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arXiv · 1304.2300

Incremental Computation of Pseudo-Inverse of Laplacian: Theory and Applications

Gyan Ranjan·Zhi-Li Zhang·Daniel Boley
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Abstract

A divide-and-conquer based approach for computing the Moore-Penrose pseudo-inverse of the combinatorial Laplacian matrix $(\bb L^+)$ 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 $\bb L^+$ and the effective resistance distance $(\Omega)$. Closed forms are provided for a novel {\em two-stage} process that helps compute the pseudo-inverse 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). An order of magnitude reduction in computational time is achieved for dynamic graphs; while in the general case, our approach performs better in practice than the standard methods, even though the worst case theoretical complexities may remain the same: an important contribution with consequences to the study of online social networks.

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Gyan Ranjan, Zhi-Li Zhang, Daniel Boley. 2013-04-08. Incremental Computation of Pseudo-Inverse of Laplacian: Theory and Applications. https://arxiv.org/abs/1304.2300

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