The Power-Law Distribution

Pareto’s Power-Law Distribution [source]

Explaining the Laws of Nature (Including the Golden Ratio)

The laws of nature are complicated and throughout time, Scientists from all corners of the world have attempted to model and reengineer what they see around them to extract some value from it. Quite often we see a pattern that comes up time and time again: be it the golden ratio, or be it that fractal spiral.

In its empirical form, the Power Law describes how a lot of the time, not much actually happens but more often than not, some patterns cover a wide range of magnitudes. Think about the following relatable examples:

  1. Number of Comments on a Post
  2. Number of Social Media Followers
  3. Money Grossed at Box Offices
  4. Books sold for Top Authors
  5. Market Capitalisation of American Companies

They all seem to fit the pattern, but we can also see these examples widely in the form of natural phenomena:

  1. Ratio of Surface Area to Volume
  2. Fractal Geometry
  3. Initial Mass Function of Stars
  4. Distribution of Wealth

There are plenty more of these examples but the thing that really begs the question is, what is actually driving this skewed phenomena? Why is there a lot of density at low values and why, at other times, do we get incidences at really far ends of the tail?

Let’s first cover the mathematics of the it, before discussing it more in detail.


Probability Distribution of Power-Laws

A random variable that characterises a power-law Distribution can be defined as follows:

where C is the normalisation constant C=(a-1)x^α-1. This equation only really makes sense if α>1, but as a more compact form, we can also write it as follows:

Note that from this functional form, we can see that if we apply a logarithmic function, the functional form becomes linear:

Scale Invariance of Power Law Distributions

So if we compare the densities at p(x) and at some other p(c x), where c is a constant, we find that they’re always proportional. That is, p(c x) ∝ p(x). This behaviour shows us that the relative likelihood between small and large events is the same, no matter what choice of “small” we make.

Note: I’m not going to go into detail about the moments of Power Distributions because the phenomena surrounding these ‘infinite’ moments is fascinating and requires an article of its own!

Let’s look back at some examples and see what we find: Pinto et al (2012) show a few examples where we can really see the distribution come into its own:

Log-log plot of distribution of wealth: Pinto et al (2012): [Source]

and here for forest fires:

Log-log plot of distribution of forest fires: Pinto et al (2012): [Source]

It’s crazy how well the power distribution fits these phenomena and how linear the log-log plots are. I encourage the reader to try this out for themselves: it’s always surprising what you find!


Golden Ratio

The Golden Ratio or ‘80–20’ rule exists as a colloquial natural phenomena. It postulates things like: 20% of the worlds population own 80% of the wealth. Let’s assume for a second that wealth is defined by the Power Law which and is characterised by some α. What fraction W of the total wealth is held by the richest fraction P of the population?

Now we can integrate the power-law function above to derive the fraction of the population whose wealth is at least x, given by the cumulative distribution function:

Moreover, the fraction wealth held by those people is given by:

where α>2. If we now solve the first equation and substitute it into the second, we find an expression that does not depend on wealth (x) at at all:

Now this is crazy to me: by making small assumptions about the distributional properties of wealth distribution, we can remove wealth from the equation and still show how wealth is spread. This extreme top-heaviness is sometimes called the “80–20 rule,” meaning that 80% of the wealth is in the hands of the richest 20% of people.

As an example, say we want to know how much wealth the top 20% of richest people own? Let’s set α=2.2, then (α-2)/(α-1) = 0.2/1.2 = 0.167. Then we set P = 20%, so W = 20% to the power of 1/3, = 76%, which is not far off 80%! Funnily enough, this is actually a pretty good fit for society.

Note: that the relationship can skew if we change the value of α, becoming more extreme as α<2, which shows that wealth is held by a single person.

It’s exactly because of this this functional form being so unique in nature and so eloquent, that we can simplify characteristics as ‘80–20’. It’s not an exact science but social science rarely is. However, deriving an α for these social dynamics goes a long way in telling us exactly how these natural phenomena realise and act.


The Power-Law Distribution is phenomenal because small insights to its functional form can lead to incredibly detailed explanations of natural phenomena.

From Statistical Physics to man-made artefacts like comments on blogs, these all seem to share an underlying level of respect. Moreover, its scale invariance and log-log view shows how accessible these complicated models can be.

I’ve only scratched the surface of this model, but please keep reading as there’s so much more out there.


Thanks again for reading! If you have any questions, please message!

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