Stochastic Processes

Listed on this page are current research projects being offered for the Vacation Scholarship Program.

For more information on this research group see: Stochastic Processes

Strongly Correlated Percolation Models

Percolation is the study of the connectivity properties of disordered media, for instance how water `percolates' through soil. Percolation models with strong correlations can behave very different to the classical models with short-range correlations. This project will explore strongly correlated percolation models, how to efficiently simulate them, and how to estimate their critical exponents.

Contact: Stephen Muirhead smui@unimelb.edu.au

Hypergraph Animal Decomposition of Complex Networks

Hypergraph animals are small sub-networks which capture the local neighbourhoods of vertices in complex hypergraphs.The combine aspects of classical lattice animals and network motifs. We understand their combinatorial properties and their frequency spectra in random hypergraphs, and a next step in their analysis is to study their frequency spectra in real-world hypergraphs. Determining empirical distributions of hypergraph animals (so-called hypergraph zoos) and comparing them between different types of networks/hypergraphs will allow us to distill their functional relevance. This project will compare hypergraph zoos corresponding to metabolic and biochemical reaction networks in different species, in order to explore the functional role of hypergraph animals in real systems.

Contact: Michael Stumpf mstumpf@unimelb.edu.au

Inference and learning with spin glass models

Spin glasses arise in statistical physics as models of systems of interacting variables, the most famous of which is the Ising model, a classical model of ferromagnetism in which magnetic sites in a lattice are coupled via spin-spin interactions. Spin glass models have been shown to be a powerful tool for understanding complex systems, and have been well studied in the machine learning literature under the name of "energy-based models". This project will focus on the problem of inferring spin glass models of gene regulation from single-cell gene expression data. Genes can be modelled as sites in an

unstructured graph which have a spin of +1 (gene is on) or -1 (gene is off). The state of a system evolves according to a spin glass Hamiltonian. We want to solve the inverse problem of inferring the coupling matrix, which in the genetic setting contains information about which pairs of genes interact, and is unknown. There are a wealth of related problems which are of interest -- for example, one can relax the state of the system to be continuous rather than discrete, giving rise to Hopfield networks. Having a continuous model allows for more tractable inference and opens the way to investigate a wider range of loss functions; and we want to use these models to understand the gene regulatory programs underpinning cellular behaviour.

Contact: Michael Stumpf mstumpf@unimelb.edu.au