Continuum Limits for Adaptive Network Dynamics

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.

Mean field limits of co-evolutionary heterogeneous networks

“…Proof. The proof is in similar spirit to that of [18,Theorem A]. For the reader's convenience, we provide a complete proof here.…”

“…The following properties of measure-valued functions from [18,26] will be used to propose a generalized coevolutionary Kuramoto network in the next section.…”

“…c.f. [18]. Moreover, not all families of parameterized measures are differentiable (e.g., (δ t ) t [18]).…”

“…Another type of macroscopic limit frequently encountered in the literature is the so-called continuum limit, defined as pointwise limit of the network model. Continuum limits of the Kuramoto model were investigated in [32] on random sparse static networks, and recently investigated in [18] on deterministic co-evolutionary networks. We mention that continuum limits of collective dynamics models with "uniform" time-varying weights (e.g., the Cucker-Smale model) were also recently studied in [2], which is restricted to the case, where the approach for particle systems coupled on static networks remains valid.…”

“…We mention that continuum limits of collective dynamics models with "uniform" time-varying weights (e.g., the Cucker-Smale model) were also recently studied in [2], which is restricted to the case, where the approach for particle systems coupled on static networks remains valid. In contrast, such a restriction was removed in [18]. 1.2.…”