Research ArticleMATHEMATICS

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

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Science Advances  27 May 2016:
Vol. 2, no. 5, e1501495
DOI: 10.1126/sciadv.1501495
  • Fig. 1 Systemic risk as a complexity problem: How to account for multiple indirect risk exposures in a financial ecosystem.

    Understanding indirect counterparty risk has gained increasing importance with the recent global financial crisis combined with the continuous rise of complex financial instruments. This paper proposes a new metric for characterizing instability with respect to agent-to-agent information in the context of a global network, and we illustrate the method by characterizing market fragility from a feedback perspective resulting from well-known “herding” phenomena during periods of financial crisis.

  • Fig. 2 An intuitive understanding of curvature.

    (A to C) We compare geodesic triangles, which are triangles in which each side is connected by the shortest (geodesic) curve for three different surfaces: (A) sphere, (B) planar surface, and (C) hyperbolic paraboloid. From the bottom row, we can see that such triangles on a sphere are puffier than their Euclidean planar surface counterparts (due to great circles being geodesics as opposed to straight lines). As we move toward negatively curved space, such triangles become skinnier. This behavior is noted by measuring the length of the (geodesic) curve connecting the midpoint xm to x2.

  • Fig. 3 Average Ricci curvature over a 15-year span of the S&P 500.

    (A) Choosing a window of T = 22 days, we see that curvature captures several financial crashes and show that, on average, market behavior is fragile. (B) We extended our analysis with a larger window of T = 132 days, and as one can see, there is an increase in Ricci curvature compared to normal fragile market behavior during periods of known financial crisis.

  • Fig. 4 Comparison of network robustness measures over a smaller time window.

    (A to D) We compare Ollivier-Ricci curvature (A) to network entropy (B), shortest average path (C), and graph diameter (D) for a shorter time scale of 22 days. As predicted, there is a notable resemblance between network entropy and network curvature. Further analysis shows that decreases in graph diameter and shortest path length result in increases in graph curvature.

  • Fig. 5 Comparison of network robustness measures over a larger time window.

    (A to D) We compare Ollivier-Ricci curvature (A) to network entropy (B), shortest average path (C), and graph diameter (D) for a longer time scale of 132 days. As predicted, there is a notable resemblance between network entropy and network curvature. Further analysis shows that decreases in graph diameter and shortest path length result in increases in graph curvature.

  • Fig. 6 Minimum risk Markowitz portfolio and Ricci curvature.

    We compute Embedded Image using weights from the minimum risk portfolio along with minimum risk. (A and B) Regions of interest are highlighted for Ricci curvature Embedded Image (A) and minimum risk (B) that can be obtained along the efficient frontier over a given window T = 122 days with a threshold of ξ = 0.85.

  • Table 1 Comparison of network robustness measures.

    Comparison of network robustness measures.. We provide the average Ricci curvature, average global network entropy, average shortest path, and average graph diameter computed over a period of 1 year beginning with 1 January of each year with a window T = 22 days and a threshold of ξ = 0.85. As seen in the graph, a correlation between curvature and well-known measures of fragility.

    MeasureJan
    1998
    Jan
    1999
    Jan
    2000
    Jan
    2001
    Jan
    2002
    Jan
    2003
    Jan
    2004
    Jan
    2005
    Jan
    2006
    Jan
    2007
    Jan
    2008
    Jan
    2009
    Jan
    2010
    Jan
    2011
    Jan
    2012
    Curvature−0.297−0.304−0.285−0.218−0.0166−0.256−0.253−0.214−0.245−0.218−0.068−0.141−0.1010.023−0.0255
    Entropy0.9410.8620.9091.1231.5051.1931.0221.1281.0661.3162.1591.7452.1052.6881.282
    Shortest path12.41213.81114.91012.9519.36210.19912.29111.84512.4139.6367.1868.0607.4696.2089.545
    Diameter31.36134.72638.21033.18625.50027.39731.25029.92931.49425.06421.04422.48020.96417.77025.144

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/5/e1501495/DC1

    fig. S1. Example of setting up the problem.

    fig. S2. Illustration of transporting regions on nonnegatively curved space.

    fig. S3. Illustration of how shortest path relates to curvature or entropy.

    fig. S4. Intuitive understanding of Ricci curvature (a different perspective).

    fig. S5. Illustrating geodesic deviations on nonnegatively curved space.

  • Supplementary Materials

    This PDF file includes:

    • fig. S1. Example of setting up the problem.
    • fig. S2. Illustration of transporting regions on nonnegatively curved space.
    • fig. S3. Illustration of how shortest path relates to curvature or entropy.
    • fig. S4. Intuitive understanding of Ricci curvature (a different perspective).
    • fig. S5. Illustrating geodesic deviations on nonnegatively curved space.

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