Rank distributions: Frequency vs. magnitude

Area of Science:

• Data analysis
• Statistical modeling
• Complex systems

Background:

• Ranked data can be analyzed by frequency (occurrences) or magnitude (size).
• Examples include moon craters, earthquakes, and billionaire wealth.
• Understanding the relationship between these two perspectives is crucial for data interpretation.

Purpose of the Study:

• To investigate the functional relationship between frequency and magnitude ranked data.
• To mathematically define this inverse relationship using probability distributions and iterated maps.
• To explore how this relationship varies across different decay rates.

Main Methods:

• Analysis of ranked data distributions (frequency vs. magnitude).
• Formulation of parent probability distributions to model data generation.
• Development of analog nonlinear iterated maps to reproduce observed distributions.
• Extension of the framework using thermodynamic and statistical-mechanical concepts.

Main Results:

• Frequency and magnitude ranked data are functional inverses of each other.
• The relationship is precisely defined by the underlying probability distribution and a nonlinear iterated map.
• For hyperbolic decay (Zipf's Law), distributions are identical (power law).
• Maximum divergence occurs with logarithmic decay and inverse exponential decay (Benford's Law).
• Generic differences in power-law exponents appear at various ranks for intermediate decay rates.

Conclusions:

• The inverse relationship between frequency and magnitude ranked data is a fundamental property.
• This framework unifies diverse phenomena, from Zipf's Law to Benford's Law.
• The inclusion of thermodynamic concepts deepens the theoretical understanding of ranked data.