ISIRA: Integrated shift–invert residual Arnoldi method for graph Laplacian matrices from big data

Abstract

The eigenvalue problem of a graph Laplacian matrix $L$ arising from a simple, connected and undirected graph has been given more attention due to its extensive applications, such as spectral clustering, community detection, complex network, image processing and so on. The associated graph Laplacian matrix is symmetric, positive semi-definite, and is usually large and sparse. Computing some smallest positive eigenvalues and corresponding eigenvectors is often of interest. However, the singularity of $L$ makes the classical eigensolvers inefficient since we need to factorize $L$ for the purpose of solving large and sparse linear systems exactly. The next difficulty is that it is usually time consuming or even unavailable to factorize a large and sparse matrix arising from real network problems from big data such as social media transactional databases, and sensor systems because there is in general not only local connections. In this paper, we propose an eignsolver based on the inexact residual Arnoldi method together with an implicit remedy of the singularity and an effective deflation for convergent eigenvalues. Numerical experiments reveal that the integrated eigensolver outperforms the classical Arnoldi/Lanczos method for computing some smallest positive eigeninformation provided the LU factorization is not available.