# A Hessenberg-type Algorithm for Computing PageRank Problems

## TL;DR

This paper introduces a Hessenberg-type algorithm tailored for efficiently solving large-scale PageRank problems, especially when the damping factor approaches one, outperforming some existing methods.

## Contribution

A novel Hessenberg-based method for PageRank computation that improves efficiency and competitiveness over Arnoldi-type algorithms in challenging scenarios.

## Key findings

- The new method is highly effective for large, difficult PageRank problems.
- It performs well when the damping factor is close to 1.
- Numerical experiments confirm its practical efficiency.

## Abstract

PageRank is a widespread model for analysing the relative relevance of nodes within large graphs arising in several applications. In the current paper, we present a cost-effective Hessenberg-type method built upon the Hessenberg process for the solution of difficult PageRank problems. The new method is very competitive with other popular algorithms in this field, such as Arnoldi-type methods, especially when the damping factor is close to $1$ and the dimension of the search subspace is large. The convergence and the complexity of the proposed algorithm are investigated. Numerical experiments are reported to show the efficiency of the new solver for practical PageRank computations.