Power-law distributions in empirical data

That Science paper https://www.science.org/doi/10.1126/science.adq7055 ought to be retracted, it is using the same ad-hoc methods that we used to wrongly fit power laws to statistical distributions when I was a grad student 30 years ago. I wrote a trash paper too and I'm still angry about not feeling psychologically safe to bring the issue up. A postdoc who was there wrote a paper many years later about the right way to do it https://arxiv.org/abs/0706.1062 I don't think anybody read it. Note that the Science paper uses the "Pearson product moment correlation" which is "can only be used to measure the relationship between two variables which are both normally distributed" -- and there's nothing normally distributed here.

For what it's worth, showing a straight line on a log-log plot isn't really enough to demonstrate existence of a power law. Most papers that test for the existence of power laws don't test goodness-of-fit for other distribtions. The lognormal distribution is often just as good a fit to the data. This paper covers a lot of the frequent problems in academic literature that tries to fit power laws: https://arxiv.org/abs/0706.1062

being roughly straight on a log-log plot is a necessary but not sufficient condition for power-law behavior Source: https://arxiv.org/abs/0706.1062, p.15 pp. 24-29 have nice log-log plots of various data sets, along with whether they can plausibly described as power law distributions. 'Wealth', interestingly enough, looks like a pretty straight line, but fails the statistical test badly. Btw, if you just take logs and run a regression to estimate the exponent, you get a highly biased estimate. In general, it's much better to do a direct statistical test, although the method described in the paper is somewhat involved.

>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 Shalizi has a short blog post describing the main results: http://bactra.org/weblog/491.html

I think log-normal distributions usually fit this kind of data as well as power laws do. Anyway, such a plot is not the best way to choose between those two models (see http://arxiv.org/abs/0706.1062). We don't actually do any such test in the paper though (or say that it is a power law or log-normal). You might want to check out the paper for more details. https://www.facebook.com/publications/244240069095667/

About the "fictive power law" thing: this is THE paper to read: http://arxiv.org/abs/0706.1062 (it's an easy read, explaining what the maximum likelihood method is, etc.). Despite what they say, fitting the log-log CDF usually gives pretty good results when done right (fitting the PDF does not)

"Mark wrote a paper about this ten years later, after physicists had published thousands of bogus papers using bogus statistics." Agreed, but to be fair a lot of non-physicists were also doing this. For example in their excellent review article http://arxiv.org/abs/0706.1062 Mark Newman, Aaron Clauset, and Cosma Shalizi show that a claimed power law in bytes received per http request was wrong. I'm not going to defend the sloppiness of physicists, but I would note that other fields don't fare as well in this either. And I wish you or Mark had written that paper earlier too!

Actually, the paper you should be reading is this: "Power-law distributions in empirical data", Aaron Clauset, Cosma Rohilla Shalizi, M. E. J. Newman http://arxiv.org/abs/0706.1062

If you follow the literature referenced in this recent blog post you should be able to find what you are looking for. Also pay close attention to the content of shalizi and newman's review article - there is a lot of notorious and well known statistical sloppiness in the literature on power law distributions. http://cs.unm.edu/~aaron/blog/archives/2009/11/power_laws_an... It's especially interesting that they call the power law distribution of wealth into question. EDIT: the link to the shalizi and newman review in the blog post is a link to a "buy this .pdf" site. Here's the mentioned article on the arxiv http://arxiv.org/abs/0706.1062

... less power It is a sore point for me that physicists have long wrongly plotted distributions on log-log paper, drawn a line, and pronounced they discovered a "universal" power law without ever once using words like "estimator", "significance", etc. My first paper was wrong because of this and when I was asked what I thought the cause of our observation was I didn't give my real answer (which I'd already had given and had rejected) which was that our analysis was bogus. Mark Newman was at our lab at this time and later on he wrote some good papers on this project. https://arxiv.org/abs/0706.1062