Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions

TL;DR

This paper numerically validates a macroscopic aggregation-diffusion model derived from microscopic particle interactions via a dynamical network, confirming phase transition behaviors and bifurcations in 1D and 2D.

Contribution

It provides a numerical validation of the macroscopic model's accuracy in representing microscopic particle dynamics with dynamical networks.

Findings

Macroscopic model accurately predicts phase transitions in particle systems.

Stationary solutions exhibit the same bifurcation types as microscopic models.

Numerical simulations in 2D align well with theoretical predictions.

Abstract

We provide a numerical study of the macroscopic model of [3] derived from an agent-based model for a system of particles interacting through a dynamical network of links. Assuming that the network remodelling process is very fast, the macroscopic model takes the form of a single aggregation diffusion equation for the density of particles. The theoretical study of the macroscopic model gives precise criteria for the phase transitions of the steady states, and in the 1-dimensional case, we show numerically that the stationary solutions of the microscopic model undergo the same phase transitions and bifurcation types as the macroscopic model. In the 2-dimensional case, we show that the numerical simulations of the macroscopic model are in excellent agreement with the predicted theoretical values. This study provides a partial validation of the formal derivation of the macroscopic model from a microscopic formulation and shows that the former is a consistent approximation of an underlying particle dynamics, making it a powerful tool for the modelling of dynamical networks at a large scale.