I'll talk about two pieces of work. The first concerns non-reversible Markov chains and their application to Monte Carlo sampling. I will spend some time discussing why non-reversible Markov chains are of interest to the sampling community, explore some simple schemes and then discuss an extension to Peskun's theorem on ordering the asymptotic variances of Markov chain ergodic averages to this case (based on joint work with Christophe Andrieu). If time permits, I'll also discuss more recent work in which we consider a new class of (reversible) Markov processes termed 'locally-balanced', and their use in Monte Carlo sampling algorithms (based on joint work with Giacomo Zanella, Jure Vogrinc and Max Hird).

In this talk we propose an extension of a classical reaction-diffusion model for cell repolarisation by adding dynamic noise. This stochastic partial differential equation (SPDE) generates data, which differ qualitatively from the deterministic PDE model corrupted by measurement errors. The dynamic noise has interesting effects on the repolarisation behaviour such as larger noise levels speeding up the time to repolarisation instead of destroying the pattern formation. Apart from a qualitative description of the SPDE model, results on parameter estimation are presented to calibrate the model to data.

Community structure is a commonly observed feature of real networks. The term refers to the presence in a network of groups of nodes (communities) that feature high internal connectivity but are poorly connected between each other. Whereas the issue of community detection has been addressed in several works, the problem of validating a partition of nodes as a good community structure for a network has received considerably less attention and remains an open issue. In this work we propose a new method for the validation of network partitions as community structures. We observe that the validation of network partitions should not just consider the distribution of nodes among clusters, but it should primarily focus on the distribution of links between the groups. When assessing the goodness of different partitions of a network, intuitively we would like to rate better partitions where a high proportion of links is allocated within communities and a low one between communities. Our method is based on a significance testing procedure for the number of links that are observed between and within the communities; the results are then combined into a community structure validation (CSV) index that provides an overall assessment of whether a partition of nodes induces a community structure in the network. Our approach provides also a practical way of comparing networks, by assessing similarity and differences in their community structures. This is joint work with Mirko Signorelli (Leiden)

There are numerous recent developments and enticing open problems concerning the question of which oriented trees (or other tree-like structures) can be found in dense directed graphs or tournaments. I will give an overview of some of these questions, before describing recent work with Tassio Naia, in which we show that every oriented tree whose maximum degree is not too large is contained in every directed graph of large minimum semidegree on the same number of vertices (as well as a similar result for certain other `tree-like’ structures). The key step in the proof of this result was the further development of randomised allocation and embedding algorithms from versions we previously used to show that uniformly-random oriented trees are unavoidable with high probability (meaning that they are contained in every tournament on the same number of vertices).