Perron-Frobenius Theory in Nearly Linear Time: Positive Eigenvectors, M-matrices, Graph Kernels, and Other Applications

TL;DR

This paper introduces nearly linear time algorithms for computing Perron vectors, solving linear systems in M-matrices, and related problems, significantly improving efficiency over previous methods.

Contribution

The paper presents novel iterative algorithms that reduce these problems to solving RCDD linear systems with nearly linear time complexity.

Findings

Algorithms run in nearly linear time relative to input size

Improved methods for computing graph kernels and Katz centrality

First efficient reduction of eigenvector problems to RCDD systems

Abstract

In this paper we provide nearly linear time algorithms for several problems closely associated with the classic Perron-Frobenius theorem, including computing Perron vectors, i.e. entrywise non-negative eigenvectors of non-negative matrices, and solving linear systems in asymmetric M-matrices, a generalization of Laplacian systems. The running times of our algorithms depend nearly linearly on the input size and polylogarithmically on the desired accuracy and problem condition number. Leveraging these results we also provide improved running times for a broader range of problems including computing random walk-based graph kernels, computing Katz centrality, and more. The running times of our algorithms improve upon previously known results which either depended polynomially on the condition number of the problem, required quadratic time, or only applied to special cases. We obtain these results by providing new iterative methods for reducing these problems to solving linear systems in Row-Column Diagonally Dominant (RCDD) matrices. Our methods are related to the classic shift-and-invert preconditioning technique for eigenvector computation and constitute the first alternative to the result in Cohen et al. (2016) for reducing stationary distribution computation and solving directed Laplacian systems to solving RCDD systems.