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We first explore methods for approximating the commute time and Katz scorebetween a pair of nodes. These methods are based on the approach of matrices,moments, and quadrature developed in the numerical linear algebra community.They rely on the Lanczos process and provide upper and lower bounds on anestimate of the pair-wise scores. We also explore methods to approximate thecommute times and Katz scores from a node to all other nodes in the graph.Here, our approach for the commute times is based on a variation of theconjugate gradient algorithm, and it provides an estimate of all the diagonalsof the inverse of a matrix. Our technique for the Katz scores is based onexploiting an empirical localization property of the Katz matrix. We adoptalgorithms used for personalized PageRank computing to these Katz scores andtheoretically show that this approach is convergent. We evaluate these methodson 17 real world graphs ranging in size from 1000 to 1,000,000 nodes. Ourresults show that our pair-wise commute time method and column-wise Katzalgorithm both have attractive theoretical properties and empiricalperformance.

Fast matrix computations for pair-wise and column-wise commute times and Katz scores