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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*Related*

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

^_^

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