We devise methods for finding approximations of the generalized inverse of
the graph Laplacian matrix, which arises in many graph-theoretic applications.
Finding this matrix in its entirety involves solving a matrix inversion problem,
which is resource demanding in terms of consumed time and memory and hence
impractical whenever the graph is relatively large. Our approximations use only
few eigenpairs of the Laplacian matrix and are parametric with respect to this
number, so that the user can compromise between effectiveness and efficiency
of the approximated solution. We apply the devised approximations to the the
problem of computing current-flow betweenness centrality on a graph. However,
given the generality of the Laplacian matrix, many other applications can be
sought. We experimentally demonstrate that the approximations are effective
already with a constant number of eigenpairs. These few eigenpairs can be
stored with a linear amount of memory in the number of nodes of the graph
and, in the realistic case of sparse networks, they can be efficiently computed
using one of the many methods for retrieving few eigenpairs of sparse matrices
that abound in the literature.
