Contents Online

# Communications in Mathematical Sciences

## Volume 21 (2023)

### Number 1

### Continuum limits for adaptive network dynamics

Pages: 83 – 106

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n1.a4

#### Authors

#### Abstract

Adaptive (or co-evolutionary) 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 non-adaptive systems, a possible way to tackle this problem is by passing to so-called continuum or mean-field 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 large-scale 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 Kuramoto-type network models. The resulting integro-differential equations allow us to incorporate a large class of co-evolving graphs with high density. Furthermore, we use a very general measure-theoretical 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.

#### Keywords

adaptive networks, co-evolutionary networks, continuum limit, graph limits, Kuramoto-type models

#### 2010 Mathematics Subject Classification

05C82, 35Q83, 35R02, 92B20

M.A.G. and C.K. gratefully thank the TUM International Graduate School of Science and Engineering (IGSSE) for support via the project “Synchronization in Co-Evolutionary Network Dynamics (SEND)”. C.K. also acknowledges partial support by a Lichtenberg Professorship funded by the VolkswagenStiftung. C.X. acknowledges support from the TUM Foundation Fellowship as well as the Alexander von Humboldt Fellowship.

Received 13 September 2021

Received revised 5 February 2022

Accepted 7 April 2022

Published 27 December 2022