Behavior of the scaling correlation functions under severe subsampling

Scale invariance is a ubiquitous observation in the dynamics of large distributed complex systems. The computation of its scaling exponents, which provide clues on its origin, is often hampered by the limited available sampling data, making an appropriate mathematical description a challenge. This work investigates the behavior of correlation functions in fractal systems under conditions of severe subsampling. Analytical and numerical results reveal a striking robustness: the correlation functions continue to capture the expected scaling exponents despite substantial data reduction. This behavior is demonstrated numerically for the random 2D Cantor set and the Sierpinski gasket, both consistent with exact analytical predictions. Similar robustness is observed in 1D time series both synthetic and experimental, as well as in high resolution images of a neuronal structure. Overall, these findings are broadly relevant for the structural characterization of biological systems under realistic sampling constraints.