Collaborative Research: Random Dynamics on Networks

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Transport and distribution networks come in a number of forms, from animal cardiovascular and respiratory systems to communication and industrial infrastructures. Practical issues abound: prediction of local spikes, estimation of perfusion, and impact of structural changes such as vessel occlusion. The complexity of such phenomena can be illustrated by the well-known Braess' paradox: adding links to a transportation network might not improve the operation of the system! In spite of recent successes, our understanding of network flows is usually limited to small deterministic problems, while most applications correspond to large uncertain ones. The goal of this project is to enable improved predictions in biological and technological transport and diffusion networks. For instance, can one predict how cerebral blood flow will be affected if one of the carotids becomes narrow or blocked? Will the vasculature allow for re-routing? If so, with what probability and how fast? The main challenge in this research project is the presence of uncertainties. For many applications, only partial information about the systems is available. For instance the size or even the presence of a specific vessel might be uncertain or the status of a router unknown. The analysis of such problems requires the creation of novel mathematical tools and numerical methods to describe how uncertainties propagate through vast and complex networks. The computational tools to be constructed will provide information, usually probabilistic in nature, regarding phenomena that are difficult, expensive, or impossible to measure.