Isospectral Reductions of Non-negative Matrices

TL;DR

This paper introduces an algorithmic approach using isospectral reductions to approximate the stationary measure of stochastic matrices, especially effective when multiple eigenvalues are near 1, outperforming traditional iterative methods.

Contribution

It proposes a novel scheme leveraging isospectral reductions for better approximation of stationary measures in complex stochastic matrices.

Findings

Scheme outperforms iterative methods when multiple eigenvalues are near 1

Isospectral reduction can improve spectral gap in certain cases

Numerical experiments support the effectiveness of the proposed approach

Abstract

Isospectral reduction is an important tool for network/matrix analysis as it reduces the dimension of a matrix/network while preserving its eigenvalues and eigenvectors. The main contribution of this manuscript is a proposed algorithmic scheme to approximate the stationary measure of a stochastic matrix based on isospectral reductions. We run numerical experiments that indicate this scheme is advantageous when there is more than one eigenvalue near 1, precisely the case where iterative methods perform poorly. We give a partial explanation why this scheme should work well, showing that in some situations isospectral reduction improves the spectral gap.