  The use of higher-order stochastic processes such as nonlinear Markov chains
or vertex-reinforced random walks is significantly growing in recent years as
they are much better at modeling high dimensional data and nonlinear dynamics
in numerous application settings. In many cases of practical interest, these
processes are identified with a stochastic tensor and their stationary
distribution is a tensor $Z$-eigenvector. However, fundamental questions such
as the convergence of the process towards a limiting distribution and the
uniqueness of such a limit are still not well understood and are the subject of
rich recent literature. Ergodicity coefficients for stochastic matrices provide
a valuable and widely used tool to analyze the long-term behavior of standard,
first-order, Markov processes. In this work, we extend an important class of
ergodicity coefficients to the setting of stochastic tensors. We show that the
proposed higher-order ergodicity coefficients provide new explicit formulas
that (a) guarantee the uniqueness of Perron $Z$-eigenvectors of stochastic
tensors, (b) provide bounds on the sensitivity of such eigenvectors with
respect to changes in the tensor and (c) ensure the convergence of different
types of higher-order stochastic processes governed by cubical stochastic
tensors. Moreover, we illustrate the advantages of the proposed ergodicity
coefficients on several example application settings, including the analysis of
PageRank vectors for triangle-based random walks and the convergence of lazy
higher-order random walks.
