Dynamical systems and complex networks: A Koopman operator perspective

This perspective article is meant to be a self-contained introduction to and review of transfer operators such as the Koopman operator and the Perron-Frobenius operator as well as an overview of different applications. We will first introduce the required foundations and then show how transfer operators can not only be used to analyze highly nonlinear dynamical systems but also complex networks. In particular, we will focus on relationships between transfer operators for continuous-time stochastic processes defined on a continuous state space--whether they be reversible, non-reversible but time-homogeneous, or time-inhomogeneous--and their discrete counterparts associated with random walks on undirected, directed, and time-evolving graphs. Transfer operators play an important role in an increasing number of research fields. A few exemplary applications are illustrated in Figure 1.