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Two of the main challenges are that the eigenvalues and eigenvectors of graph Laplacians associated with directed graphs are in general complex-valued and that there is no universally accepted definition of clusters in directed graphs. We first exploit relationships between the graph Laplacian and transfer operators and in particular between clusters in undirected graphs and metastable sets in stochastic dynamical systems and then use a generalization of the notion of metastability to derive clustering algorithms for directed and time-evolving graphs. The resulting clusters can be interpreted as <jats:italic>coherent sets<\/jats:italic>, which play an important role in the analysis of transport and mixing processes in fluid flows.\n<\/jats:p>\n              <\/jats:sec><jats:sec>\n                <jats:title>Graphic Abstract<\/jats:title>\n                \n              <\/jats:sec>","DOI":"10.1007\/s00332-022-09863-0","type":"journal-article","created":{"date-parts":[[2022,11,8]],"date-time":"2022-11-08T00:03:41Z","timestamp":1667865821000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Koopman-Based Spectral Clustering of Directed and Time-Evolving Graphs"],"prefix":"10.1007","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9672-3806","authenticated-orcid":false,"given":"Stefan","family":"Klus","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nata\u0161a","family":"Djurdjevac Conrad","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,11,7]]},"reference":[{"issue":"9","key":"9863_CR1","doi-asserted-by":"publisher","DOI":"10.1063\/1.4922968","volume":"25","author":"MR Allshouse","year":"2015","unstructured":"Allshouse, M.R., Peacock, T.: Lagrangian based methods for coherent structure detection. 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