I am most grateful to Dr. Massimiliano Ignaccolo for referring me to an arXiv paper on the use and abuse of power-law distributions in empirical data.

It has become oddly customary to recognize power laws everywhere, from finance to biology, from politics to earthquakes. Now, Clauset (Univ. of New Mexico, Albuquerque), Shalizi (Carnegie Mellon) and Newman (Univ. of Michigan, Ann Arbor) clarify that reasearchers sometimes employ too liberal means for qualifying a statistical distribution as a power-law one:

The common practice of identifying and quantifying power-law distributions by the approximately straight-line behavior of a histogram on a doubly logarithmic plot should not be trusted: such straight-line behavior is a necessary but by no means sufficient condition for true power-law behavior.

According to these authors, examples of distributions wrongly purported as power laws include, e.g.:

  • the size of files transmitted over the internet
  • the intensities of California earthquakes 1910-1922
  • the number of links to web sites
  • the distribution of human wealth
  • the degrees of metabolites in the netaboilic network of Escherichia Coli

A number of other presumed power-law distributions are found, in this study, to correlate equally well with other statistical models, such as log-normal or stretched-exponential. I.e., in order to classify them as power-law distributed we should investigate the underlying mechanisms more deeply (examples: severity of terrorist attacks; populations of US cities; distinct sigthings of bird species; number of aherents to religious denominations; citations of a scientific paper over a period of time).

Furthermore, there are distributions that, while seemingly following a power law, are merely heavy-tailed (i.e., p(x) goes like x at the a power only for values of x greater than some minimum threshold). And in many such cases, different types of statistical distributions are a better fit than the power-law one.

It is a pop-complex author’s favorite sport to refer you to some underlying “power law” (especially since they read of the Black Swan), as if this necessarily were an indication of underlying non-linearity (which it is not): next time they do it to you, just refer them to this paper.

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Comments
  1. Massimiliano Ignaccolo says:

    Here is another one about one of the “darling” of complexity science: multifractality
    http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.85.6884

    All the best,
    Massi

  2. […] no longer a Gaussian but rather other stable distributions (although not necessarily a “power law“, as is believed in popular versions of the […]

  3. […] How fast can you say power law? […]

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