# Testing statistical laws in complex systems

## TL;DR

This paper examines how correlations in complex system data affect the testing of statistical laws, proposing a conservative method to improve the reliability of such tests.

## Contribution

It introduces a new testing approach that accounts for correlations, reducing false rejections of statistical laws in complex systems analysis.

## Key findings

- Correlations cause standard tests to falsely reject statistical laws.
- The proposed method reduces false rejections and increases confidence in law validation.
- Accounting for correlations leads to more accurate statistical law assessments.

## Abstract

The availability of large datasets requires an improved view on statistical laws in complex systems, such as Zipf's law of word frequencies, the Gutenberg-Richter law of earthquake magnitudes, or scale-free degree distribution in networks. In this paper we discuss how the statistical analysis of these laws are affected by correlations present in the observations, the typical scenario for data from complex systems. We first show how standard maximum-likelihood recipes lead to false rejections of statistical laws in the presence of correlations. We then propose a conservative method (based on shuffling and under-sampling the data) to test statistical laws and find that accounting for correlations leads to smaller rejection rates and larger confidence intervals on estimated parameters.