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Efficient Algorithms for Relevant Quantities of Friedkin-Johnsen Opinion Dynamics Model

Gengyu Wang, Runze Zhang, Zhongzhi Zhang

TL;DR

This work tackles the challenge of efficiently computing the Friedkin-Johnsen equilibrium vector $\boldsymbol{z}$ and related social-measurements on large-scale networks. It introduces a deterministic local iteration framework with relative-error guarantees, augmented by successive over-relaxation (SOR) to accelerate convergence, and extends to both directed and undirected graphs. Key contributions include the BoundLocalIter algorithm, a robustness-enhanced ImprovedBLI, and BoundLocalIterSOR, all backed by theoretical error guarantees and practical scalability to networks with tens of millions of nodes; extensive experiments show substantial speedups over traditional Laplacian-solvers and forest-sampling approaches while maintaining high accuracy. The methods enable efficient, scalable analysis of disagreement, polarization, and controversy in real-world social networks, supporting large-scale studies of opinion dynamics and intervention strategies.

Abstract

Online social networks have become an integral part of modern society, profoundly influencing how individuals form and exchange opinions across diverse domains ranging from politics to public health. The Friedkin-Johnsen model serves as a foundational framework for modeling opinion formation dynamics in such networks. In this paper, we address the computational task of efficiently determining the equilibrium opinion vector and associated metrics including polarization and disagreement, applicable to both directed and undirected social networks. We propose a deterministic local algorithm with relative error guarantees, scaling to networks exceeding ten million nodes. Further acceleration is achieved through integration with successive over-relaxation techniques, where a relaxation factor optimizes convergence rates. Extensive experiments on diverse real-world networks validate the practical effectiveness of our approaches, demonstrating significant improvements in computational efficiency and scalability compared to conventional methods.

Efficient Algorithms for Relevant Quantities of Friedkin-Johnsen Opinion Dynamics Model

TL;DR

This work tackles the challenge of efficiently computing the Friedkin-Johnsen equilibrium vector and related social-measurements on large-scale networks. It introduces a deterministic local iteration framework with relative-error guarantees, augmented by successive over-relaxation (SOR) to accelerate convergence, and extends to both directed and undirected graphs. Key contributions include the BoundLocalIter algorithm, a robustness-enhanced ImprovedBLI, and BoundLocalIterSOR, all backed by theoretical error guarantees and practical scalability to networks with tens of millions of nodes; extensive experiments show substantial speedups over traditional Laplacian-solvers and forest-sampling approaches while maintaining high accuracy. The methods enable efficient, scalable analysis of disagreement, polarization, and controversy in real-world social networks, supporting large-scale studies of opinion dynamics and intervention strategies.

Abstract

Online social networks have become an integral part of modern society, profoundly influencing how individuals form and exchange opinions across diverse domains ranging from politics to public health. The Friedkin-Johnsen model serves as a foundational framework for modeling opinion formation dynamics in such networks. In this paper, we address the computational task of efficiently determining the equilibrium opinion vector and associated metrics including polarization and disagreement, applicable to both directed and undirected social networks. We propose a deterministic local algorithm with relative error guarantees, scaling to networks exceeding ten million nodes. Further acceleration is achieved through integration with successive over-relaxation techniques, where a relaxation factor optimizes convergence rates. Extensive experiments on diverse real-world networks validate the practical effectiveness of our approaches, demonstrating significant improvements in computational efficiency and scalability compared to conventional methods.

Paper Structure

This paper contains 22 sections, 9 theorems, 15 equations, 2 figures, 8 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Notations
  5. Graphs and Related Matrices
  6. Friedkin-Johnsen Model
  7. Measures in FJ Model
  8. Existing Algorithms
  9. Algorithm Based on Random Walk
  10. Algorithm Based on Laplacian Solver
  11. Local Iteration Algorithm
  12. Simple Local Iteration Algorithm
  13. Random Walk Interpretation
  14. Deterministic Local Iteration Algorithm
  15. Robustness Enhancement
  16. ...and 7 more sections

Key Result

lemma 1

As $t$ approaches infinity, $\boldsymbol{\mathit{z}}^{(t)}$ converges to an equilibrium vector $\boldsymbol{\mathit{z}} = (z_1,z_2,\cdots,z_n)^\top$ satisfying $\boldsymbol{\mathit{z}} = (\boldsymbol{\mathit{I}}+\bm{\mathit{L}})^{-1}\boldsymbol{\mathit{s}}$.

Figures (2)

  • Figure 1: Actual relative-error v.s. execution time (seconds) on two real-world datasets: (a) YoutubeSnap, (b) Flixster.
  • Figure 2: Impact of different values of relaxation factor $\omega$ on the number of updates performed by the BLISOR. The experiments are conducted on four real-world datasets: (a) Email-Enron, (b) DBLP, (c) YoutubeSnap and (d) Flixster.

Theorems & Definitions (9)

  • lemma 1: BiKlOr15
  • lemma 2
  • lemma 3
  • lemma 4
  • theorem 1
  • lemma 5
  • theorem 2
  • theorem 3
  • lemma 6