Perturbation and Inverse Problems of Stochastic Matrices

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-02-08 DOI:10.1137/22m1489162

Joost Berkhout, Bernd Heidergott, Paul Van Dooren

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引用次数: 0

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 553-584, March 2024.
Abstract. It is a classical task in perturbation analysis to find norm bounds on the effect of a perturbation [math] of a stochastic matrix [math] to its stationary distribution, i.e., to the unique normalized left Perron eigenvector. A common assumption is to consider [math] to be given and to find bounds on its impact, but in this paper, we rather focus on an inverse optimization problem called the target stationary distribution problem (TSDP). The starting point is a target stationary distribution, and we search for a perturbation [math] of the minimum norm such that [math] remains stochastic and has the desired target stationary distribution. It is shown that TSDP has relevant applications in the design of, for example, road networks, social networks, hyperlink networks, and queuing systems. The key to our approach is that we work with rank-1 perturbations. Building on those results for rank-1 perturbations, we provide heuristics for the TSDP that construct arbitrary rank perturbations as sums of appropriately constructed rank-1 perturbations.

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随机矩阵的扰动和逆问题

SIAM 矩阵分析与应用期刊》，第 45 卷，第 1 期，第 553-584 页，2024 年 3 月。摘要。随机矩阵[math]的扰动[math]对其静态分布的影响，即对唯一归一化左佩伦特征向量的影响，是扰动分析中的一项经典任务。一个常见的假设是将[math]视为给定的，并找出其影响的边界，但在本文中，我们更关注一个反向优化问题，即目标静态分布问题（TSDP）。起点是一个目标静态分布，我们寻找一个最小规范的扰动[math]，使[math]保持随机，并具有所需的目标静态分布。研究表明，TSDP 可应用于道路网络、社交网络、超链接网络和排队系统等的设计。我们方法的关键在于我们使用的是秩-1扰动。基于这些针对秩-1扰动的结果，我们为 TSDP 提供了启发式方法，将任意秩扰动构造为适当构造的秩-1 扰动之和。

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来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.