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Graphs and networks play an important role in modeling and analyzing complexinterconnected systems such as transportation networks, integrated circuits,power grids, citation graphs, and biological and artificial neural networks.Graph clustering algorithms can be used to detect groups of strongly connectedvertices and to derive coarse-grained models. We define transfer operators suchas the Koopman operator and the Perron-Frobenius operator on graphs, studytheir spectral properties, introduce Galerkin projections of these operators,and illustrate how reduced representations can be estimated from data. Inparticular, we show that spectral clustering of undirected graphs can beinterpreted in terms of eigenfunctions of the Koopman operator and proposenovel clustering algorithms for directed graphs based on generalized transferoperators. We demonstrate the efficacy of the resulting algorithms on severalbenchmark problems and provide different interpretations of clusters.

Transfer operators on graphs: Spectral clustering and beyond