Fast Matrix Computations for Pairwise and Columnwise Commute Times and Katz Scores

We explore methods for approximating the commute time and Katz score between 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 an estimate of the pairwise scores. We also explore methods to approximate the commute 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 the conjugate gradient algorithm, and it provides an estimate of all the diagonals of the inverse of a matrix. Our technique for the Katz scores is based on exploiting an empirical localization property of the Katz matrix. We adapt algorithms used for personalized PageRank computing to these Katz scores and theoretically show that this approach is convergent. We evaluate these methods on 17 real-world graphs ranging in size from 1000 to 1,000,000 nodes. Our results show that our pairwise commute-time method and columnwise Katz algorithm both have attractive theoretical properties and empirical performance.