  Measuring systemic risk or fragility of financial systems is a ubiquitous
task of fundamental importance in analyzing market efficiency, portfolio
allocation, and containment of financial contagions. Recent attempts have shown
that representing such systems as a weighted graph characterizing the complex
web of interacting agents over some information flow (e.g., debt, stock
returns, shareholder ownership) may provide certain keen insights. Here, we
show that fragility, or the ability of system to be prone to failures in the
face of random perturbations, is negatively correlated with geometric notion of
Ricci curvature. The key ingredient relating fragility and curvature is
entropy. As a proof of concept, we examine returns from a set of stocks
comprising the S\&P 500 over a 15 year span to show that financial crashes are
more robust compared to normal "business as usual" fragile market behavior -
i.e., Ricci curvature is a "crash hallmark." Perhaps more importantly, this
work lays the foundation of understanding of how to design systems and policy
regulations in a manner that can combat financial instabilities exposed during
the 2007-2008 crisis.
