Barré J, Carrillo JA, Degond P, Peurichard D, Zatorska E. Particle Interactions Mediated by Dynamical Networks: Assessment of Macroscopic Descriptions.
JOURNAL OF NONLINEAR SCIENCE 2017;
28:235-268. [PMID:
29367810 PMCID:
PMC5756578 DOI:
10.1007/s00332-017-9408-z]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/30/2017] [Accepted: 08/01/2017] [Indexed: 06/07/2023]
Abstract
We provide a numerical study of the macroscopic model of Barré et al. (Multiscale Model Simul, 2017, to appear) derived from an agent-based model for a system of particles interacting through a dynamical network of links. Assuming that the network remodeling 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 one-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 two-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 modeling of dynamical networks at a large scale.
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