Nonnegative and Combinatorial Matrix Theory

我的研究计划的首要目标是为非负矩阵理论和组合矩阵理论开发工具和见解，并将它们应用于促进对马尔可夫链的理解和揭示网络结构。马尔可夫链是一种在科学和工程中普遍存在的特定类型的概率模型，在计算药物设计、网页排序和无线网络设计等不同领域中找到了应用。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。我提出的这一领域的研究计划发展了支持Kemeny常数的理论基础和对Kemeny常数的深入理解，从而对具有理想效率特性的马尔可夫链的设计有了深入的了解。马尔可夫链的长期行为由所谓的平稳分布来描述，而接近该平稳分布的速度由称为特征值的量来控制。事实证明，平稳分布本身对逼近它的速度施加了限制。在我的研究计划中，这方面将对平稳分布和特征值之间的关系进行详细的调查。这将有助于揭示马尔可夫链的长期特性以及它们所模拟的现象。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。具体地说，双模网络产生于对社交网络的研究，可以被认为是对个人及其所属群体的记录。衡量个人和群体重要性的一种方法涉及用两个相关的网络取代原来的两种模式网络，一个只基于个人，另一个只基于群体。事实证明，这种替代有时可以忽略原有双模网络的结构，这就是所谓的数据丢失。我在这个领域的研究计划将通过识别确定数据丢失的双模网络来调查这种数据丢失，并以一种互补的方式，识别不会发生数据丢失的网络。这一系列调查的结果将会告诉我们，根据一对相关网络来分析单个双模网络技术的实用性。我提出的研究计划是以HQP培训为重点的。具体地说，这项研究计划将为两名博士生和一名理科硕士学生的培训奠定基础。学生，一名博士后研究员和两名博士后。

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.