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Opinion dynamics on non-sparse networks with community structure

Panagiotis Andreou, Mariana Olvera-Cravioto

TL;DR

This model is akin to the popular Friedkin-Johnsen model, and is able to incorporate factors such as stubbornness, confirmation bias, selective exposure, and multiple topics, which are believed to play an important role in the formation of opinions.

Abstract

We study the evolution of opinions on a directed network with community structure. Individuals update their opinions synchronously based on a weighted average of their neighbors' opinions, their own previous opinions, and external media signals. Our model is akin to the popular Friedkin-Johnsen model, and is able to incorporate factors such as stubbornness, confirmation bias, selective exposure, and multiple topics, which are believed to play an important role in the formation of opinions. Our main result shows that, in the large graph limit, the opinion process concentrates around its mean-field approximation for any level of edge density, provided the average degree grows to infinity. Moreover, we show that the opinion process exhibits propagation of chaos. We also give results for the trajectories of individual vertices and the stationary version of the opinion process, and prove that the limits in time and in the size of the network commute. The mean-field approximation is explicit and can be used to quantify consensus and polarization.

Opinion dynamics on non-sparse networks with community structure

TL;DR

This model is akin to the popular Friedkin-Johnsen model, and is able to incorporate factors such as stubbornness, confirmation bias, selective exposure, and multiple topics, which are believed to play an important role in the formation of opinions.

Abstract

We study the evolution of opinions on a directed network with community structure. Individuals update their opinions synchronously based on a weighted average of their neighbors' opinions, their own previous opinions, and external media signals. Our model is akin to the popular Friedkin-Johnsen model, and is able to incorporate factors such as stubbornness, confirmation bias, selective exposure, and multiple topics, which are believed to play an important role in the formation of opinions. Our main result shows that, in the large graph limit, the opinion process concentrates around its mean-field approximation for any level of edge density, provided the average degree grows to infinity. Moreover, we show that the opinion process exhibits propagation of chaos. We also give results for the trajectories of individual vertices and the stationary version of the opinion process, and prove that the limits in time and in the size of the network commute. The mean-field approximation is explicit and can be used to quantify consensus and polarization.
Paper Structure (12 sections, 13 theorems, 171 equations)

This paper contains 12 sections, 13 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. The Model
  3. The opinion process on a fixed graph
  4. The network
  5. Main Results
  6. Related Literature
  7. Proofs
  8. The semi-sparse to dense regimes
  9. The subcritical to sparse regimes
  10. Coupling with a multi-type Galton-Watson tree
  11. Proof of Theorem \ref{['MFA_generalthm']} for semi-sparse graphs
  12. Proofs of Theorems \ref{['thm_LWCprob']} and \ref{['thm_convergence_in_prob']}

Key Result

Theorem 3.1

Define the process $\{\mathcal{R}^{(k)}: k \geq 0\}$ according to eq:Limit. Suppose $\theta_n \geq (6H \Lambda_n)^2 \Delta_n \log n$. Then, there exists a constant $\Gamma < \infty$ such that where $\mathcal{E}_n := \max_{1\leq r,s \leq K} \left| \frac{\pi_s^{(n)} \pi_r - \pi_s \pi_r^{(n)}}{\pi_r^{(n)} \pi_s} \right|$. Moreover, for any sequence $\theta_n$ satisfying $\theta_n\rightarrow\infty$ a

Theorems & Definitions (29)

  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Theorem 3.3
  • Remark 3.2
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • Proposition 5.1
  • ...and 19 more