Mean field limits of coevolutionary heterogeneous networks
Abstract
Many science phenomena are modelled as interacting particle systems (IPS) coupled on static networks. In reality, network connections are far more dynamic. Connections among individuals receive feedback from nearby individuals and make changes to better adapt to the world. Hence, it is reasonable to model myriad realworld phenomena as coevolutionary (or adaptive) networks. These networks are used in different areas including telecommunication, neuroscience, computer science, biochemistry, social science, as well as physics, where Kuramototype networks have been widely used to model interaction among a set of oscillators. In this paper, we propose a rigorous formulation for limits of a sequence of coevolutionary Kuramoto oscillators coupled on heterogeneous coevolutionary networks, which receive feedback from the dynamics of the oscillators on the networks. We show under mild conditions, the mean field limit (MFL) of the coevolutionary network exists and the sequence of coevolutionary Kuramoto networks converges to this MFL. Such MFL is described by solutions of a generalized Vlasov type equation. We treat the graph limits as graph measures, motivated by the recent work in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261349]. Under a mild condition on the initial graph measure, we show that the graph measures are positive over a finite time interval. In comparison to the recently emerging works on MFLs of IPS coupled on noncoevolutionary networks (i.e., static networks or timedependent networks independent of the dynamics of the IPS), our work seems the first to rigorously address the MFL of a coevolutionary network model.
 Publication:

arXiv eprints
 Pub Date:
 February 2022
 DOI:
 10.48550/arXiv.2202.01742
 arXiv:
 arXiv:2202.01742
 Bibcode:
 2022arXiv220201742A
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Analysis of PDEs;
 Mathematics  Classical Analysis and ODEs;
 Mathematics  Probability;
 35R02;
 92C42;
 60B10