Not all powerlaws have long tails

Many people seem to suffer from the misconception that every power law needs to have a long tail. This is simply not true.

To begin with, let's clarify the real item of interest: what we care about is the area under the tail, not the length of the tail itself. Tails can stretch on forever without mattering a whit, unless they subtend a large fraction of the total area under the curve.

Let's take a power law of the form: p(x) = Cx^-a, where p(x) dx is the probability that a random item has a value (e.g. wealth, usage) between x and x+dx. Let's take those individuals who represent the wealthiest fraction P of the population (where P lies between 0 and 1). It is elementary math to compute the fraction of the "wealth" that is concentrated among these individuals, and it turns out to be P^(a-2)/(a-1).

Here is a plot that shows the concentration at P=0.2 for differen values of a. For values of a that are less than 2, or just greater than 2, the concentration is very marked. For a = 2.1, we get the traditional 80/20 rule: 20% of people account for 80% of wealth.

For more details, I recommend this excellent discussion of the mathematics of power laws, from where I've taken the graph on the left.

Comments

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