Topological Complexity and Phase Space Stability: A Persistent Homology Approach to Cryptocurrency Risk

Paper: arXiv 2604.13311

Authors: Gabriel Santana, Jemirson Ramirez

Abstract

Traditional risk measures in finance, predominantly based on the second moment of return distributions or tail risk heuristics (VaR/CVaR), fail to account for the intrinsic geometric structure of market dynamics. This paper introduces a rigorous mathematical framework utilizing Topological Data Analysis (TDA) to quantify risk as the structural instability of the reconstructed phase space. By applying Takens’ Delay Embedding Theorem to cryptocurrency log-returns, we generate a point cloud representation of the underlying attractor. We analyze the evolution of the filtration of Vietoris-Rips complexes to compute persistent homology groups $H_k$. We define a “Topological Persistence Norm” to characterize market regimes and propose a leverage calibration heuristic based on the persistence of 1-dimensional cycles. This approach provides a coordinate-free, stability-invariant metric for risk assessment that is robust to high-frequency noise.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents an advanced, highly novel mathematical framework applying Topological Data Analysis and persistent homology to cryptocurrency risk. However, it lacks extensive empirical validation, real-world backtesting against standard risk metrics, and accessible implementation details for practical replication.

Research Flowchart

  flowchart TD
    A["Research Goal: Quantify Crypto Risk via Phase Space Topology"] --> B["Data Input: High-Frequency Cryptocurrency Log-Returns"]
    B --> C["Phase Space Reconstruction<br>(Takens' Delay Embedding Theorem)"]
    C --> D["Topological Data Analysis<br>(Vietoris-Rips Filtration & Persistent Homology)"]
    D --> E["Calculate Topological Persistence Norm"]
    E --> F["Outcome: Noise-Robust Risk Metric & Leverage Calibration"]