Paper: arXiv 2602.00383

Authors: Samuel W. Akingbade

Abstract

Financial markets exhibit temporal organization that is not fully captured by volatility measures or linear correlation structure. We study a null validated topological approach for quantifying market complexity and apply it to Bitcoin daily log returns. The analysis uses the $L^1$ norm of persistence landscapes computed from sliding-window delay embeddings. This quantity shows strong co-movement with stochastic volatility during periods of market stress, but remains intermittently elevated during low volatility regimes, indicating dynamical structure beyond fluctuation scale. Rolling correlation analysis reveals that the dependence between geometry and volatility is not stationary. Surrogate based null models provide statistical validation of these observations. Rejection of shuffle surrogates rules out explanations based on marginal distributions alone, while departures from phase randomized surrogates indicate sensitivity to nonlinear and phase dependent temporal organization beyond linear correlations. These results demonstrate that persistence landscape norms provide complementary information about market dynamics across market conditions.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a mathematically solid application of persistent homology (TDA) to Bitcoin time series, using persistence landscape norms to study financial dynamics. It features good empirical rigor through surrogate-based statistical null models (shuffle and phase-randomized) to disentangle genuine non-linear geometry from volatility and marginal effects, though code availability is not explicitly specified in the excerpt.

Research Flowchart

  flowchart TD
    A["Research Question:<br>Quantify market complexity beyond volatility & linear correlation"] --> B["Data Input:<br>Bitcoin Daily Log Returns"]
    B --> C["Methodology:<br>Sliding-Window Delay Embeddings"]
    C --> D["Topological Analysis:<br>Compute L1 Norm of Persistence Landscapes"]
    D --> E["Statistical Validation:<br>Surrogate Null Models<br>(Shuffle & Phase-Randomized)"]
    E --> F["Key Findings:<br>1. Co-movement with stochastic volatility in stress periods<br>2. Elevates in low-volatility regimes (Nonlinear structure)<br>3. Non-stationary geometry-volatility correlation"]

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1 of 111 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.