Spectral algorithms for temporal graph cuts

Abstract

The sparsest cut problem consists of identifying a small set of edges that breaks the graph into balanced sets of vertices. The normalized cut problem balances the total degree, instead of the size, of the resulting sets. Applications of graph cuts include community detection and computer vision. However, cut problems were originally proposed for static graphs, an assumption that does not hold in many modern applications where graphs are highly dynamic. In this paper, we introduce sparsest and normalized cuts in temporal graphs, which generalize their standard definitions by enforcing the smoothness of cuts over time. We propose novel formulations and algorithms for computing temporal cuts using spectral graph theory, divide-and-conquer and low-rank matrix approximation. Furthermore, we extend temporal cuts to dynamic graph signals, where vertices have attributes. Experiments show that our solutions are accurate and scalable, enabling the discovery of dynamic communities and the analysis of dynamic graph processes.