Beyond Signal and Noise: Unraveling Scale Invariance in Neuroscience and Financial Networks with Topological Data Analysis

Abstract

Topological Data Analysis (TDA) is increasingly crucial in investigating the shape of complex data structures across scientific fields, particularly in neuroscience and finance. This study delves into persistent homology, a TDA component initially aimed at differentiating between signal and noise. We explore two methodologies: the conventional cycle length approach and the novel death-birth ratio method proposed by Bobrowski and Skraba. Analyzing rs-fMRI data from the Human Connectome Project and daily $S\&P 500$ financial networks, our study compares these methods in identifying significant cycles. A key discovery is a robust relationship between z-score thresholds applied to bar lengths or ratios and behavioural traits in brain networks and market volatility in financial networks. In the brain, this is evident in the strong correlation between significant 1-cycles, brain volumes, and sex-based differences. In financial markets, a fractal pattern emerges, where market volatility negatively correlates with the number of significant cycles, indicating that more complex market topologies are associated with increased stability. Our findings also imply a fractal nature of 1-cycles at both population levels and across multiple days in the stock market. The distribution of significant loops, marked by high z-scores, remains consistent across various z-score thresholds, revealing a scale-invariant, fractal structure in both data sets. Given the scale invariance in these fractal structures, the traditional TDA distinction between signal and noise becomes less meaningful. This suggests that all scales of cycle length are relevant, challenging the conventional approach of segregating signal from noise and broadening the scope of TDA to reveal intricate, scale-invariant relationships in complex systems.