Nonnegative and Combinatorial Matrix Theory

非负和组合矩阵理论

• 批准号: RGPIN-2019-05408
• 负责人: Kirkland, Stephen
• 金额: $ 1.53万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738255
• 关键词: Nonnegative; Combinatorial; Matrix; Theory

The overarching objective of my research program is to develop tools and insights for nonnegative matrix theory and combinatorial matrix theory, and to apply them in order to advance the understanding of Markov chains and to expose the structure of networks. A Markov chain is a certain type of probabilistic model that is ubiquitous in science and engineering, finding applications in such diverse areas as computational drug design, ranking of web pages, and wireless network design. Kemeny's constant is a key quantity associated with a Markov chain which provides an overall measure of the short-term efficiency of the Markov chain. My proposed program of research in this area develops the theory underpinning, and the understanding of, Kemeny's constant, thus yielding insight into the design of Markov chains with desirable efficiency properties. The long-term behaviour of a Markov chain is described by the so-called stationary distribution, and the rate of approach to that stationary distribution is governed by quantities called eigenvalues. It turns out that the stationary distribution itself places constraints on the rate of approach to it. This aspect of my program of research will undertake a detailed investigation of the relationship between the stationary distribution and the eigenvalues. This will shed useful light on the long-term properties of Markov chains and the phenomena that they model. I will also bring insights from nonnegative and combinatorial matrix theory to the study of certain networks. Specifically, two-mode networks arise in the study of social networks, and can be thought of as a record of individuals and the groups to which they belong. One approach to measuring the importance of the individuals and groups involves replacing the original two-mode network by two related networks, one based only on the individuals and the other based only on the groups. It has been shown that sometimes this replacement can overlook the structure of the orginal two-mode network, and this is known as data loss. My program of research in this domain will investigate this data loss by identifying two-mode networks where data loss is certain, and in a complementary manner, networks where data loss cannot take place. The results of this line of inquiry will inform the utility of the technique of analysing a single two-mode network in terms of a pair of related networks. My proposed program of research is conceived with HQP training as a key priority. Specifically, this research program will underpin the training of two Ph.D. students, one M.Sc. student, one Postdoctoral Fellow and two USRAs.

我的研究计划的总体目标是开发工具和见解的非负矩阵理论和组合矩阵理论，并应用它们，以促进马尔可夫链的理解，并揭示网络的结构。马尔可夫链是一种概率模型，在科学和工程中无处不在，在计算药物设计、网页排名和无线网络设计等不同领域都有应用。Kemeny常数是与马尔可夫链相关的一个关键量，它提供了一个整体的衡量短期效率的马尔可夫chains.My在这一领域的研究提出的计划发展的理论基础，和理解，Kemeny常数，从而产生洞察力的马尔可夫链的设计与理想的效率属性。 马尔可夫链的长期行为由所谓的平稳分布描述，而接近平稳分布的速率由称为特征值的量控制。结果表明，平稳分布本身对接近它的速度有限制。我的研究计划的这一方面将详细研究平稳分布和特征值之间的关系。这将有助于揭示马尔可夫链的长期性质及其所模拟的现象。 我也将从非负和组合矩阵理论的见解，以某些网络的研究。具体来说，双模式网络出现在社交网络的研究中，可以被认为是个人和他们所属的群体的记录。衡量个人和群体重要性的一种方法是用两个相关的网络取代原来的双模式网络，一个只基于个人，另一个只基于群体。已经表明，有时这种替换可以忽略原始双模网络的结构，这被称为数据丢失。我在这一领域的研究计划将通过识别数据丢失是肯定的双模网络来调查这种数据丢失，并以补充的方式，数据丢失不会发生的网络。这条调查线的结果将告知公用事业的技术分析一个单一的双模式网络的一对相关的网络。 我所提出的研究计划是以HQP培训为重点的。具体来说，这项研究计划将支持两个博士的培训。学生，一名硕士生，一名博士后研究员和两名USRA。