Clustering Time-Evolving Networks Using the Dynamic Graph Laplacian

Time-evolving graphs arise frequently when modeling complex dynamical systems such as social networks, traffic flow, and biological processes. Developing techniques to identify and analyze communities in these time-varying graph structures is an important challenge. In this work, we generalize existing spectral clustering algorithms from static to dynamic graphs using canonical correlation analysis (CCA) to capture the temporal evolution of clusters. Based on this extended canonical correlation framework, we define the dynamic graph Laplacian and investigate its spectral properties. We connect these concepts to dynamical systems theory via transfer operators, and illustrate the advantages of our method on benchmark graphs by comparison with existing methods. We show that the dynamic graph Laplacian allows for a clear interpretation of cluster structure evolution over time for directed and undirected graphs.

Transfer operators on graphs: Spectral clustering and beyond

Graphs and networks play an important role in modeling and analyzing complex interconnected 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 connected vertices and to derive coarse-grained models. We define transfer operators such as the Koopman operator and the Perron-Frobenius operator on graphs, study their spectral properties, introduce Galerkin projections of these operators, and illustrate how reduced representations can be estimated from data. In particular, we show that spectral clustering of undirected graphs can be interpreted in terms of eigenfunctions of the Koopman operator and propose novel clustering algorithms for directed graphs based on generalized transfer operators. We demonstrate the efficacy of the resulting algorithms on several benchmark problems and provide different interpretations of clusters.