Ricci curvature: An economic indicator for market fragility and systemic risk.

Quantifying the systemic risk and fragility of financial systems is of vital importance in analyzing market efficiency, deciding on portfolio allocation, and containing financial contagions. At a high level, financial systems may be represented as weighted graphs that characterize the complex web of interacting agents and information flow (for example, debt, stock returns, and shareholder ownership). Such a representation often turns out to provide keen insights. We show that fragility is a system-level characteristic of "business-as-usual" market behavior and that financial crashes are invariably preceded by system-level changes in robustness. This was done by leveraging previous work, which suggests that Ricci curvature, a key geometric feature of a given network, is negatively correlated to increases in network fragility. To illustrate this insight, we examine daily returns from a set of stocks comprising the Standard and Poor's 500 (S&P 500) over a 15-year span to highlight the fact that corresponding changes in Ricci curvature constitute a financial "crash hallmark." This work lays the foundation of understanding how to design (banking) systems and policy regulations in a manner that can combat financial instabilities exposed during the 2007-2008 crisis.