Power Law

Definition and classic examples

A power law is a relationship in which one variable is proportional to a power of another. Taking logarithms of both sides yields a straight line, so the exponent α is read as a slope on a log–log plot. The economist Vilfredo Pareto (1896) found income distributions across countries had this shape, the source of the Pareto principle that the top 20% hold 80%. The linguist George Kingsley Zipf (1949) described "Zipf's law": a word's frequency is inversely proportional to its rank, a pattern also found in city-size rankings. The Gutenberg–Richter law for earthquake magnitude and frequency (1944), citation counts (Price, 1965) and the biological scaling law that metabolic rate rises with body mass to the 3/4 power (West et al., 1997) are other classics.

Fat tails and generating mechanisms

Unlike the normal distribution, where values such as height cluster around the mean, power-law distributions produce values hundreds or thousands of times the mean, rarely but persistently. With small exponents the variance or even the mean can be infinite, so the usual form of the central limit theorem may not apply. Benoit Mandelbrot (1963) showed cotton price changes had far fatter tails than a normal distribution, with lasting influence on financial risk assessment. A leading explanation is cumulative advantage, the rich getting richer (Simon, 1955; Price, 1965), the same principle as the Matthew effect. Barabási and Albert (1999) recast it as preferential attachment in networks, and Bak, Tang and Wiesenfeld (1987) proposed self-organized criticality, in which systems like sandpiles drive themselves to a critical state.

Statistical testing and debate

The practice of calling a distribution a power law because it looks straight on a log–log plot has been heavily criticized. Clauset, Shalizi and Newman (2009) proposed methods based on maximum-likelihood estimation and goodness-of-fit tests and showed that many reported power laws cannot be distinguished from other fat-tailed distributions such as the lognormal. Broido and Clauset (2019) analyzed nearly 1,000 real networks and reported that strongly scale-free networks are rare, sparking debate. Even so, as Gabaix (2009) reviews, near-power-law regularities in firm size, city size and the tails of stock returns remain important clues in economics and complexity science. Newman's (2005) review is the standard introduction.

FAQ

Is a power law the same as the Pareto principle?

The Pareto principle (80/20 rule) is an empirical summary of one consequence of a power-law distribution. The power law is the more general mathematical relationship, and depending on the exponent the split may be quite different from 80/20.

If a log–log plot is a straight line, is it a power law?

Not necessarily. Other distributions such as the lognormal can look straight over parts of their range, so maximum-likelihood estimation, goodness-of-fit tests and comparison with alternatives are needed.

Sources reviewed: 2026-10-26

References

1. Pareto, V. (1896). Cours d'économie politique. Lausanne: F. Rouge.
2. Zipf, G. K. (1949). Human Behavior and the Principle of Least Effort. Cambridge, MA: Addison-Wesley.
3. Gutenberg, B., & Richter, C. F. (1944). Frequency of earthquakes in California. Bulletin of the Seismological Society of America, 34(4), 185–188.
4. Simon, H. A. (1955). On a class of skew distribution functions. Biometrika, 42(3/4), 425–440.
5. de Solla Price, D. J. (1965). Networks of scientific papers. Science, 149(3683), 510–515.
6. Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business, 36(4), 394–419.
7. Bak, P., Tang, C., & Wiesenfeld, K. (1987). Self-organized criticality: An explanation of the 1/f noise. Physical Review Letters, 59(4), 381–384.
8. Barabási, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286(5439), 509–512.
9. Newman, M. E. J. (2005). Power laws, Pareto distributions and Zipf's law. Contemporary Physics, 46(5), 323–351.
10. Gabaix, X. (2009). Power laws in economics and finance. Annual Review of Economics, 1, 255–294.
11. Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51(4), 661–703.
12. Broido, A. D., & Clauset, A. (2019). Scale-free networks are rare. Nature Communications, 10, 1017.
13. West, G. B., Brown, J. H., & Enquist, B. J. (1997). A general model for the origin of allometric scaling laws in biology. Science, 276(5309), 122–126.