Markov Chains and Spectral Graph Theory: Interactions and Applications

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A Markov chain is a certain type of probabilistic model, and these models are ubiquitous in science and engineering, finding applications in such diverse areas as computational drug design, vehicle traffic networks, and ranking of web pages. The Kemeny constant is a key quantity associated with a Markov chain, and it provides, in some sense, a global measure of the overall short term efficiency of the Markov chain. Despite having been introduced more than 50 years ago, relatively little is known about the Kemeny constant, and the proposed research programme will undertake a thorough investigation of that quantity. By illuminating the mathematical properties of the Kemeny constant, I hope to provide insights into the design of Markov chains with desirable efficiency properties. A graph (or network) is a mathematical structure that records information about objects (called vertices) that are related in some way. Examples include Facebook users that are related by 'friendship', proteins in a cell that perform some function together, and terminals in an electrical network that are connected by resistors. There is a natural way to associate a Markov chain with any graph, and it turns out that certain parameters of the Markov chain can be used to measure how central each vertex is in the graph. This Markov chain centrality has been investigated empirically, but at present there is little in the way of rigorous theory on the topic. The proposed programme of research will develop the mathematical theory of this Markov chain centrality, thus enhancing the understanding of the graph-theoretic properties that are reflected by that centrality, and informing the use of that centrality in practical applications. The notion of a quantum walk arises in a so-called quantum wire -- a model for information transport inside a quantum computer. The fidelity of a quantum walk measures the probability that the information is transferred correctly along the quantum wire. This research programme will also investigate the sensitivity of the fidelity in terms of the time taken for the information to transfer, and the physical parameters associated with the quantum wire. The results will help to inform the design and implementation of quantum walks. This programme of research is expected to have impact in several different domains. Researchers working on Markov chains from the theoretical side, as well as scientists and engineers applying Markov chain techniques in practical settings (such as traffic modelling, migration models and network analysis) will benefit from the research on the Kemeny constant, as that quantity will become more deeply understood, and will hence be a more useful tool. The research on Markov chain centrality will, by developing the mathematical theory of that centrality, have an impact on network science and its numerous applications. Finally the research on quantum walks will have an impact in the area of quantum computing.