MCMC (and other advanced sampling methods) for investigating non-linear emergent network properties
While naive sampling approaches can reveal the dominant behavior of a complex system, they do not allow us to efficiently understand rare events. This project will develop and test advanced optimization and sampling methods that purposefully drive the system towards rare failure modes. Such methods will allow us to efficiently investigate and model complicated emergent phenomena such as cascading failures in power grids and other networks.
A secondary challenge for analytically inclined MA students is to develop new MCMC methods for this context.
Clustering trajectories in dynamical systems, unsupervised machine learning (MA)
In order to understand the behavior of high dimensional systems under parameter change, their bifurcation structure, we need novel methods. Monte-Carlo basin bifurcation studies the bifurcation structure from the point of view of changes in the basin volume. At the core of this method is the clustering of trajectories, a challenging problem in unsupervised ML. There is considerable scope for more analytic and semi-analytic work in this context.
Quasi-local loop base control
Loops or cycles play a fundamental role in the behavior of networked systems. A basis of the possible flow states in a networked system is given by a basis of cycles. This project will investigate the potential of controlling a networks dynamical behaviors through the cycle flows.
Another component here is to look at complex formulations of flow networks and study the solution spaces using algebraic methods.
Low dimensional approximations to oscillator networks
For many phenomena observed in high dimensional systems we can find simplified low dimensional models. We will explore the behavior of some of these models analytically and semi-analytically and develop new ones.
