In mathematics and science, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent: one quantity varies as a power of another. The change is independent of the initial size of those quantities.
For instance, the area of a square has a power law relationship with the length of its side, since if the length is doubled, the area is multiplied by 2, while if the length is tripled, the area is multiplied by 3, and so on.
Empirical examples The distributions of a wide variety of physical, biological, and human-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, cloud sizes, 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, volcanic eruptions, human judgments of stimulus intensity and many other quantities. Empirical distributions can only fit a power law for a limited range of values, because a pure power law would allow for arbitrarily large or small values. 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. Properties
Statistical incompleteness The power-law model does not obey the treasured paradigm of statistical completeness. Especially probability bounds, the suspected cause of typical bending and/or flattening phenomena in the high- and low-frequency graphical segments, are parametrically absent in the standard model.Van Droogenbroeck, Frans J., 'Completion of the standard power-law model by including upper and lower probability bounds' (2023).
Scale invariance One attribute of power laws is their scale invariance. Given a relation , scaling the argument by a constant factor causes only a proportionate scaling of the function itself. That is,
where denotes direct proportionality. That is, scaling by a constant simply multiplies the original power-law relation by the constant . Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both and , and the straight-line on the log–log plot is often called the signature of a power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation. In fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws. Thus, accurately fitting and validating power-law models is an active area of research in statistics; see below.
Lack of well-defined average value A power-law has a well-defined mean over only if , and it has a finite variance only if ; most identified power laws in nature have exponents such that the mean is well-defined but the variance is not, implying they are capable of black swan behavior. This is exemplified by the effect, on the mean income, of the inclusion of the person with the highest income in the world into a sample of individuals of otherwise similar income. Income is distributed according to a power-law known as the Pareto distribution (for example, the net worth of Americans is distributed according to a power law with an exponent of 2).
On the one hand, this makes it incorrect to apply traditional statistics that are based on variance and standard deviation (such as regression analysis). On the other hand, this also allows for cost-efficient interventions. For example, given that car exhaust is distributed according to a power-law among cars (very few cars contribute to most contamination) it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially.
The median does exist, however: for a power law x –k, with exponent , it takes the value 21/(k – 1)xmin, where xmin is the minimum value for which the power law holds.
Universality The equivalence of power laws with a particular scaling exponent can have a deeper origin in the dynamical processes that generate the power-law relation. In physics, for example, phase transitions in thermodynamic systems are associated with the emergence of power-law distributions of certain quantities, whose exponents are referred to as the critical exponents of the system. Diverse systems with the same critical exponents—that is, which display identical scaling behaviour as they approach criticality—can be shown, via renormalization group theory, to share the same fundamental dynamics. For instance, the behavior of water and CO2 at their boiling points fall in the same universality class because they have identical critical exponents. In fact, almost all material phase transitions are described by a small set of universality classes. Similar observations have been made, though not as comprehensively, for various self-organized critical systems, where the critical point of the system is an attractor. Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.
Power-law functions Scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them. The demonstration of a power-law relation in some data can point to specific kinds of mechanisms that might underlie the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems; see also universality above. The ubiquity of power-law relations in physics is partly due to dimensional constraints, while in complex systems, power laws are often thought to be signatures of hierarchy or of specific stochastic processes. A few notable examples of power laws are Pareto's law of income distribution, structural self-similarity of fractals, scaling laws in biological systems, and scaling laws in cities. Research on the origins of power-law relations, and efforts to observe and validate them in the real world, is an active topic of research in many fields of science, including physics, computer science, linguistics, geophysics, neuroscience, systematics, sociology, economics and more.
However, much of the recent interest in power laws comes from the study of probability distributions: The distributions of a wide variety of quantities seem to follow the power-law form, at least in their upper tail (large events). The behavior of these large events connects these quantities to the study of theory of large deviations (also called extreme value theory), which considers the frequency of extremely rare events like stock market crashes and large natural disasters. It is primarily in the study of statistical distributions that the name "power law" is used.
In empirical contexts, an approximation to a power-law often includes a deviation term , which can represent uncertainty in the observed values (perhaps measurement or sampling errors) or provide a simple way for observations to deviate from the power-law function (perhaps for stochastic reasons):
Mathematically, a strict power law cannot be a probability distribution, but a distribution that is a truncated power function is possible: for where the exponent (Greek letter alpha, not to be confused with scaling factor used above) is greater than 1 (otherwise the tail has infinite area), the minimum value is needed otherwise the distribution has infinite area as x approaches 0, and the constant C is a scaling factor to ensure that the total area is 1, as required by a probability distribution. More often one uses an asymptotic power law – one that is only true in the limit; see power-law probability distributions below for details. Typically the exponent falls in the range , though not always.
Examples 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: Artificial Intelligence Neural scaling law Astronomy Kepler's third law The initial mass function of stars The differential energy spectrum of cosmic-ray nuclei The M–sigma relation Solar flares
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 The size of forest patches globally follows a power law The species–area relationship relating the number of species found in an area as a function of the size of the area
Chemistry Rate law
Climate science Sizes of cloud areas and perimeters, as viewed from space The size of rain-shower cells Energy dissipation in cyclones Diameters of dust devils on Earth and Mars
General science 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)Sutton, J. (1997), "Gibrat's Legacy", Journal of Economic Literature XXXV, 40–59. 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 Associated with exponential growth: Tails in statistical distributions for exponential growth processes with random observation (or killing) Progress through exponential growth and exponential diffusion of innovations
Economics Population sizes of cities in a region or urban network, Zipf's law. Distribution of artists by the average price of their artworks. Income distribution in a market economy. Distribution of degrees in banking networks. Firm-size distributions. Scaling laws of socio-economic quantities with respect to population size (see urban scaling).
Finance
Returns for high-risk venture capital investments The mean absolute change of the logarithmic mid-prices Large price changes, volatility, and transaction volume on stock exchanges Average waiting time of a directional change Average waiting time of an overshoot
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
Physics The Angstrom exponent in aerosol optics The frequency-dependency of acoustic attenuation in complex media The Stefan–Boltzmann law