Eigenvectors for clustering: Unipartite, bipartite, and directed graph cases

TL;DR

This paper provides a unified spectral clustering framework for unipartite, bipartite, and directed graphs, including theoretical insights and generalizations of key theorems to support various clustering approaches.

Contribution

It introduces a unified treatment for different graph types by transforming bipartite and directed graphs into unipartite graphs and generalizes the Ky Fan theorem for rectangular matrices.

Findings

Eigenvector-based solutions for bipartite graph co-clustering

Theoretical equivalence of row and column clustering in bipartite graphs

Generalization of Ky Fan theorem to rectangular matrices

Abstract

This paper presents a concise tutorial on spectral clustering for broad spectrum graphs which include unipartite (undirected) graph, bipartite graph, and directed graph. We show how to transform bipartite graph and directed graph into corresponding unipartite graph, therefore allowing a unified treatment to all cases. In bipartite graph, we show that the relaxed solution to the $K$-way co-clustering can be found by computing the left and right eigenvectors of the data matrix. This gives a theoretical basis for $K$-way spectral co-clustering algorithms proposed in the literatures. We also show that solving row and column co-clustering is equivalent to solving row and column clustering separately, thus giving a theoretical support for the claim: ``column clustering implies row clustering and vice versa''. And in the last part, we generalize the Ky Fan theorem---which is the central theorem for explaining spectral clustering---to rectangular complex matrix motivated by the results from bipartite graph analysis.