The complexity of financial markets arise from the strategic interactions among agents trading stocks, which manifest in the form of vibrant correlation patterns among stock prices. Over the past few decades, complex financial markets have often been represented as networks whose interacting pairs of nodes are stocks, connected by edges that signify the correlation strengths. However, we often have interactions that occur in groups of three or more nodes, and cannot be described simply by pairwise interactions. Only recently, researchers have started devoting attention to the higher-order architecture of complex financial systems, that can significantly enhance our ability to estimate systemic risk as well as measure the robustness of financial systems in terms of market efficiency, etc. Geometric measures, such as the Forman-Ricci curvature, Ollivier-Ricci curvature, etc. can be used to capture the network fragility and continuously monitor financial dynamics. Here, we review such computed Ricci-type curvatures designed to characterize the structure of the financial systems and use them as generic indicators of the market instability. For this purpose, we examine the daily returns from a set of stocks comprising the US S&P-500 and the Japanese Nikkei-225 over a 32-year period, and monitor the changes in the edge-centric geometric curvatures. We find that the different geometric measures capture well the system-level features of the market and hence we can distinguish between the normal or “business-as-usual” periods and all the major market crashes. This can be very useful in strategic designing of financial systems and regulating the markets in order to tackle financial instabilities.
