Continuum Limits for Adaptive Network Dynamics
Abstract
Adaptive (or coevolutionary) network dynamics, i.e., when changes of the network/graph topology are coupled with changes in the node/vertex dynamics, can give rise to rich and complex dynamical behavior. Even though adaptivity can improve the modelling of collective phenomena, it often complicates the analysis of the corresponding mathematical models significantly. For nonadaptive systems, a possible way to tackle this problem is by passing to socalled continuum or meanfield limits, which describe the system in the limit of infinitely many nodes. Although fully adaptive network dynamic models have been used a lot in recent years in applications, we are still lacking a detailed mathematical theory for largescale adaptive network limits. For example, continuum limits for static or temporal networks are already established in the literature for certain models, yet the continuum limit of fully adaptive networks has been open so far. In this paper we introduce and rigorously justify continuum limits for sequences of adaptive Kuramototype network models. The resulting integrodifferential equations allow us to incorporate a large class of coevolving graphs with high density. Furthermore, we use a very general measuretheoretical framework in our proof for representing the (infinite) graph limits, thereby also providing a structural basis to tackle even larger classes of graph limits. As an application of our theory, we consider the continuum limit of an adaptive Kuramoto model directly motivated from neuroscience and studied by Berner et al.~in recent years using numerical techniques and formal stability analysis.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.05898
 Bibcode:
 2021arXiv210905898A
 Keywords:

 Mathematics  Dynamical Systems
 EPrint:
 22 pages