高阶随机过程的遍历系数
Ergodicity coefficients for higher-order stochastic processes
摘要：
近年来，高阶随机过程（如非线性马尔可夫链或顶点增强随机游动）的使用显著增长，因为它们在许多应用环境中更好地建模高维数据和非线性动力学。在许多实际感兴趣的情况下，这些过程与随机张量和它们的平稳分布是一个张量$Z$ -特征向量。然而，基本的问题，如收敛的过程对一个限制分布和唯一性这样的限制仍然没有很好地理解，是丰富的最近的文献的主题。随机阵的遍历系数为分析标准一阶马尔可夫过程的长期行为提供了一个有价值的和广泛使用的工具。在这项工作中，我们扩展了一类重要的遍历系数的设置随机张量。我们表明，建议的高阶遍历系数提供了新的明确的公式，（a）保证Perron $Z$ -随机张量的特征向量的唯一性，（b）提供的界限，这样的特征向量的灵敏度相对于张量的变化和（c）确保不同类型的高阶随机过程的收敛由立方随机张量。此外，我们说明了几个示例应用程序设置，包括基于三角形的随机游动和懒惰的高阶随机游动的收敛PageRank向量的分析，建议遍历系数的优点。
英文摘要：
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.
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