The Logic of Fractal Randomness (with a Warning)
Fractal distributions (power laws) exhibit scalability where the ratio of exceedances above two thresholds is given by the ratio of the thresholds to the power of the negative exponent, allowing estimation of rare events from known data.
THE IDEA
For a power-law distribution, the ratio of the numbers of observations above two thresholds equals the threshold ratio raised to the negative power-law exponent. For example, if only 96 books a year sell more than 250,000 copies (the exceedance at 250,000), and the exponent is 1.5, then:
- Books selling more than 500,000 copies: 96 × (500,000 / 250,000)^(-1.5) ≈ 34.
- Books selling more than 1,000,000 copies: 96 × (1,000,000 / 250,000)^(-1.5) ≈ 12. This property holds for any power-law-distributed quantity.
When to use it
Predicting the number of earthquakes above a given magnitude using the Gutenberg-Richter law.
Estimating the number of billionaires in a country from wealth distribution data.
Forecasting the frequency of extreme stock market moves based on historical tail behavior.
Assessing the risk of very large losses in insurance (e.g., natural catastrophes).
FROM THEORY TO EVERYDAY LIFE
In practice
Estimating book sales exceedances
Assume 96 books sell more than 250,000 copies per year, with a power-law exponent of 1.5. Then the estimated number selling more than 500,000 copies is 96 × (500,000/250,000)^(-1.5) = 34, and more than 1,000,000 is 96 × (1,000,000/250,000)^(-1.5) = 12.
A MOMENT TO REFLECT
Put it to work.
Can you think of a real-world dataset (e.g., city populations, website traffic) that might follow a power law? How would you estimate the exponent?
What are the dangers of assuming a power law when the underlying distribution is actually Gaussian?
In what situations might the warning in the title 'with a warning' be most relevant?
CONTINUE THE CONVERSATION
Source texts
- The Black Swan The Impact of the Highly Improbable With a new section On Robustness and Fragility (Nassim Nicholas Taleb)