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This exploratory study delves into persistent homology, an integral componentof TDA that initially aimed at differentiating between signal and noise. Weexplore two methodologies for this differentiation: the conventional cyclelength approach and the novel death-birth ratio method proposed by Bobrowskiand Skraba. Analyzing comprehensive rs-fMRI data from the Human ConnectomeProject and daily S\&P 500 financial networks, our study compares these methodsin identifying significant cycles. A pivotal discovery of this paper is arobust relationship between z-score thresholds applied to bar lengths or ratiosand, respectively, behavioural traits in brain networks and market volatilityin financial networks. In the brain, this is evident in the strong correlationbetween the number of significant 1-cycles in brain networks, brain volumes,and sex-based differences. In financial markets, a fractal pattern emerges,where market volatility consistently negatively correlates with the number ofsignificant cycles, indicating that more intricate market topologies areassociated with increased stability and less susceptibility to rapid shifts.Our findings also imply a fractal nature of 1-cycles at both population levelsand across multiple days in the stock market. Notably, the distribution ofsignificant loops, marked by high z-scores, remains consistent across variousz-score thresholds, revealing a scale-invariant, fractal structure in both datasets. Given the scale invariance in these fractal structures, the traditionalTDA distinction between signal and noise becomes less meaningful. The fractalnature of 1-cycles suggests that all scales of cycle length are relevant,challenging the conventional approach of segregating signal from noise. Thisrealization broadens the scope of TDA, underscoring its potential to revealintricate, scale-invariant relationships in complex systems.

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