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

TL;DR

This paper introduces fast approximation methods for computing pair-wise and column-wise commute times and Katz scores in large graphs, leveraging numerical linear algebra techniques like Lanczos and conjugate gradient, with proven convergence and tested on large real-world networks.

Contribution

It presents novel, efficient algorithms for approximating commute times and Katz scores, combining theoretical guarantees with empirical validation on large-scale graphs.

Findings

Methods are effective on graphs with up to 1 million nodes.

Algorithms provide bounds and estimates with theoretical convergence.

Empirical results demonstrate attractive performance and scalability.

Abstract

We first 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 pair-wise 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 adopt 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 pair-wise commute time method and column-wise Katz algorithm both have attractive theoretical properties and empirical performance.