  Geometry-inspired measures (such as discrete Ricci curvatures) and
topological data analysis (TDA) based methods (such as persistent homology)
have become attractive tools for characterizing the higher-order structure of
networks representing the financial systems. In this study, our goal is to
perform a comparative analysis of both these approaches, especially by
assessing the fragility and systemic risk in the Indian stock markets, which is
known for its high volatility and risk. To achieve this goal, we analyze the
time series of daily log-returns of stocks comprising the National Stock
Exchange (NSE) and the Bombay Stock Exchange (BSE). Specifically, our aim is to
monitor the changes in standard network measures, edge-centric discrete Ricci
curvatures, and persistent homology based topological measures computed from
cross-correlation matrices of stocks. In this study, the edge-centric discrete
Ricci curvatures have been employed for the first time in the analysis of the
Indian stock markets. The Indian stock markets are known to be less diverse in
comparison to the US market, and hence provides us an interesting example. Our
results point that, among the persistent homology based topological measures,
persistent entropy is simple and more robust than $L^1$-norm and $L^2$-norm of
persistence landscape. In a broader comparison between network analysis and
TDA, we highlight that the network analysis is sensitive to the way of
constructing the networks (threshold or minimum spanning tree), as well as the
threshold values used to construct the correlation-based threshold networks. On
the other hand, the persistent homology is a more robust approach and is able
to capture the higher-order interactions and eliminate noisy data in financial
systems, since it does not take into account a single value of threshold but
rather a range of values.
