This project is focused on a novel application of recently developed tools in computational topology to certain problems of risk in financial networks. Complex problems often benefit from being viewed in a new light; the investigators use persistent homology to study behavior of financial trading networks. Persistent homology focuses on how a family of simplicial complexes D(t) vary with a real parameter t. Typically one starts with a discrete data set (point cloud data) corresponding to t=0, and then as t varies, points are replaced by balls of radius t, centered at the original point. Thus varying t yields a filtered family of simplicial complexes, the Rips-Vietoris complex of the associated topological space. In the context of trading networks, the parameter t represents margin; a low value for t corresponds to allowing highly leveraged trading. The aim of the project is to understand how changes in the topology of the D(t) associated to a trading network correspond in some way to systemic collapse. Put simply, the investigators explore the circumstances that cause a local event to cause global contagion.
The problem of understanding interconnectedness is one of today's premier challenges. Connectivity is ubiquitous; interconnections often allow a system to be tapped to its full potential and enhance its robustness. The flip side of connectivity is that it gives rise to pathways for unexpected emergent behavior ("black swan" events) and even systemic collapse. The world is awash in data: how can we extract meaning (and understand connectivity) from it? One answer is via the mathematical discipline of algebraic topology, which is a tool to distill the study of complex objects or spaces into a simple form. For example, how can we distinguish between an orange and a donut? The obvious answer is that the donut has a "hole"; algebraic topology makes the natural intuition (which is not so natural in higher dimensions!) rigorous. In particular, algebraic topology is built to probe the relationship between local and global structures; this makes it an ideal tool for studying financial trading networks. The proliferation of problems where there is a confluence of local/global transitions and a large experimental data set has given rise to the field of applied algebraic topology, yielding a systematic way to unfold connections. The investigators interpret various notions of collapse of financial networks via this unfolding perspective. The work focuses on clearing networks, which encode trades and liabilities, and bring the tools developed in applied topology to bear on unfolding the interconnections in such networks. In particular, the investigators study how local events propagate: under what circumstances does contagion and systemic collapse result from a local event?
