stubbornness of some users (particularly, neutral/unbiased users) can reduce the polarization and disagreement. We characterize specific conditions under which this phenomenon occurs. Finally, we conduct an extensive set of experiments on real-world network data to corroborate and complement our theoretical findings.

Paper Structure

This paper contains 17 sections, 12 theorems, 72 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Definitions
  4. Friedkin-Johnson Dynamics
  5. Polarization and Disagreement.
  6. Our contribution
  7. Related Work
  8. Matrix Formulation and Bounds
  9. $\mathcal{PD}$ in homogeneous stubbornness
  10. Stochastic Block Model Analysis
  11. $\mathcal{PD}$ in inhomogeneous stubbornness
  12. Experiments
  13. Setup
  14. Findings
  15. Conclusion
  16. ...and 2 more sections

Key Result

lemma 1

The iterative averaging process defined by the update rule in Equation FJ converges to the Nash equilibrium $z^* = ( \mathbf{L} + \mathbf{K} )^{-1} \mathbf{K} s$.

Figures (5)

  • Figure 1: Variation in the provided polarization upper bound in terms of the stubbornness factor.
  • Figure 2: Variation in $\mathcal{PD}$ (in logarithmic scale) as a function of network's stubbornness, within the SBM (block size = $1000$, outgoing connection probability $q = 0.1$).
  • Figure 3: The range of innate opinion of node $C$, for which increasing stubbornness from $k=1$ to $k=2$ causes reduction in the $\mathcal{PD}$.
  • Figure 4: The change in the $\mathcal{PD}$ (in logarithmic scale) with variation in (homogeneous) stubbornness across different networks.
  • Figure 5: The average change in the $\mathcal{PD}$ in the SBM with two positive/negative bubbles when the stubbornness of nodes with opposing opinions in each bubble is increased.

Theorems & Definitions (27)

  • lemma 1
  • proof
  • definition 1: Disagreement
  • definition 2: Polarization
  • definition 3: Polarization-Disagreement
  • lemma 2
  • proof
  • theorem 1
  • proof
  • theorem 2
  • ...and 17 more