Koopman-based spectral clustering of directed and time-evolving graphs

TL;DR

This paper introduces a novel spectral clustering method for directed and time-evolving graphs by leveraging transfer operators and metastability concepts, enabling the identification of coherent sets in complex dynamical systems.

Contribution

It proposes a new clustering algorithm for directed graphs based on transfer operators and metastability, extending spectral clustering to more complex graph types.

Findings

Effective clustering of directed graphs demonstrated

Identification of coherent sets in fluid flow analysis

Extension of spectral clustering to time-evolving graphs

Abstract

While spectral clustering algorithms for undirected graphs are well established and have been successfully applied to unsupervised machine learning problems ranging from image segmentation and genome sequencing to signal processing and social network analysis, clustering directed graphs remains notoriously difficult. 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 coherent sets, which play an important role in the analysis of transport and mixing processes in fluid flows.