A note on certain ergodicity coefficients

TL;DR

This paper explores and extends ergodicity coefficients for matrices, providing new theoretical insights and a novel approach to eigenvector problems, including applications to PageRank centrality.

Contribution

It introduces a new generalization of the ergodicity coefficient $ au_{n-1}$ and demonstrates its use in reformulating eigenvector problems as M-matrix linear systems.

Findings

Limit of powers of the generalized ergodicity coefficient $ au_{n-1}$ established.

Reformulation of eigenvector problem as M-matrix system generalizes PageRank formulations.

Provides theoretical properties of ergodicity coefficients for complex matrices.

Abstract

We investigate two ergodicity coefficients $\phi_{\|\, \|}$ and $\tau_{n-1}$, originally introduced to bound the subdominant eigenvalues of nonnegative matrices. The former has been generalized to complex matrices in recent years and several properties for such generalized version have been shown so far. We provide a further result concerning the limit of its powers. Then we propose a generalization of the second coefficient $\tau_{n-1}$ and we show that, under mild conditions, it can be used to recast the eigenvector problem $Ax=x$ as a particular M-matrix linear system, whose coefficient matrix can be defined in terms of the entries of $A$. Such property turns out to generalize the two known equivalent formulations of the Pagerank centrality of a graph.