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; luckily, most real networks are sparse, meaning that the number
of edges (binary connections among nodes) are few with respect to the maximum number
of possible edges. In this paper we experimentally compare three important 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.
