# Integral Betti signatures of brain, climate and financial networks compared to hyperbolic, Euclidean and spherical models

## TL;DR

This paper uses topological tools to compare the geometry of brain, climate, and financial networks with hyperbolic, Euclidean, and spherical models.

## Contribution

The study introduces Betti curves to distinguish geometric matrices and reveals hyperbolic characteristics in real-world networks.

## Key findings

- Betti curves can distinguish spherical, hyperbolic, and Euclidean matrices from random ones.
- Real-world networks in neuroscience, finance, and climate show hyperbolic geometry.
- Standard network construction methods can produce spurious spherical geometry.

## 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. We show that low-dimensional Betti curve approximations effectively distinguish not only Euclidean, but also spherical and hyperbolic geometric matrices, both from purely random matrices as well as among themselves. We proved this by analysing the behaviour of Betti curves for various geometric matrices – i.e distance matrices of points randomly distributed on manifolds given by the Euclidean space, the sphere, and the hyperbolic space. We further show that the standard approach to network construction gives rise to (spurious) spherical geometry, and document the role of sample size and dimension to assess real-world connectivity matrices. Finally, we observe that real-world datasets coming from neuroscience, finance and climate seem to exhibit a hyperbolic character. The potential confounding “hyperbologenic effect” of intrinsic low-rank modular structures is evaluated.