A power law is any polynomial relationship that exhibits the property of scale invariance. The most common power laws relate two variables and have the form
where and are constants, and is of . Here, is typically called the "scaling exponent", the word "scaling" denoting the fact that a power-law function satisfies where is a constant. That is, a rescaling of the function's argument changes the constant of proportionality but preserves the shape of the function itself. This point becomes clearer if we take the logarithm of both sides:
Notice that this expression has the form of a linear relationship with slope . Rescaling the argument produces a linear shift of the function up or down but leaves both the basic form and the slope unchanged.
Power-law relations characterize a staggering number of naturally occurring phenomena, and this is one of the principal reasons why they have attracted interest. For instance, inverse-square laws, such as gravitation and the Coulomb force, are power laws, as are many common mathematical formulae such as the quadratic law of area of the circle. However it is mainly in the study of probability distributions that power laws have attracted recent interest. A wide variety of observed probability distributions appear, at least approximately, to have tails asymptotically following power-law forms, an observation connected closely with 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 other areas the power-law functional form is more often referred to simply as a polynomial form or polynomial function.
Scientific interest in power law relations also derives from the ease with which certain general classes of mechanisms can generate them, so that the observation of a power-law relation in data often points to specific kinds of mechanisms that might underly the natural phenomenon in question, and can indicate a deep connection with other, seemingly unrelated systems (see the reference by Simon and the subsection on universality below). 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 the Gutenberg-Richter law for earthquake sizes, Pareto's law of income distribution, structural self-similarity of fractals, and scaling laws in biological systems. 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, sociology, economics and more.
Properties of power laws
The main property of power laws that makes them interesting is their scale invariance. Given a relation , or, indeed any homogeneous polynomial, scaling the argument by a constant factor causes only a proportionate scaling of the function itself. That is,
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 both logarithms are taken of both and , and the straight-line on the log-log plot is often called the "signature" of a power law. Notably, however, with real data, such straightness is necessary, but not a 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.
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.
The general power-law function follows the polynomial form given above, and is a ubiquitous form throughout mathematics and science. Notably, however, not all polynomial functions are power laws because not all polynomials exhibit the property of scale invariance. Typically, power-law functions are polynomials in a single variable, and are explicitly used to model the scaling behavior of natural processes. For instance, allometric scaling laws for the relation of biological variables are some of the best known power-law functions in nature. In this context, the term is most typically replaced by 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 no power-law function (perhaps for stochastic reasons):
Estimating the exponent from empirical data
There are many methods for fitting power-law functions to data, and the best option typically depends strongly on the kind of question being asked. For instance, prediction-type questions should rely on nonlinear regression, while descriptive-type summary questions, such as those found in allometry, should use a method that allows for uncertainty in both the and measurements. If the residuals are log normally distributed, e.g. if the spread in is multiplicative (increasing proportionally with ), a simple least-squares linear regression on log-transformed data can be performed, since the log transformed residues are normally distributed after transformation. Otherwise, the logarithmic transformation produces residuals that are log-normally distributed, while the least squares method requires normally distributed errors. In this latter context, the method of standardized major axis (SMA) regression (sometimes called "reduced major axis", but this term should be avoided) is preferred.
The major axis is the linear equation that minimizes the sum of squares of the shortest (perpendicular) distance between data points and the equation. This axis is equivalent to the first principal component axis of the covariance matrix. From this observation, the estimator for the slope can be derived
where and are the sample means of the and data, respectively.
More about this method, and the conditions under which it can be used, can be found in the Warton reference below. Further, Warton's comprehensive review article also provides [http://web.maths.unsw.edu.au/~dwarton/programs.html usable code] (C++, R, and Matlab) for estimation and testing routines for power-law functions.
Examples of power law functions
*The Stefan-Boltzmann law
*The Gompertz Law of Mortality
*The Ramberg-Osgood stress-strain relationship
*The Inverse-square law of Newtonian gravity
*The Initial mass function
*Gamma correction relating light intensity with voltage
*Kleiber's law relating animal metabolism to size, and allometric laws in general
*Behaviour near second-order phase transitions involving critical exponents
*Proposed form of experience curve effects
*The differential energy spectrum of cosmic-ray nuclei
A power-law distribution is any that, in the most general sense, has the form
where , and is a slowly varying function, which is any function that satisfies with constant. This property of follows directly from the requirement that be asymptotically scale invariant; thus, the form of only controls the shape and finite extent of the lower tail. For instance, if is the constant function, then we have a power-law that holds for all values of . In many cases, it is convenient to assume a lower bound from which the law holds. Combining these two cases, and where is a continuous variable, the power law has the form
where the constant is necessary to guarantee that the distribution is properly normalized. Briefly, we can consider several properties of this distribution.
In general, the moments of this distribution are given by
which is only well defined for . That is, all moments diverge: when , the average and all higher-order moments are infinite; when