Kinetic Equations for Processes on Co-evolving Networks

Burger, Martin

doi:10.48550/arxiv.2104.13815

Cited by 2 publications

(3 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also, new dynamical states such as multi-clusters can be induced by taking into adaptivity [5]. Although a number of formal asymptotic methods or moment-closure schemes exist to study adaptive networks mathematically [8,18,19], a rigorous mathematical development of the field has just been started. One natural way to study networks rigorously is to exploit the large-scale limit of an infinite number of nodes/vertices to derive continuum or mean-field differential equations.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, if there is no feedback between network topology and node dynamics but just a given time-dependent family of networks, continuum limits were considered in [3]. A view via moment hierarchies and formal moment closure can be found in [8]. In summary, the myriad application results for finite N for adaptive Kuramoto-type models and the existing results for static/temporal network continuum limits for dense graphs, naturally pose the question, whether one can prove continuum limits for fully adaptive Kuramoto-type models on various classes of graph limits?…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Continuum Limits for Adaptive Network Dynamics

Gkogkas,

Kuehn,

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Continuum Limits for Adaptive Network Dynamics

Gkogkas,

Kuehn,

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…However, when the weights are not "uniform", rigorous works are lacking. MFLs of IPS were discussed also in [11], where kinetic equations for large finite population size were obtained, but the MFL was not rigorously characterized. Letting the number N of population size tend to infinity and the small time scale ε tend to zero simultaneously for sparse co-evolutionary networks using fast-slow arguments, a non-MFL result was investigated in [4], where the underlying graph is assumed to be independent of the dynamics of the particle system and hence the dynamics of the particle system plus the weights of the underlying graph is decoupled.…”

mentioning

confidence: 99%

Mean field limits of co-evolutionary heterogeneous networks

Gkogkas¹,

Kuehn²,

Xu³

2022

Preprint

View full text Add to dashboard Cite

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. These networks are used in different areas including telecommunication, neuroscience, computer science, biochemistry, social science, as well as physics, where Kuramoto-type 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 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].

show abstract

Kinetic Equations for Processes on Co-evolving Networks

Cited by 2 publications

References 48 publications

Continuum Limits for Adaptive Network Dynamics

Continuum Limits for Adaptive Network Dynamics

Mean field limits of co-evolutionary heterogeneous networks

Contact Info

Product

Resources

About