PhDs in Probability

My research interests are in stochastic processes and applied probability, often towards situations in which finding a reasonable model is a difficult mathematical problem.

A common theme throughout much of my work is genealogies. Genealogies record relationships, and transfers, of information across time and space - such as a family tree, or the spread of a news story. I study the random geometric structures that record these transfers; when viewed over large space-time scales they are often both beautiful and mathematically interesting. Some examples:

Population genetics studies the dynamics of evolution, over long time-scales and for large numbers of organisms. Here, genetic information is passed from parent to child. My work in this area has focused on spatial effects related to natural selection, in particular on the relationship between spatial structure and the rate at which natural selection progresses.

Preferential attachment processes are randomly evolving graphs which seek to capture a 'rich get richer' mechanism. For example, already popular holiday destinations tend to be viewed as (disproportionately) more attractive, and therefore attract more new visitors; new users joining a social network are more likely to become friends with existing users who already have many friends. In this area, I am interested in the interaction between the underlying appeal and the random network effects that contribute towards information 'going viral'.

Recently, I have also been working on extensions of the Brownian web. This work involves trying to make sense of very dense systems (in space-time) of randomly moving particles, which have non-trivial interaction with one another – such as coalescing when they meet, or repelling if they get close. These systems tend to exhibit fractal-like structures, and can react very sensitively to small changes in the motions of a small fraction of the particles.

For some pictures, see http://nicfreeman.staff.shef.ac.uk/research.html

The preferential attachment model was popularised as a network model by the work of Barabási and Albert in 1999, with rigorous mathematical work on the model following shortly afterwards. I have been particularly interested in variations of this model where the preferential attachment mechanism interacts with intrinsic properties of the vertices; examples of such properties may include fitness, where fitter vertices are preferred for attachment, or location in some metric space, where for example new vertices may prefer to connect to vertices close to them in the spatial sense. Questions which may be asked about these models include how much the interaction between the two mechanisms disrupts the known properties of the preferential attachment model, such as its "power law" degree distribution. Some of these models display an interesting property known as "condensation" where, roughly speaking, a small (tending to zero) fraction of the vertices acquires a positive proportion of edges in the limit. 

Another topic is preferential attachment or similar graphs where the vertices acquire "types" as they join the graph, under the influence of their neighbours; these may be considered as modelling brand preferences, for example. Natural questions here include whether all types persist in the limit, or whether one may come to dominate. In models where vertices have spatial locations, a question is whether different regions of space might show different patterns of types, for example a different type dominating in the limit.

Much of this research uses the method known as stochastic approximation: the random behaviour of the system can be related to that of some dynamical system, and the long term behaviour of the dynamical system gives insight into that of the stochastic process. This is a method which is also useful in studying urns and other stochastic processes with reinforcement. Other methods I have used in papers on random networks have included embedded branching processes and Markov chains.

My research work is related in some fashion to the study of random trees - this includes both discrete trees, which are the usual connected graphs without cycles, and continuum trees, which are metric spaces which display some tree-like properties. Random continuum trees first appeared in the work of David Aldous in the early nineties, where he showed that the family trees of some population models converge to the now well-known brownian continuum random tree.

Two specific families of random trees which have interested me particularly are Markov branching trees and their continuum analogues, fragmentation trees. These describe genealogies where individuals have an abstract mass which is spread out to their children. Typical questions one would ask about these models are the convergence of Markov branching trees to fragmentation trees, applying those abstract models to actual examples, and studying interesting geometric properties of fragmentation trees. These are often answered by studying properties of the underlying Lévy processes.

I am also interested in more combinatorial properties of Galton-Watson trees, especially multi-type ones, and their local limits. Like the monotype case, critical multi-type Galton-Watson trees have as local limits when they get large an infinite tree based around a spine where the offspring distribution is size-biased. This is linked to limiting properties of some related complex random walks. Interesting questions around this would include finding general conditions under which those limits stay valid, or special cases when they are not (such as the so-called nongeneric subcritical case, where a phenomenon known as condensation creates a finite spine instead of an infinite one).

Another topic I have been recently interested in is the study of large directed graphs, and their possible scaling limits. This is still a very new area, but there are already results. Techniques are expected to be related to those used for non-directed graphs, such as the use of exploration processes and using branching processes.

My research interests are in probabilistic and extremal combinatorics, model theory, and related areas.

Random graphs is a fascinating and actively developing area of probabilistic combinatorics that is of interest both for mathematicians and computer scientists. Binomial random graphs are probably the most known and most actively and comprehensively studied models. In these graphs, every pair of vertices is adjacent with probability p independently of all the others. I am interested in extremal properties of random graphs, limit distributions of extremal statistics, reconstruction and isomorphism of random graphs, bootstrap percolation on edges of random graphs, logical limit laws, and many other related topics.