A subtlety to note is that this list includes two different types of reported power laws:
power law as bivariate function like allometric scaling; and
power law as probability density function like the paper publication rates.
- The derivation of a power law suggests that—in a certain (“critical”) regime—phenomena do not possess a preferred scale in space, time, or something else: They are, in a sense, “scale free.” (See Heavy Tailed Distribution)
- The fact that heavy-tailed distributions occur in complex systems is certainly important (because it implies that extreme events occur more frequently than would otherwise be the case), and statistically sound empirical ﬁts of event data, when used with caution, can help in data interpretation (as it is certainly useful to estimate how often extreme events occur in a given system).
- However, although power laws have been reported in areas ranging from finance and molecular biology to geophysics and the Internet, the data are typically insufﬁcient and the mechanistic insights are almost always too limited for the identification of power-law behavior to be scientifically useful.
- Numerous scholars have neglected to apply careful statistical tests to data that were reported to exhibit power-law relationships.
- Even if a reported power law surmounts the statistical hurdle, it often lacks a generative mechanism.
Indeed, the same power law (that is, with the same value of λ) can arise from many different mechanisms.
- However, a statistically sound power law is no evidence of universality without a concrete underlying theory to support it.
- However, as Philip Anderson pointed out in 1972, one must be cautious when claiming power-law behavior in ﬁnite systems, and it is not clear whether power laws are relevant or useful in so-called “complex systems”
- Moreover, knowledge of whether or not a distribution is heavy-tailed is far more important than whether it can be ﬁt using a power law.
- The central limit theorem also holds ubiquitously, including in situations in which random variables are drawn from heavy-tailed distributions; in such cases, however, power laws replace the Gaussian distribution as the limiting situation. [Ubiquity of power laws]
- Finally, and perhaps most importantly, even if the statistics of a purported power law have been done correctly, there is a theory that underlies its generative process, and there is ample and uncontroversial empirical support for it, a critical question remains: What genuinely new insights have been gained by having found a robust, mechanistically supported, and in-all-other-ways superb power law? We believe that such insights are very rare.
- Michael P. H. Stumpf and Mason A. Porter, Critical Truths About Power Laws, Science 10 February 2012: 335 (6069), 665-666.
- A. Clauset, C. R. Shalizi, M. E. J. Newman, Power-Law Distributions in Empirical Data, SIAM Rev.
51, 661 (2009).
- L. Bettencourt, G. West, Nature 467, 912 (2010).
Online Materials: scaling laws