# Besides the Pareto and Zipfian distributions, which distributions obey the power-law?

I need a list of distributions that obey the power-law, beside the commonly used Pareto and Zipfian distributions. A comprehensive list or a reference to a comprehensive list will be particularly appreciated.

That's basically the complete list in your question, (the Pareto and the zeta/Zipf).

A power law is one where the pdf/pmf is proportional to $x^{-p}\,$ ($1$).

People use power laws for either continuous ($x> k$) or discrete ($x=1,2,...$) data; the continuous case proportional to $x^{-p}$ is the Pareto, the discrete case proportional to $x^{-p}$ is the zeta.

They are sometimes truncated, which would give a truncated Pareto or a truncated zeta (a right-truncated zeta is a Zipf) respectively.

There are numerous variants of power laws -- there are ones that modify the tail with say an exponential term or include a kink (essentially two different power laws in two different parts) and so on. These are not exactly a power law but a modified power law -- and the number of different possibilities for modifying power laws is vast. You could probably fill a small book with variants that have been tried at different times, and it could take an infinity of books listing possible variants that have yet to be tried.

If you're working with power laws, you may like to read Cosma Shalizi's discussion of power laws and the paper (Clauset, Shalizi and Newman, "Power-law distributions in empirical data") he links to there.

(1) See Clauset et al equation 1.1

Since OP now mentions the paper they were looking at, it seems we are actually dealing with a more general definition of "power law" that the one in Clauset et al.:

The definition in that reference is $P(X\geq x) \sim c x ^{-\alpha}$.

There are two main distinctions from Clauset et al:

1. This one is defined in terms of the tail probability rather than the pdf. That would still leave us with the Pareto and the zeta but also generalizes to include some marginal cases (such as mixtures of discrete and continuous power-law variates). However, in practice such models would be unlikely to be used much, if at all, on the kind of physical data people tend to look for power laws on.

2. The use of "$\sim$"; $f(x)\sim g(x)$ means $\lim_{x\to\infty} f(x)/g(x) \to 1$

The second change means that instead of the log of the tail probability being linear in log(x) it's only asymptotically approaching something that is linear in log(x).

This admits an infinity of distributions. Consider the Pareto:

$P(X\geq x) =P(X> x) = S_{_P}(x) = 1-(x/\theta)^{-\alpha}$

with density $p_{_P}(x) = \frac{\alpha}{\theta} (\frac{x}{\theta})^{-\alpha-1}\,,\:x>\theta$

Now consider, for example, some continuous function $r(x)\,,\:x>\theta$ where $r(\theta)=1$, $r(x)\sim 1$, $r(x)>0$, and such that $S_{_P}(x+t)r(x+t) \leq S_{_P}(x)r(x)$ for all $x>\theta$ and al $t>0$. Then I think $S(x) = S_{_P}(x)r(x)$ should be a valid survivor function with the property that $S(x)\sim S_{_P}(x)$.

We could invent as many of these $r$ functions as we wish.

As an example, if we look at the Pareto pdf on the log-log scale (a straight line, $c+\beta (x-\theta)$) and imagine adding a small but decreasing-in-amplitude sine-wave wiggle to it ( $\delta x^{-s} \sin(\kappa (x-\theta))$), then the corresponding survivor function should have the right characteristics as long as $s$ is large enough (I haven't checked but any positive $s$ might even suffice).

• Thanks. It is sentences like this "one specific commonly used power law distribution is the Pareto distribution" (here) that give the impression that there are many such distributions. Nov 12, 2016 at 9:27
• A piece of advice --- if you're reading a paper and they give a definition of a term you want to ask about, like "power law", mention either the definition or the reference, preferably both. Note that the paper you now mention offers a particular definition of power law that's different from the one in Clauset et al (from which my definition was taken). The definition in your paper yields an infinity of distributions which are only asymptotically power laws by the definition of Clauset et al. .... I have now edited my answer to address the issue. Nov 12, 2016 at 9:40