Algebraic Multilevel Methods for Markov Chains

TL;DR

This paper introduces an algebraic multilevel algorithm leveraging deflation and aggregation techniques to efficiently compute the second eigenvector of column-stochastic matrices, demonstrating good convergence on example problems.

Contribution

It presents a novel multilevel algorithm combining deflation and aggregation methods for eigenvector computation in Markov chains.

Findings

Good convergence properties demonstrated on example problems

Effective application of square and stretch approach in multilevel framework

Algorithm improves computational efficiency for second eigenvector calculation

Abstract

A new algebraic multilevel algorithm for computing the second eigenvector of a column-stochastic matrix is presented. The method is based on a deflation approach in a multilevel aggregation framework. In particular a square and stretch approach, first introduced by Treister and Yavneh, is applied. The method is shown to yield good convergence properties for typical example problems.