Mean field limits of co-evolutionary heterogeneous networks

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 real-world phenomena as co-evolutionary (or adaptive) networks. In this paper, we propose a rigorous formulation for limits of a sequence of co-evolutionary Kuramoto oscillators coupled on heterogeneous co-evolutionary 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 co-evolutionary network exists and the sequence of co-evolutionary 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, arXiv:2107.08419, 2021]. 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 non-co-evolutionary networks (i.e., static networks or time-dependent networks independent of the dynamics of the IPS), our work is the first to rigorously address the MFL of a co-evolutionary network model. The approach is based on our formulation of a generalization of the co-evolutionary network as a hybrid system of ODEs and measure differential equations parametrized by a vertex variable, together with an analogue of the variation of parameters formula, as well as the generalized Neunzert's in-cell-particle method developed in [Kuehn, Xu. Vlasov equations on digraph measures, arXiv:2107.08419, 2021]