Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions

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.

References in corpus (13)

• A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
• The derivation of Swarming models: Mean-Field Limit and Wasserstein distances
• Collective Thermotaxis of Thermally Active Colloids
• Cross-link governed dynamics of biopolymer networks
• Existence of Compactly Supported Global Minimisers for the Interaction Energy
• Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics
• Existence of Ground States of Nonlocal-Interaction Energies
• Phase Transitions in a kinetic flocking model of Cucker-Smale type
• Filamin cross-linked semiflexible networks: Fragility under strain
• The McKean-Vlasov Equation in Finite Volume
• Phase transition and diffusion among socially interacting self-propelled agents
• Trail formation based on directed pheromone deposition
• Kinetic theory of particle interactions mediated by dynamical networks

Cited by in corpus (8)

• Long-time behaviour and phase transitions for the McKean--Vlasov equation on the torus
• Opinion modeling on social media and marketing aspects
• Singular Cucker-Smale Dynamics
• Structure preserving schemes for the continuum Kuramoto model: phase transitions
• Existence of ground states for aggregation-diffusion equations
• A Mean-Field Approach to Evolving Spatial Networks, with an Application to Osteocyte Network Formation
• Breaking Consensus in Kinetic Opinion Formation Models on Graphons
• Modelling pattern formation through differential repulsion