Kinetic equations for processes on co-evolving networks

Martin Burger

doi:10.3934/krm.2021051

Article Contents

2022, Volume 15, Issue 2: 187-212. Doi: 10.3934/krm.2021051

This issue Previous Article On spectral gaps of growth-fragmentation semigroups in higher moment spaces Next Article Global hypocoercivity of kinetic Fokker-Planck-Alignment equations

Kinetic equations for processes on co-evolving networks

Martin Burger

Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, Cauerstr. 11, D 91058 Erlangen, Germany

Received: May 2021

Revised: December 2021

Early access: April 2022

Published: April 2022

The author acknowledges partial financial support by European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 777826 (NoMADS) and the German Science Foundation (DFG) through CRC TR 154 "Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks", Subproject C06

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Abstract

The aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to processes on graphs (or networks), where both the states of particles in nodes as well as the weights between them are updated in time. In our derivation we follow the basic paradigm of statistical mechanics: We start from paradigmatic microscopic models and derive a Liouville-type equation in a high-dimensional space including not only the node states in the network (corresponding to positions in mechanics), but also the edge weights between them. We then derive a natural (finite) marginal hierarchy and pass to an infinite limit.

We will discuss the closure problem for this hierarchy and see that a simple mean-field solution can only arise if the weight distributions between nodes of equal states are concentrated. In a more interesting general case we propose a suitable closure at the level of a two-particle distribution (including the weight between them) and discuss some properties of the arising kinetic equations. Moreover, we highlight some structure-preserving properties of this closure and discuss its analysis in a minimal model. We discuss the application of our theory to some agent-based models in literature and discuss some open mathematical issues.

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Mathematics Subject Classification: Primary: 35Q91, 91D30; Secondary: 35Q83.

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