Integral Betti signature confirms the hyperbolic geometry of brain, climate, and financial networks

ArXiv ID: 2406.15505 “View on arXiv”

Authors: Unknown

Abstract

This paper extends the possibility to examine the underlying curvature of data through the lens of topology by using the Betti curves, tools of Persistent Homology, as key topological descriptors, building on the clique topology approach. It was previously shown that Betti curves distinguish random from Euclidean geometric matrices - i.e. distance matrices of points randomly distributed in a cube with Euclidean distance. In line with previous experiments, we consider their low-dimensional approximations named integral Betti values, or signatures that effectively distinguish not only Euclidean, but also spherical and hyperbolic geometric matrices, both from purely random matrices as well as among themselves. To prove this, we analyse the behaviour of Betti curves for various geometric matrices – i.e. distance matrices of points randomly distributed on manifolds of constant sectional curvature, considering the classical models of curvature 0, 1, -1, given by the Euclidean space, the sphere, and the hyperbolic space. We further investigate the dependence of integral Betti signatures on factors including the sample size and dimension. This is important for assessment of real-world connectivity matrices, as we show that the standard approach to network construction gives rise to (spurious) spherical geometry, with topology dependent on sample dimensions. Finally, we use the manifolds of constant curvature as comparison models to infer curvature underlying real-world datasets coming from neuroscience, finance and climate. Their associated topological features exhibit a hyperbolic character: the integral Betti signatures associated to these datasets sit in between Euclidean and hyperbolic (of small curvature). The potential confounding ``hyperbologenic effect’’ of intrinsic low-rank modular structures is also evaluated through simulations.

Keywords: Persistent Homology, Betti Curves, Topological Data Analysis, Geometric Matrices, Low-rank Modular Structures, Multi-Asset (Equities/Bonds)

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper employs advanced topological and geometric mathematics, specifically persistent homology (Betti curves) and Riemannian geometry (constant curvature manifolds), resulting in a high math complexity score. While it applies these tools to real-world datasets (brain, finance, climate), the empirical implementation is primarily comparative analysis using topological signatures rather than backtesting or data-heavy quantitative finance modeling, leading to a lower empirical rigor score.

  flowchart TD
    A["Research Goal<br>Establish hyperbolic geometry of<br>brain, climate, and financial networks"] --> B["Methodology: Integral Betti Signatures"]
    
    B --> C["Data Inputs<br>Geometric Matrices: Euclidean, Spherical, Hyperbolic<br>Real-World Networks: Brain, Climate, Finance"]
    
    C --> D["Computational Process<br>Persistent Homology & Betti Curves<br>Low-rank Structure Evaluation"]
    
    D --> E["Key Finding 1<br>Signatures distinguish all geometric models"]
    D --> F["Key Finding 2<br>Sample size affects topology (spurious spherical geometry)"]
    D --> G["Key Finding 3<br>Real networks show hyperbolic character"]
    D --> H["Key Finding 4<br>Low-rank structures do not explain hyperbolicity"]