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Evolution of Discordance

F. den Hollander

Abstract

The present paper is a brief overview of random opinion dynamics on random graphs based on the Ising Lecture given by the author at the World Congress in Probability and Statistics, 12--16 August 2024, Bochum, Germany. The content is a snapshot of an interesting area of research that is developing rapidly.

Evolution of Discordance

Abstract

The present paper is a brief overview of random opinion dynamics on random graphs based on the Ising Lecture given by the author at the World Congress in Probability and Statistics, 12--16 August 2024, Bochum, Germany. The content is a snapshot of an interesting area of research that is developing rapidly.

Paper Structure

This paper contains 11 sections, 5 theorems, 26 equations, 11 figures.

Table of Contents

  1. Introduction
  2. One-way feedback
  3. Voter model on a graph
  4. Voter model on the complete graph
  5. Voter model on the random regular graph
  6. Voter model on a directed random graph
  7. Voter model on the random regular graph with random rewiring
  8. Two-way feedback
  9. Attempts at capturing co-evolution
  10. A model where co-evolution can be fully described
  11. Simulations

Key Result

Theorem 2.1

(Avena, Baldasso, Hazra, den Hollander, Quattropani ABHdHQ24.) Fix $u\in(0,1)$. Then, for any $t_N \in [0,\infty)$ as $N\to\infty$, with where ${\bf P}^{\mathcal{T}_d}$ is the law of two independent random walks on the infinite $d$-regular tree $\mathcal{T}_d$ starting from the endpoints of an edge, and $\tau_{\rm meet}$ denotes their first meeting time. Moreover, for every $\varepsilon>0$,

Figures (11)

  • Figure 1: Two communities of opinions on a friendship network.
  • Figure 2: A configuration of opinions on a graph. The graph stays fixed, the opinions evolve via random selection and adoption.
  • Figure 3: The complete graph on $N=7$ vertices.
  • Figure 4: A realisation of the Fisher-Wright diffusion, which eventually gets trapped at $0$ or $1$.
  • Figure 5: Random regular graph with $d=3$, $N=8$, $M=12$.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1