# Ergodicity coefficients for higher-order stochastic processes

## TL;DR

This paper introduces ergodicity coefficients for stochastic tensors to analyze the long-term behavior of higher-order stochastic processes, ensuring convergence and uniqueness of stationary distributions in complex models.

## Contribution

It extends ergodicity coefficients from matrices to tensors, providing explicit formulas for uniqueness, sensitivity bounds, and convergence guarantees for higher-order stochastic processes.

## Key findings

- Guarantees uniqueness of Perron Z-eigenvectors for stochastic tensors
- Provides bounds on eigenvector sensitivity to tensor changes
- Ensures convergence of higher-order stochastic processes

## Abstract

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.