Spanning Tree Recovery via Random Walks in a Riemannian Manifold

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 rownormalisation, 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.