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>A large number of natural distributions follow a power-law distribution. This is surprisingly debatable. People like to find power-law/scale-free distribution

>A large number of natural distributions follow a power-law distribution. This is surprisingly debatable. People like to find power-law/scale-free distributions because it implies a neat generative story, but a lot of the "evidence" for power-law distributions is pretty weak. For example, you cannot just show that a log-log plot is linear--lots of other distributions can produce similar plots. Clauset, Shalizi, and Newman have a very readable paper where they describe 1) how to properly test for a power-law distribution, and 2) use those tests to assess the validity of some claims from the literature (spoiler: not many have "good" statistical support). Here is the paper: http://arxiv.org/abs/0706.1062 http://arxiv.org/abs/0706.1062 Shalizi has a short blog post describing the main results: http://bactra.org/weblog/491.html http://bactra.org/weblog/491.html

dalke 10y ago
Agreed. The old saying is "anything looks linear when plotted on a log-log graph." However, the OP was giving a rough approximation in the first place, in saying that all natural distributions follow a Gaussian curve. But Gaussian curves go from -∞ to +∞. Many of the real-world distributions must only have positive values, like heights and weights. Although for real-world purposes, they can usually be approximated as Gaussian. With that roughness in mind, I think it's okay to say that the Stefan–Boltzmann law, the inverse-square laws of Newtonian gravity and electrostatics, or Kleiber's law, which are all listed in the Wikipedia link are close to a power law to be acceptable counter-examples. (Stefan–Boltzmann assumes a perfect black body, the inverse-square laws ignore relativity, and Kleiber's law is a rule-of-thumb in the first place.)
- mattkrause 10y ago
  Oh sure. When there's a plausible reason for thinking something is scale-free, I'm all for it. For a while though, it was en vogue to find power law distributions in all sorts of weird places (email response times, numbers of friends), and that's what I was attempting to object to!

dalke 10y ago

Agreed. The old saying is "anything looks linear when plotted on a log-log graph." However, the OP was giving a rough approximation in the first place, in saying that all natural distributions follow a Gaussian curve. But Gaussian curves go from -∞ to +∞. Many of the real-world distributions must only have positive values, like heights and weights. Although for real-world purposes, they can usually be approximated as Gaussian. With that roughness in mind, I think it's okay to say that the Stefan–Boltzmann law, the inverse-square laws of Newtonian gravity and electrostatics, or Kleiber's law, which are all listed in the Wikipedia link are close to a power law to be acceptable counter-examples. (Stefan–Boltzmann assumes a perfect black body, the inverse-square laws ignore relativity, and Kleiber's law is a rule-of-thumb in the first place.)

mattkrause 10y ago
Oh sure. When there's a plausible reason for thinking something is scale-free, I'm all for it. For a while though, it was en vogue to find power law distributions in all sorts of weird places (email response times, numbers of friends), and that's what I was attempting to object to!