Nonnegative and Combinatorial Matrix Theory

点击翻译按钮获取中文摘要

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.