The Push Algorithm for Spectral Ranking

TL;DR

This paper generalizes the push algorithm for spectral ranking of nonnegative matrices, providing a simple, elementary approach with new convergence estimates and clarifying its relation to existing methods.

Contribution

It introduces a simple, general framework for the push algorithm applicable to any nonnegative matrix, with improved convergence analysis and connections to prior work.

Findings

The push algorithm can be applied to spectral ranking of any nonnegative matrix.

New methods for estimating convergence rates are proposed.

The framework simplifies understanding of existing spectral ranking algorithms.

Abstract

The push algorithm was proposed first by Jeh and Widom in the context of personalized PageRank computations (albeit the name "push algorithm" was actually used by Andersen, Chung and Lang in a subsequent paper). In this note we describe the algorithm at a level of generality that make the computation of the spectral ranking of any nonnegative matrix possible. Actually, the main contribution of this note is that the description is very simple (almost trivial), and it requires only a few elementary linear-algebra computations. Along the way, we give new precise ways of estimating the convergence of the algorithm, and describe some of the contribution of the existing literature, which again turn out to be immediate when recast in our framework.