I am a Lecturer in the School of Mathematics and Statistics at the University of Sheffield. I obtained my DPhil from the Mathematical Institute in the University of Oxford, associated with St. Anne's College and working with Neil O'Connell and Ben Hambly; my thesis (PDF format) was titled Renormalisation of random hierarchical systems.Publications and preprints
My current main research interest is in random graphs. The theory of random graphs dates back to Erdős and Rényi in 1959, but there has been an explosion of interest since the late 1990s, with applications arising in fields such as computer science and biology. One family of models I have been particularly interested in is the so-called preferential attachment models based on the original model of Barabási and Albert, in which new vertices join the graph and are connected to existing vertices; existing vertices are more likely to pick up new neighbours if they already have high degree. These models tend to show an asymptotically power law (or "scale free") degree sequence. I have been particularly interested in variations on this model with a geometric element, with three papers:
I have also looked at other models of random graphs which may show some similar features, such as duplication graphs, preferential duplication and random reproducing graphs.
Some of the pictures below show some of the models I have worked on.
From 2006 to 2010 I was involved with the Amorph project. I am available to supervise PhD topics in the general area of random graphs, in particular where related to the models discussed above. I may also consider supervising topics related to random graphs with a more applied flavour, possibly in partnership with other researchers interested in this area.
I have also been interested in fractals, in particular random fractals and the spectral properties of fractal graphs. Random fractal graphs were a large part of my PhD thesis (and see also the paper on the "series-parallel network"). The spectral properties of Laplacians (defined as matrices closely related to the generator matrices of continuous time random walks) of some fractal graphs (such as those related to the Sierpiński gasket) show a property called "spectral decimation", by which the eigenvalues of one stage in the construction may be found by solving equations of the form f(x)=λ, where λ is an eigenvalue at the previous stage. Some of my work has looked at variations and extensions of this property; click on the pictures below for some examples of my work in this area.