Nonnegative and Combinatorial Matrix Theory

我的研究计划的总体目标是开发非负矩阵理论和组合矩阵理论的工具和见解，并应用它们来推进对马尔可夫链的理解，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，它提供了马尔可夫链短期效率的总体度量。我在这一领域提出的研究计划发展了理论基础，并理解了凯梅尼常数，从而深入了解了具有理想效率特性的马尔可夫链的设计。马尔可夫链的长期行为用所谓的平稳分布来描述，而接近平稳分布的速率由称为特征值的量来控制。事实证明，平稳分布本身限制了接近它的速度。在我的研究计划中，这方面将对平稳分布与特征值之间的关系进行详细的研究。这将有助于阐明马尔可夫链的长期性质及其所模拟的现象。我还将把非负和组合矩阵理论的见解带到某些网络的研究中。具体来说，双模式网络出现在社会网络的研究中，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。研究表明，有时这种替换会忽略原有双模网络的结构，这就是数据丢失。我在这一领域的研究计划将通过识别数据丢失是确定的双模式网络和以互补的方式，数据丢失不会发生的网络来调查这种数据丢失。这条调查路线的结果将告知在一对相关网络方面分析单个双模网络的技术的效用。我提出的研究计划以HQP培训为重点。具体而言，该研究计划将支持培养两名博士生，一名硕士研究生，一名博士后和两名usra。

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.