Detectability limits of scaling laws
Abstract
Power law scaling relations between size and output are central to quantitative theories of cities, organisms, and other complex systems. Competing theories predict scaling exponents that differ by small fractions, but there is no existing theory for verifying whether a given dataset can even distinguish exponents at the required resolution to address such discrepancies. Here we derive a resolution limit for scaling exponents, giving the smallest exponent difference that any method of analysis can detect. We find that the Hurst exponents governing the evolution of systems' sizes and deviations from the scaling law determine how long a record of growing systems must be before it can separate competing scaling theories. Empirical results suggest that many available data panels are insufficient for reliable scaling model selection.
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BibTeX
@misc{kirkley2026detectability,
title = {{Detectability limits of scaling laws}},
author = {Alec Kirkley},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21638},
eprint = {2609.21638},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21638},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Detectability limits of scaling laws
AU - Alec Kirkley
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21638
UR - https://arxiv.org/abs/2609.21638
AB - Power law scaling relations between size and output are central to quantitative theories of cities, organisms, and other complex systems. Competing theories predict scaling exponents that differ by small fractions, but there is no existing theory for verifying whether a given dataset can even distinguish exponents at the required resolution to address such discrepancies. Here we derive a resolution limit for scaling exponents, giving the smallest exponent difference that any method of analysis can detect. We find that the Hurst exponents governing the evolution of systems' sizes and deviations from the scaling law determine how long a record of growing systems must be before it can separate competing scaling theories. Empirical results suggest that many available data panels are insufficient for reliable scaling model selection.
ER -