Beyond Zipf's Law: The Lavalette Rank Function and Its Properties.

Area of Science:

• Quantitative Social Sciences
• Statistical Physics
• Information Theory

Background:

• Zipf's law accurately models many natural and social phenomena.
• Deviations from Zipf's law occur at the extremes of ranked data.
• The Beta rank function was previously introduced to address these deviations.

Purpose of the Study:

• To introduce and analyze the Lavalette rank function, a special case of the Beta rank function.
• To analytically derive the probability density function of the Lavalette distribution.
• To compare the Lavalette distribution with the lognormal distribution.

Main Methods:

• Analytical derivation of the probability density function for the Lavalette distribution.
• Computational analysis to compare Lavalette and lognormal distributions.
• Application and illustration of the Lavalette rank function on diverse datasets.

Main Results:

• The Lavalette rank function is derived analytically when Beta function parameters are equal.
• The Lavalette distribution is shown to be an approximation of the lognormal distribution.
• The utility of the Lavalette rank function is demonstrated across multiple datasets.

Conclusions:

• The Lavalette rank function provides an effective method for fitting data that deviates from Zipf's law.
• The close approximation to the lognormal distribution offers insights into underlying data generation processes.
• Further analysis addresses statistical testing and comparative studies involving these distributions.