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There is more than a power law in Zipf
Matthieu Cristelli1, Michael Batty, Luciano Pietronero
1Department of Physics, University of Rome La Sapienza, Rome, Italy.
Scientific Reports
|November 10, 2012
Summary
Zipf's Law describes many real-world phenomena, but standard sampling methods do not accurately predict it for large cities. This study introduces
Area of Science:
- Quantitative social science
- Statistical physics
- Urban studies
Background:
- Zipf's Law is a rank-size rule observed in various phenomena, including city sizes, word frequencies, and wealth distribution.
- It is often considered a manifestation of underlying power laws, such as Pareto's or Benford's Law.
- However, direct application of Zipf's Law to empirical data, like city sizes from random sampling, often yields inaccurate results for the largest entities.
Purpose of the Study:
- To investigate the discrepancy between theoretical Zipf's Law predictions and empirical observations, particularly for large cities.
- To identify the underlying statistical property responsible for the inaccurate fitting of Zipf's Law.
- To propose a method for accurately fitting Zipf's Law by accounting for this property.
Main Methods:
- Analysis of city size distributions and word frequency data.
- Examination of the functional form of Zipf's Law in relation to the number of events (N).
- Introduction and definition of a new statistical property termed 'coherence' and its relation to 'screening' effects within sample distributions.
Main Results:
- Demonstration that simple random sampling from typical distributions does not yield Zipf's Law for the largest cities.
- Identification of 'coherence' as a crucial property of sample distributions that affects Zipf's Law fitting.
- The functional form of Zipf's Law is shown to be dependent on the number of events (N), highlighting the need for accounting for coherence.
Conclusions:
- Zipf's Law fitting requires accounting for the 'coherence' of the sample distribution, which represents 'screening' effects between elements.
- Standard statistical assumptions may not be sufficient for accurately applying Zipf's Law to empirical data, especially in complex systems like urban populations.
- The proposed method addresses a fundamental limitation in applying Zipf's Law, leading to more accurate modeling of rank-size relationships.
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