Fractal (Power Law) Distributions
In fractal or power law distributions, statistical self-similarity means the degree of inequality remains constant across scales, unlike Gaussian distributions where it decreases. The exponent parameter critically determines the concentration of values in the tail, with small changes leading to dramatic differences in top shares.
THE IDEA
The pure fractal statistical mountain illustrates that inequality is scale-invariant: disparities in wealth (or any quantity) are the same at all levels—billionaires are not more equal relative to each other than millionaires are, and so on. This is statistical self-similarity.
Table 2 shows assumed exponents for various phenomena, e.g., book sales (1.5), wealth (1.1), company size (1.5), market moves (3). These exponents are approximate and hard to measure precisely.
| TABLE 2: ASSUMED EXPONENTS FOR VARIOUS PHENOMENA* | |
| Phenomenon | Assumed Exponent (vague approximation) |
| Frequency of use of words | 1.2 |
| Number of hits on websites | 1.4 |
| Number of books sold in the U.S. | 1.5 |
| Telephone calls received | 1.22 |
| Magnitude of earthquakes | 2.8 |
| Diameter of moon craters | 2.14 |
| Intensity of solar flares | 0.8 |
| Intensity of wars | 0.8 |
| Net worth of Americans | 1.1 |
| Number of persons per family name | 1 |
| Population of U.S. cities | 1.3 |
| Market moves | 3 (or lower) |
| Company size | 1.5 |
| People killed in terrorist attacks | 2 (but possibly a much lower exponent) |
| *Source: M.E.J. Newman (2005) and the author’s own calculations. |
Table 3 demonstrates the impact of the exponent on inequality. The lower the exponent, the larger the share of the top 1% and top 20%. Small changes in the exponent (e.g., from 1.1 to 1.3) cause huge differences (top 1% share drops from 66% to 34%).
| TABLE 3: THE MEANING OF THE EXPONENT | ||
| Exponent | Share of the top 1% | Share of the top 20% |
| 1 | 99.99%* | 99.99% |
| 1.1 | 66% | 86% |
| 1.2 | 47% | 76% |
| 1.3 | 34% | 69% |
| 1.4 | 27% | 63% |
| 1.5 | 22% | 58% |
| 2 | 10% | 45% |
| 2.5 | 6% | 38% |
| 3 | 4.6% | 34% |
These exponents are estimates with large sampling error; the crossover point (where the power law begins) is also uncertain. Even without a precise exponent, possible scaling can matter for a decision. Whether a power law fits, and over which range, still needs examination.
When to use it
Insurance companies use power law models to estimate the probability of catastrophic events like large earthquakes or pandemics.
Economists apply power law distributions to model wealth inequality and predict the effects of tax policies on the top shares.
Risk managers can compare heavy-tail models for portfolio losses and examine how uncertainty about the tail changes reserve estimates.
Internet companies analyze website hits or social media popularity using power laws to optimize server capacity and marketing strategies.
FROM THEORY TO EVERYDAY LIFE
In practice
Book Sales Distribution
The source gives an approximate power-law exponent of 1.5 for U.S. book sales. In that model, a few books account for many sales, and the relative concentration persists as the sales threshold rises.
Wealth Inequality in the United States
Using an assumed exponent of 1.1, the source’s illustration assigns the top 1% about 66% of wealth and the top 20% about 86%. These figures illustrate the model’s sensitivity to the exponent; they are not presented here as current measured wealth shares.
A MOMENT TO REFLECT
Put it to work.
How does the assumption of a power law distribution change your risk assessment for rare events compared to a Gaussian assumption?
What are some real-world phenomena you encounter that might follow a fractal distribution, and how could you test for self-similarity?
Why is it difficult to precisely estimate the exponent of a power law, and what are the practical consequences of that uncertainty?
Often confused with
References
- Power laws, Pareto distributions and Zipf's law — M.E.J. Newman (2005)
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)