Abstract
Although Zipf’s law is widespread in natural and social data, one often encounters situations where one or both ends of the ranked data deviate from the power-law function. Previously we proposed the Beta rank function to improve the fitting of data which does not follow a perfect Zipf’s law. Here we show that when the two parameters in the Beta rank function have the same value, the Lavalette rank function, the probability density function can be derived analytically. We also show both computationally and analytically that Lavalette distribution is approximately equal, though not identical, to the lognormal distribution. We illustrate the utility of Lavalette rank function in several datasets. We also address three analysis issues on the statistical testing of Lavalette fitting function, comparison between Zipf’s law and lognormal distribution through Lavalette function, and comparison between lognormal distribution and Lavalette distribution.
Highlights
It is said that a certain quantity follows a power law if the probability of observing it varies inversely as a power of this quantity
Power laws are observed in epidemic systems: beginning with the observation that epidemic sizes and durations are well characterized by power laws [9], this scale free behavior has been used to model patch sizes during an epidemic spread [10], as well as other relevant spatial patterns in theoretical ecology [11]
We have presented a novel probability distribution function and showed that it is a good alternative for data that does not follow a perfect Zipf ’s law
Summary
It is said that a certain quantity follows a power law if the probability of observing it varies inversely as a power of this quantity. Power laws in data collected from natural or social phenomena are well documented [1]. The asymptotic occurrence of power laws in critical phenomena and statistical physics has been widely studied [2]. Power law tails have been reported in the distribution of word frequency [3], city sizes [4], fluctuations in financial market indexes [5], firm sizes in the U.S [6], scientific citations [7, 8] et caetera.
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