Eigenvector-based identification of bipartite subgraphs

TL;DR

This paper explores eigenvector-based methods for identifying large bipartite subgraphs in various graph models, comparing their effectiveness to existing algorithms and introducing new bipartivity indices.

Contribution

It introduces eigenvector-based techniques for bipartite subgraph detection and proposes new bipartivity indices, demonstrating their effectiveness over traditional methods.

Findings

Eigenvector methods perform comparably to local switching algorithms.

Normalized Laplacian and adjacency eigenvector methods yield slightly better results.

Iterative removal based on bipartivity indices improves bipartite subgraph detection.

Abstract

We report our experiments in identifying large bipartite subgraphs of simple connected graphs which are based on the sign pattern of eigenvectors belonging to the extremal eigenvalues of different graph matrices: adjacency, signless Laplacian, Laplacian, and normalized Laplacian matrix. We compare the performance of these methods to a local switching algorithm based on the Erdos bound that each graph contains a bipartite subgraph with at least half of its edges. Experiments with one scale-free and three random graph models, which cover a wide range of real-world networks, show that the methods based on the eigenvectors of the normalized Laplacian and the adjacency matrix yield slightly better, but comparable results to the local switching algorithm. We also formulate two edge bipartivity indices based on the former eigenvectors, and observe that the method of iterative removal of edges with maximum bipartivity index until one obtains a bipartite subgraph, yields comparable results to the local switching algorithm, and significantly better results than an analogous method that employs the edge bipartivity index of Estrada and Gomez-Gardenes.