A note on certain ergodicity coefficients
Theoretical contribution for researchers working on matrix analysis and graph centrality, but incremental in nature.
The paper studies two ergodicity coefficients for bounding subdominant eigenvalues of nonnegative matrices, provides a limit result for a generalized version, and proposes a generalization of the second coefficient that recasts the eigenvector problem as an M-matrix linear system, generalizing Pagerank formulations.
We investigate two ergodicity coefficients $ϕ_{\|\, \|}$ and $τ_{n-1}$, 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 $τ_{n-1}$ 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.