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Incremental Computation of Pseudo-Inverse of Laplacian

  • Gyan RanjanEmail author
  • Zhi-Li Zhang
  • Daniel Boley
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8881)

Abstract

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.

Keywords

Undirected Graph Online Social Network Effective Resistance High Degree Node Rich Club 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgment

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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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringUniversity of MinnesotaMinneapolisUSA

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