Zipf's law, 1/f noise, and fractal hierarchy

TL;DR

This paper explores the deep connections between fractals, 1/f noise, and Zipf's law, revealing their shared mathematical structures and hierarchical frameworks that unify various scaling phenomena across physical and social systems.

Contribution

It provides a theoretical derivation showing the analogy and common mathematical form of fractal patterns, 1/f noise, and Zipf's law, proposing a unified hierarchical framework.

Findings

Multifractal processes follow generalized Zipf's law.

1/f spectrum is mathematically identical to Zipf's law.

Hierarchical structures unify various scaling phenomena.

Abstract

Fractals, 1/f noise, Zipf's law, and the occurrence of large catastrophic events are typical ubiquitous general empirical observations across the individual sciences which cannot be understood within the set of references developed within the specific scientific domains. All these observations are associated with scaling laws and have caused a broad research interest in the scientific circle. However, the inherent relationships between these scaling phenomena are still pending questions remaining to be researched. In this paper, theoretical derivation and mathematical experiments are employed to reveal the analogy between fractal patterns, 1/f noise, and the Zipf distribution. First, the multifractal process follows the generalized Zipf's law empirically. Second, a 1/f spectrum is identical in mathematical form to Zipf's law. Third, both 1/f spectra and Zipf's law can be converted into a self-similar hierarchy. Fourth, fractals, 1/f spectra, Zipf's law, and the occurrence of large catastrophic events can be described with similar exponential laws and power laws. The self-similar hierarchy is a more general framework or structure which can be used to encompass or unify different scaling phenomena and rules in both physical and social systems such as cities, rivers, earthquakes, fractals, 1/f noise, and rank-size distributions. The mathematical laws on the hierarchical structure can provide us with a holistic perspective of looking at complexity such as self-organized criticality (SOC).