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Spanning Tree Recovery via Random Walks in a Riemannian Manifold

  • Antonio Robles-Kelly
  • Edwin R. Hancock
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3287)

Abstract

In this paper, we describe the use of Riemannian geometry and graph-spectral methods for purposes of minimum spanning tree recovery. We commence by showing how the sectional curvature can be used to model the edge-weights of the graph as a dynamic system in a manifold governed by a Jacobi field. With this characterisation of the edge-weights at hand, we proceed to recover an approximation for the minimum spanning tree. To do this, we present a random walk approach which makes use of a probability matrix equivalent, by row-normalisation, to the matrix of edge-weights. We show the solution to be equivalent, up to scaling, to the leading eigenvector of the edge-weight matrix. We approximate the minimum spanning tree making use of a brushfire search method based upon the rank-order of the eigenvector coefficients and the set of first-order neighbourhoods for the nodes in the graph. We illustrate the utility of the method for purposes of network optimisation.

Keywords

Random Walk Riemannian Manifold Span Tree Sectional Curvature Curvature Tensor 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Antonio Robles-Kelly
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkYorkUK

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