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Opinion formation on evolving network. The DPA method applied to a nonlocal cross-diffusion PDE-ODE system

Simone Fagioli, Gianluca Favre

TL;DR

The Deterministic Particle Approximation (DPA) method is applied to a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network to prove the existence of solutions under suitable assumptions on the interactions between agents.

Abstract

We study a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network. The PDEs are coupled with a system of ODEs that describe the time evolution of the agents on the network. Firstly, we apply the Deterministic Particle Approximation (DPA) method to the aforementioned system in order to prove the existence of solutions under suitable assumptions on the interactions between agents. Later on, we present an explicit model for opinion formation on an evolving network. The opinions evolve based on both the distance between the agents on the network and the 'attitude areas,' which depend on the distance between the agents' opinions. The position of the agents on the network evolves based on the distance between the agents' opinions. The goal is to study radicalization, polarization, and fragmentation of the population while changing its open-mindedness and the radius of interaction.

Opinion formation on evolving network. The DPA method applied to a nonlocal cross-diffusion PDE-ODE system

TL;DR

The Deterministic Particle Approximation (DPA) method is applied to a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network to prove the existence of solutions under suitable assumptions on the interactions between agents.

Abstract

We study a system of nonlocal aggregation cross-diffusion PDEs that describe the evolution of opinion densities on a network. The PDEs are coupled with a system of ODEs that describe the time evolution of the agents on the network. Firstly, we apply the Deterministic Particle Approximation (DPA) method to the aforementioned system in order to prove the existence of solutions under suitable assumptions on the interactions between agents. Later on, we present an explicit model for opinion formation on an evolving network. The opinions evolve based on both the distance between the agents on the network and the 'attitude areas,' which depend on the distance between the agents' opinions. The position of the agents on the network evolves based on the distance between the agents' opinions. The goal is to study radicalization, polarization, and fragmentation of the population while changing its open-mindedness and the radius of interaction.
Paper Structure (17 sections, 11 theorems, 89 equations, 9 figures)

This paper contains 17 sections, 11 theorems, 89 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Modelling motivation: the social context
  3. Mathematical tools
  4. Rigorous formulation, assumptions and main result
  5. Deterministic Particle Approximation (DPA)
  6. Preliminaries and assumptions
  7. Continuous reconstruction and main result
  8. Proof of the main result
  9. Basic estimates
  10. Convergence to weak solutions
  11. Modelling and simulation
  12. Attitude areas
  13. Diffusion
  14. Evolution of the network
  15. Radicalization, polarization, and fragmentation
  16. ...and 2 more sections

Key Result

Theorem 2.2

Given $M\in\mathbb{N}$ and $T>0$ fixed, consider $\mathbf{A}^{\mathcal{ij}}$, $\mathbf{K}^{\mathcal{ij}}$, $\mathbf{V}$, and $\Phi^\mathcal{i}$ under assumptions $(\mathbf{A})$, $(\mathbf{K})$, $(\textbf{V})$, and $(Dif)$ respectively, for all $\mathcal{i},\mathcal{j}\in \mathcal{M}$. Let $\bar{\rho

Figures (9)

  • Figure 1: Attraction/repulsion function
  • Figure 2: Initial network condition
  • Figure 3: Local initial network interaction
  • Figure 4: Initial opinion distribution
  • Figure 5: Final opinion distribution
  • ...and 4 more figures

Theorems & Definitions (22)

  • Definition 2.1: Weak solution
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2: Ordering preservation
  • proof
  • Remark 3.3
  • Lemma 3.4: Velocity boundedness
  • proof
  • Theorem 3.5
  • Proposition 3.6
  • ...and 12 more