Computing the smallest eigenpairs of the graph Laplacian

TL;DR

This paper compares three advanced algorithms for efficiently computing the smallest eigenpairs of large, sparse graph Laplacian matrices, demonstrating that the Jacobi-Davidson method outperforms others in real-world network tests.

Contribution

It provides a systematic experimental comparison of state-of-the-art eigenpair algorithms on diverse real-world networks, highlighting the superior performance of the Jacobi-Davidson method.

Findings

Jacobi-Davidson method requires fewer matrix-vector products.

Jacobi-Davidson achieves lower CPU times.

The study covers biological, technological, information, and social networks.

Abstract

The graph Laplacian, a typical representation of a network, is an important matrix that can tell us much about the network structure. In particular its eigenpairs (eigenvalues and eigenvectors) incubate precious topological information about the network at hand, including connectivity, partitioning, node distance and centrality. Real networks might be very large in number of nodes (actors); luckily, most real networks are sparse, meaning that the number of edges (binary connections among actors) are few with respect to the maximum number of possible edges. In this paper we experimentally compare three state-of-the-art algorithms for computation of a few among the smallest eigenpairs of large and sparse matrices: the Implicitly Restarted Lanczos Method, which is the current implementation in the most popular scientific computing environments (Matlab \R), the Jacobi-Davidson method, and the Deflation Accelerated Conjugate Gradient method. We implemented the algorithms in a uniform programming setting and tested them over diverse real-world networks including biological, technological, information, and social networks. It turns out that the Jacobi-Davidson method displays the best performance in terms of number of matrix-vector products and CPU time.