Paper: arXiv 2510.15942

Authors: Bhargavi Srinivasan

Abstract

We use the discrete Ollivier-Ricci graph curvature with Ricci flow to examine the intrinsic geometry of financial markets through the empirical correlation graph of the NASDAQ 100 index. Our main result is the development of a technique to perform surgery on the neckpinch singularities that form during the Ricci flow of the empirical graph, using the behavior and the lower bound of curvature of the fully connected graph as a starting point. We construct an algorithm that uses the curvature generated by intrinsic geometric flow of the graph to detect hidden hierarchies, community behavior, and clustering in financial markets despite the underlying challenges posed by a highly connected geometry.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced differential geometry concepts like Ricci flow and Ollivier-Ricci curvature, requiring deep mathematical formalism, while it also applies these methods to real-world NASDAQ 100 data, demonstrating empirical implementation and a specific algorithm for community detection.

Research Flowchart

  flowchart TD
  A["Research Goal: Analyze intrinsic market geometry via correlation graphs, handling Ricci flow singularities"] --> B["Input: NASDAQ 100 Correlation Graph"]
  B --> C["Compute Ollivier-Ricci Curvature"]
  C --> D["Apply Ricci Flow"]
  D --> E{"Singularities Detected?"}
  E -- Yes --> F["Surgery: Neckpinch Removal"]
  F --> C
  E -- No --> G["Outcome: Stable Geometric State"]
  G --> H["Findings: Hidden Hierarchies & Community Structure"]

Cited References (49)

  1. Clustering the NASDAQ-100 through Ricci Flow (2026)
  2. NVIDIA: Comparison of Clustering of Complete Graphs from Ricci Flow and MST and PCA (2026)
  3. Ricci curvature of graphs (2011)
  4. Ricci curvature and eigenvalue estimate on locally finite graphs (2010)
  5. Community detection on networks with ricci flow (2019)
  6. Ollivier-Ricci Curvature-Based Method to Community Detection in Complex Networks (2019) — link
  7. COmmunity Detection in Graphs (2010)
  8. Community Structure in Social and Biological Networks (2002)
and 41 more
  1. Finding and Evaluating Community Structure in Networks (2004)
  2. Characterizing complex networks with Forman-Ricci curvature and associated geometric flows (2017)
  3. Network geometry and market instability (2021)
  4. Ricci curvature: An economic indicator for market fragility and systemic risk (2016) — link
  5. The hyperbolic geometry of financial networks (2021) — link
  6. Forman curvature for complex networks (2016)
  7. Detecting the coarse geometry of networks (2018) — link
  8. Discrete curvatures and network analysis (2018)
  9. A Review of Two Decades of Correlations, Hierarchies, Networks and Clustering in Financial Markets (2021) — link
  10. Accelerated Evaluation of Ollivier-Ricci Curvature Lower Bounds: Bridging Theory and Computation (2024) — link
  11. Unfolding the multiscale structure of networks with dynamical Ollivier-Ricci curvature (2021)
  12. A Review of Two Decades of Correlations, Hierarchies, Networks and Clustering in Financial Markets (2017) — link
  13. Least squares quantization in PCM (1982)
  14. Multivariate observations (1967)
  15. Three-dimensiional manifolds, Kleinian groups and hyperbolic geometry (1982)
  16. Hierarchically nested factor model from multivariate data (2007)
  17. Princeton Math. Series (1997)
  18. Spectral Graph Theory (1997)
  19. Three-manifolds with positive ricci curvature (1982)
  20. The formation of singularities in the Ricci flow (1982)
  21. An example of neckpinching for Ricci flow on $S^ n+1 $ (2004)
  22. The entropy formula for the ricci flow and its geometric applications (2002)
  23. Ricci flow with surgery on Three-Manifolds (2003)
  24. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds (2003)
  25. Bochner’s Method for Cell Complexes and Combinatorial Ricci Curvature (2003)
  26. Ollivier’s Ricci Curvature, Local Clustering and Curvature-Dimension Inequalities on Graphs (2014)
  27. Riemannian geometry and geometric analysis (2011)
  28. Ricci curvature of metric spaces (2007)
  29. Ricci curvature for metric-measure spaces via optimal transport (2009)
  30. Ricci curvature of markov chains on metric spaces (2009)
  31. Probabilistic Approach to Geometry (2010) — link
  32. Optimal Transport: Old and New (2008) — link
  33. Optimal Transport: Old and New (2008) — link
  34. Histoire de l’Acad´emie Royale des Sciences de Paris (1781)
  35. On the translocation of masses (1942)
  36. Ricci curvature: An economic indicator for market fragility and systemic risk (2016)
  37. Introduction to econophysics: correlations and complexity in finance (1999)
  38. Hierarchical structure in financial markets (1999)
  39. Identification of clusters of companies in stock indices via Potts super-paramagnetic transitions (2000)
  40. Noise Dressing of Financial Correlation Matrices (1999) — link
  41. Ollivier-Ricci curvature convergence in random geometric graphs (2021) — link

0 of 49 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.