Power law distribution

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A power law distribution is a continuous probability distribution with a long right tail. This means that there is a small number of extreme values that are much larger than the majority of the values. Power law distributions are often used to model phenomena that have a “rich get richer” or “winner takes all” dynamic, such as the distribution of wealth, income, and city sizes.

Density plot of 100000 random draws from a power law distribution (exponent = 3) [1].

What is the power law?

The power law, also called the scaling law, refers to a phenomenon where only a few items hold a majority of resources – typically around 95%. This means that while rare, larger occurrences do sometimes happen, but mostly, a small number of occurrences are more common. For instance, when we consider income distribution, billionaires are few and far between, and the majority of people possess modest nest eggs.

The power law asserts that a relative change in one quantity gives rise to a proportional relative change in another. The law can be demonstrated with a simple square, where a doubling of a side length (e.g., from 2 to 4 inches) leads to quadrupling of the area (in this example, 4 to 16 sq. inches).

The power law follows the formula

Y = k Xα,

where k is a constant and α is the exponent of the law.

Any relationship inverse in nature such as Y = X-1 is also considered a power law as a change in one amount leads to a negative change in the other.

If you plot two quantities against each other with logarithmic axes and they show a linear relationship, this is a sign that the two quantities have a power law distribution.

Log-log plot of the cumulative distribution of 100000 random draws from a power-law distribution (exponent = 3) [1]

Power law distribution uses

Power law distributions are often used in finance to model the distribution of asset prices. For example, the distribution of stock prices is often found to be power law distributed [2]. This means that there are a small number of stocks that are much more valuable than the majority of stocks.

Power law distributions are also used in other areas where extreme values are common, such as the distribution of :

The Pareto Principle (or Pareto Law) is an unscientific “law” that refers to a special type of distribution where 80% of effects come from 20% of causes. This means that the majority of our actions have little effect. Another related law is Zipf’s law, which describes the frequency of an event relative to its rank. Specifically, the law states that given a list of the most frequent words in a book, the most frequent word appears twice as often as the second most frequent word, which appears twice as often as the third most frequent and so on. Zipf’s law is a form of power law and gives rise to the Zeta Distribution.


[1] Giuseppe.Vittucci, CC BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via Wikimedia Commons

[2] Stanley, E. The Distribution of Stock Prices.

[3] Cordoba, J. On the distribution of city sizes. Journal of Urban Economics. Volume 63, Issue 1, January 2008, Pages 177-197

[4] Mahanti, A et al. A Tale of the Tails: Power-laws in Internet Measurements. Retrieved May 20, 2023 from: https://www.ida.liu.se/~nikca89/papers/IEEENetwork2013.pdf

[5] Zalohar, J. Chapter 14 – Gutenberg-Richter’s Law. Developments in Structural Geology and Tectonics. Volume 2, 2018, Pages 173-178

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