Power law

POWER-LAW DISTRIBUTIONS IN EMPIRICAL DATA
AARON CLAUSET∗ , COSMA ROHILLA SHALIZI† , AND M. E. J. NEWMAN‡ Abstract. Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in thetail of the distribution—the part of the distribution representing large but rare events— and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurateanswers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic and likelihood ratios. We evaluate theeffectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a powerlaw distribution. In some cases we find these conjectures to be consistent with the data while in others the power law is ruled out.Key words. Power-law distributions; Pareto; Zipf; maximum likelihood; heavy-tailed distributions; likelihood ratio test; model selection AMS subject classifications. 62-07, 62P99, 65C05, 62F99

arXiv:0706.1062v2 [physics.data-an] 2 Feb 2009

1. Introduction. Many empirical quantities cluster around a typical value. The speeds of cars on a highway, the weights of apples in a store, air pressure,sea level, the temperature in New York at noon on Midsummer’s Day. All of these things vary somewhat, but their distributions place a negligible amount of probability far from the typical value, making the typical value representative of most observations. For instance, it is a useful statement to say that an adult male American is about 180cm tall because no one deviates very far from this size.Even the largest deviations, which are exceptionally rare, are still only about a factor of two from the mean in either direction and hence the distribution can be well-characterized by quoting just its mean and standard deviation. Not all distributions fit this pattern, however, and while those that do not are often considered problematic or defective for just that reason, they are at the same timesome of the most interesting of all scientific observations. The fact that they cannot be characterized as simply as other measurements is often a sign of complex underlying processes that merit further study. Among such distributions, the power law has attracted particular attention over the years for its mathematical properties, which sometimes lead to surprising physical consequences, and forits appearance in a diverse range of natural and man-made phenomena. The populations of cities, the intensities of earthquakes, and the sizes of power outages, for example, are all thought to have power-law distributions. Quantities such as these are not well characterized by their typical or average values. For instance, according to the 2000 US Census, the average population of a city, town, or∗ Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA and Department of Computer Science, University of New Mexico, Albuquerque, NM 87131, USA † Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 15213, USA ‡ Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA

1

2

A. Clauset, C. R. Shalizi...