Powerlaw: a Python package for analysis of heavy-tailed distributions

Abstract

Power laws are theoretically interesting probability distributions that are also frequently used to describe empirical data. In recent years, effective statistical methods for fitting power laws have been developed, but appropriate use of these techniques requires significant programming and statistical insight. In order to greatly decrease the barriers to using good statistical methods for fitting power law distributions, we developed the powerlaw Python package. This software package provides easy commands for basic fitting and statistical analysis of distributions. Notably, it also seeks to support a variety of user needs by being exhaustive in the options available to the user. The source code is publicly available and easily extensible.

Figure 1. Basic steps of analysis for heavy-tailed distributions: visualizing, fitting, and comparing.

Example data for power law fitting are a good fit (left column), medium fit (middle column) and poor fit (right column). Data and methods described in text. a) Visualizing data with probability density functions. A typical histogram on linear axes (insets) is not helpful for visualizing heavy-tailed distributions. On log-log axes, using logarithmically spaced bins is necessary to accurately represent data (blue line). Linearly spaced bins (red line) obscure the tail of the distribution (see text). b) Fitting to the tail of the distribution. The best fit power law may only cover a portion of the distribution's tail. Dotted green line: power law fit starting at  = 1. Dashed green line: power law fit starting from the optimal (see Basic Methods: Identifying the Scaling Range). c) Comparing the goodness of fit. Once the best fit to a power law is established, comparison to other possible distributions is necessary. Dashed green line: power law fit starting from the optimal . Dashed red line: exponential fit starting from the same .

Figure 2. Probability density function (, blue) and complemenatary cumulative distribution function (, red) of word frequencies from “Moby Dick”.

Figure 3. Complemenatary cumulative distribution functions of the empirical word frequency data and fitted power law distribution, with and without an upper limit .

Figure 4. Complemenatary cumulative distribution functions of word frequency data and fitted power law and lognormal distributions.

Figure 5. Example of multiple local minima of Kolmogorov-Smirnov distance

across . As a power law is fitted to data starting from different , the goodness of fit between the power law and the data is measured by the Kolmogorov-Smirnov distance , with the best minimizing this distance. Here fitted data is the population sizes affected by blackouts. While there exists a clear absolute minima for at 230, and thus 230 is the optimal additional restrictions could exclude this fit. Parameter requirements such as or would restrict the values considered, leading to the identification of a different, smaller at 50.