In the last years, we have witnessed a surge in general academic interest in the field of Machine Learning, not least from researchers in mathematics and statistics. Despite this prolific expansion in the research literature, there remain plenty of challenges, many of which are closely related to industrial interests. One vibrant area within Machine Learning is the development of algorithms for understanding abstract properties of graphs: collections of vertices connected by edges that describe some measure of association between the nodes. ‘Graph clustering’ is the general name for algorithms in which similar vertices are grouped together, while nodes representing dissimilar objects are separated into different groups. Graph clustering has countless applications, ranging from computational biology to blockchain.
Despite their ubiquity and utility, the majority of clustering approaches are developed for undirected networks where all the edge weights are positive (positively weighted networks). The best ways to extend these tools to more general settings (i.e. signed or directed graphs) are open research questions and these will form the basis for the research programme.
The focus of this PhD topic is the development of novel algorithms for graph clustering by drawing intuition from physics. One well known example is the celebrated MBO scheme [1] which relies on a discrete solution of the Ginzburg-Landau functional originally devised in solid state physics to describe the optimal density profile separating different phases of matter
Extensions of the MBO scheme, or blends of this approach with more traditional spectral clustering techniques, offer new directions to address the generalisations described above.
The ideal scientific background for a student would be theoretical physics or mathematical sciences, but applications are welcome from candidates from adjacent disciplines with a strong personal interest in the topic. Programming experience would be highly advantageous.
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