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Behavior and Sublinear Algorithm for Opinion Disagreement on Noisy Social Networks

Wanyue Xu, Yubo Sun, Mingzhe Zhu, Zuobai Zhang, Zhongzhi Zhang

Abstract

The phenomenon of opinion disagreement has been empirically observed and reported in the literature, which is affected by various factors, such as the structure of social networks. An important discovery in network science is that most real-life networks, including social networks, are scale-free and sparse. In this paper, we study noisy opinion dynamics in sparse scale-free social networks to uncover the influence of power-law topology on opinion disagreement. We adopt the popular discrete-time DeGroot model for opinion dynamics in a graph, where nodes' opinions are subject to white noise. We first study opinion disagreement in many realistic and model networks with a scale-free topology, which approaches a constant, indicating that a scale-free structure is resistant to noise in the opinion dynamics. Moreover, existing algorithms for estimating opinion disagreement are computationally impractical for large-scale networks due to their high computational complexity. To solve this challenge, we introduce a sublinear-time algorithm to approximate this quantity with a theoretically guaranteed error. This algorithm efficiently simulates truncated random walks starting from a subset of nodes while preserving accurate estimation. Extensive experiments demonstrate its efficiency, accuracy, and scalability.

Behavior and Sublinear Algorithm for Opinion Disagreement on Noisy Social Networks

Abstract

The phenomenon of opinion disagreement has been empirically observed and reported in the literature, which is affected by various factors, such as the structure of social networks. An important discovery in network science is that most real-life networks, including social networks, are scale-free and sparse. In this paper, we study noisy opinion dynamics in sparse scale-free social networks to uncover the influence of power-law topology on opinion disagreement. We adopt the popular discrete-time DeGroot model for opinion dynamics in a graph, where nodes' opinions are subject to white noise. We first study opinion disagreement in many realistic and model networks with a scale-free topology, which approaches a constant, indicating that a scale-free structure is resistant to noise in the opinion dynamics. Moreover, existing algorithms for estimating opinion disagreement are computationally impractical for large-scale networks due to their high computational complexity. To solve this challenge, we introduce a sublinear-time algorithm to approximate this quantity with a theoretically guaranteed error. This algorithm efficiently simulates truncated random walks starting from a subset of nodes while preserving accurate estimation. Extensive experiments demonstrate its efficiency, accuracy, and scalability.

Paper Structure

This paper contains 29 sections, 11 theorems, 20 equations, 5 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Graph and Matrix Notation
  5. Random Walks on a Graph
  6. Discrete-Time Noisy DeGroot Model
  7. Noisy DeGroot model
  8. Spectral Expression for Opinion Disagreement
  9. Opinion Disagreement in Networks
  10. Opinion Disagreement in Power-Law Real-World Networks
  11. Opinion Disagreement in Model Networks
  12. Result Analysis
  13. Fast Approximation Algorithms for Estimating Opinion Disagreement
  14. Modified Algorithm Based on Laplacian Solvers
  15. Alternative Expression for Relevant Quantity
  16. ...and 14 more sections

Key Result

Lemma 6.1

XuShZhKaZh20 For a connected undirected weighted graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ with or without self-loops, let $\boldsymbol{\mathit{L}}^{\dagger}$ and $\boldsymbol{\mathcal{L}}^{\dagger}$ be the Moore-Penrose inverse of its Laplacian matrix $\boldsymbol{\mathit{L}}$ and normalized La

Figures (5)

  • Figure 1: Numerical results for the opinion disagreement on BA networks and random Apollonian networks.
  • Figure 2: Numerical results for the opinion disagreement on Growing Small-World networks.
  • Figure 3: Construction of the first several iterations of the PSFW.
  • Figure 4: Running time of different algorithms with varying error parameter $\epsilon$ on real-world graphs: Protein (a), web-EPA (b), Brightkite (c), soc-delicious (d), soc-hyves (e), and delaunay-n24 (f).
  • Figure 5: Mean relative error of different algorithms with varying error parameter $\epsilon$ on the pseudofractal scale-free webs: $\mathcal{F}'_{12}$, $\mathcal{F}'_{13}$, $\mathcal{F}'_{14}$, and $\mathcal{F}'_{15}$.

Theorems & Definitions (14)

  • Definition 3.1
  • Definition 4.1
  • Lemma 6.1
  • Lemma 6.2
  • Lemma 6.3
  • Theorem 6.4
  • Lemma 6.5
  • Lemma 6.6
  • Lemma 6.7
  • Lemma 6.8
  • ...and 4 more