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 \((\mathbf L^+)\) 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 \(\mathbf L^+\) and the effective resistance distance \((\varOmega )\). Closed forms are provided for a novel 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.
KeywordsUndirected Graph Online Social Network Effective Resistance High Degree Node Rich Club
This research was supported in part by DTRA grant HDTRA1-09-1-0050, DoD ARO MURI Award W911NF-12-1-0385, and NSF grants IIS-0916750, CNS-10171647, CNS-1017092, CNS-1117536, IIS-1319749 and CRI-1305237.
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