Universality of Zipf's law

Area of Science:

• Statistical Physics
• Complex Systems
• Information Theory

Background:

• Zipf's law describes scaling behavior observed in various systems, including cities, populations, and firms.
• Despite its prevalence, a general theoretical explanation for the universality of Zipf's law remains elusive.
• Existing models fail to provide a universally applicable framework for understanding this phenomenon.

Purpose of the Study:

• To demonstrate that Zipf's law is an inevitable outcome of a broad class of stochastic systems.
• To provide a general, model-free theoretical explanation for the ubiquity of Zipf's law.
• To connect Zipf's law to fundamental principles of Algorithmic Information Theory.

Main Methods:

• Utilizing concepts from Algorithmic Information Theory.
• Analyzing symbolic sequences generated by successive observations of stochastic systems with unbounded states.
• Assuming a description complexity for the system consistent with a stable state between order and disorder.

Main Results:

• Zipf's law emerges as an inevitable consequence of the analyzed stochastic systems.
• The derivation is based on a minimal set of physically relevant assumptions.
• The findings offer a general, model-free explanation for Zipf's law.

Conclusions:

• The universality of Zipf's law can be explained by its emergence from general stochastic processes.
• The theoretical framework presented provides a unified explanation for Zipf's law across diverse systems.
• This research bridges information theory and statistical physics to explain a fundamental scaling law.