Dynamical systems and complex networks: A Koopman operator perspective

TL;DR

This paper explores how Koopman operator theory can be applied to analyze complex networks, providing a data-driven, linear perspective on nonlinear dynamical systems and revealing connections with graph Laplacians.

Contribution

It demonstrates the application of Koopman operator methods to complex networks and elucidates their relationship with graph Laplacians, advancing the analysis of nonlinear systems.

Findings

Koopman operators can effectively analyze complex networks.

Connections between Koopman operators and graph Laplacians are established.

The framework offers insights without detailed system models.

Abstract

The Koopman operator has entered and transformed many research areas over the last years. Although the underlying concept$\unicode{x2013}$representing highly nonlinear dynamical systems by infinite-dimensional linear operators$\unicode{x2013}$has been known for a long time, the availability of large data sets and efficient machine learning algorithms for estimating the Koopman operator from data make this framework extremely powerful and popular. Koopman operator theory allows us to gain insights into the characteristic global properties of a system without requiring detailed mathematical models. We will show how these methods can also be used to analyze complex networks and highlight relationships between Koopman operators and graph Laplacians.