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

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.