Ergodicity Coefficients for Higher-Order Stochastic Processes
| Autor: | Dario Fasino, Francesco Tudisco |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Markov chain
Stochastic process Probability (math.PR) Ergodicity MathematicsofComputing_NUMERICALANALYSIS Numerical Analysis (math.NA) Dynamical Systems (math.DS) 010103 numerical & computational mathematics 15B51 65F35 60J10 65C40 Random walk 01 natural sciences 010104 statistics & probability Nonlinear system Order (business) FOS: Mathematics Mathematics - Numerical Analysis Statistical physics Mathematics - Dynamical Systems 0101 mathematics Mathematics - Probability Mathematics |
| Zdroj: | SIAM Journal on Mathematics of Data Science. 2:740-769 |
| ISSN: | 2577-0187 |
| Popis: | 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. Comment: New title, shorter proof of main results and improved experimental section |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|