Power law


In statistics, 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.

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 can be seen in the following thought experiment: imagine a room with your friends and estimate the average monthly income in the room. Now imagine the world's richest person entering the room, with a monthly income of about 1 billion US$. What happens to the average income in the room? Income is distributed according to a power-law known as the Pareto distribution.
On the one hand, this makes it incorrect to apply traditional statistics that are based on variance and standard deviation. 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 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/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. The behavior of these large events connects these quantities to the study of theory of large deviations, 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 or provide a simple way for observations to deviate from the power-law function :
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 is greater than 1, 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, biology, and the social sciences. Among them are:

Artificial Intelligence