Incremental Computation of Pseudo-Inverse of Laplacian

A divide-and-conquer based approach for computing the Moore-Penrose pseudo-inverse of the combinatorial Laplacian matrix (L+) of a simple, undirected graph is proposed. The nature of the underlying sub-problems is studied in detail by means of an elegant interplay between L+ and the effective resistance distance (Ω). Closed forms are provided for a novel two-stage process that helps compute the pseudoinverse incrementally. Analogous scalar forms are obtained for the converse case, that of structural regress, which entails the breaking up of a graph into disjoint components through successive edge deletions. The scalar forms in both cases, show absolute element-wise independence at all stages, thus suggesting potential parallelizability. Analytical and experimental results are presented for dynamic (time-evolving) graphs as well as large graphs in general (representing real-world networks).