Power Law Distributions

Charlottesville, Virginia

April 26, 2020

I think this post falls under the category of "if you want to feel smart." I had never heard of power law distributions prior to this week. I guess I fell into the trap of assuming more subjects qualify as normal distribution than is true. Bummer.

In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.

The distributions of a wide variety of physical, biological, and man-made phenomena approximately follow a power law over a wide range of magnitudes: these include the sizes of craters on the moon and of solar flares, the foraging pattern of various species, the sizes of activity patterns of neuronal populations, the frequencies of words in most languages, frequencies of family names, the species richness in clades of organisms, the sizes of power outages, criminal charges per convict, volcanic eruptions, human judgements of stimulus intensity and many other quantities. Few empirical distributions fit a power law for all their values, but rather follow a power law in the tail. Acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media. Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature.

More examples [Lord knows I don't know what most of these are!]

More than a hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income). Among them are:

Astronomy

• Kepler's third law

• The initial mass function of stars

• The differential energy spectrum of cosmic-ray nuclei

• The M-sigma relation

Criminology

• number of charges per criminal offender

Physics

• The Angstrom exponent in aerosol optics

• The frequency-dependency of acoustic attenuation in complex media

• Stevens's power law of psychophysics

• The Stefan–Boltzmann law

• The input-voltage–output-current curves of field-effect transistors and vacuum tubes approximate a square-law relationship, a factor in "tube sound".

• Square-cube law (ratio of surface area to volume)

• A 3/2-power law can be found in the plate characteristic curves of triodes.

• The inverse-square laws of Newtonian gravity and electrostatics, as evidenced by the gravitational potential and Electrostatic potential, respectively.

• Self-organized criticality with a critical point as an attractor

• Model of van der Waals force

• Force and potential in simple harmonic motion

• Gamma correction relating light intensity with voltage

• Behaviour near second-order phase transitions involving critical exponents

• The safe operating area relating to maximum simultaneous current and voltage in power semiconductors.

• Supercritical state of matter and supercritical fluids, such as supercritical exponents of heat capacity and viscosity.

• The Curie-von Schweidler law in dielectric responses to step DC voltage input.

• The damping force over speed relation in antiseismic dampers calculus

• Radioactive decay equation

Biology

• Kleiber's law relating animal metabolism to size, and allometric laws in general

• The two-thirds power law, relating speed to curvature in the human motor system.

• The Taylor's law relating mean population size and variance of populations sizes in ecology

• Neuronal avalanches

• The species richness (number of species) in clades of freshwater fishes

• The Harlow Knapp effect, where a subset of the kinases found in the human body compose a majority of published research

Meteorology

• The size of rain-shower cells, energy dissipation in cyclones, and the diameters of dust devils on Earth and Mars

General science

• Exponential growth and random observation (or killing)

• Progress through exponential growth and exponential diffusion of innovations

• Highly optimized tolerance

• Proposed form of experience curve effects

• Pink noise

• The law of stream numbers, and the law of stream lengths (Horton's laws describing river systems)

• Populations of cities (Gibrat's law)

• Bibliograms, and frequencies of words in a text (Zipf's law)

• 90–9–1 principle on wikis (also referred to as the 1% rule)

• Richardson's Law for the severity of violent conflicts (wars and terrorism)

• The relationship between a CPU's cache size and the number of cache misses follows the power law of cache misses.

• The spectral density of the weight matrices of deep neural networks

Mathematics

• Fractals

• Pareto distribution and the Pareto principle also called the "80–20 rule"

• Zipf's law in corpus analysis and population distributions amongst others, where frequency of an item or event is inversely proportional to its frequency rank (i.e. the second most frequent item/event occurs half as often as the most frequent item, the third most frequent item/event occurs one third as often as the most frequent item, and so on).

• Zeta distribution (discrete)

• Yule–Simon distribution (discrete)

• Student's t-distribution (continuous), of which the Cauchy distribution is a special case

• Lotka's law

• The scale-free network model

Economics

• Population sizes of cities in a region or urban network, Zipf's law.

• Distribution of artists by the average price of their artworks.

• Distribution of income in a market economy.

• Distribution of degrees in banking networks.

Finance

• The mean absolute change of the logarithmic mid-prices

• Number of tick counts over time

• Size of the maximum price move

• Time of directional change

• Time of overshoot