NationMaster - Encyclopedia: Power law

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 In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In physics, scale invariance is the feature of physical objects of laws that do not change if the space is magnified, i. ...

where

a

and

k

are constants, and

o (x k)

is an asymptotically small function of

x

. Here,

k

is typically called the scaling exponent, denoting the fact that a power-law function (or, more generally, a

k

th order homogeneous polynomial) satisfies the criteria where

c

is a constant. That is, scaling the function's argument changes the constant of proportionality as a function of the scale change, but preserves the shape of the function itself. This relationship becomes more clear if we take the logarithm of both sides (or, graphically, plotting on a log-log graph) For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... In mathematics, a homogeneous polynomial is a polynomial whose terms are monomials all having the same total degree; or are elements of the same dimension. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... A log-log plot of y=x (green), y=x^2 (blue), and y=x^3 (red). ...

Notice that this expression has the form of a linear relationship with slope

k

., and scaling the argument induces a linear shift (up or down) of the function, and leaves both the form and slope

k

unchanged. â€œLineâ€ redirects here. ...

Power-law relations characterize a staggering number of natural patterns, and it is primarily in this context that the term power law is used rather than polynomial function. 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. Also, many probability distributions have tails that asymptotically follow power-law relations, a topic that connects tightly with the 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. This diagram shows how the law works. ... â€œGravityâ€ redirects here. ... In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrical force that one stationary, electrically charged substance of small volume (ideally, a point source) exerts on another. ... Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... In mathematics and applications, particularly the analysis of algorithms, asymptotic analysis is a method of classifying limiting behaviour, by concentrating on some trend. ... Extreme value theory is a branch of statistics dealing with the extreme deviations from the median of probability distributions. ... Extreme value theory is a branch of statistics dealing with the extreme deviations from the median of probability distributions. ... Black Monday (1987) on the Dow Jones Industrial Average A stock market crash is a sudden dramatic decline of stock prices across a significant cross-section of a stock market. ... Mount Pinatubo eruption, 1991 A natural disaster is the consequence of a natural hazard (e. ...

Scientific interest in power law relations, whether functions or distributions, comes primarily from the ease with which certain general classes of mechanisms can generate them. That is, the observation of a power-law relation in data often points to specific kinds of mechanisms that underly the natural phenomenon in question, and can often indicate a deep connection with other, seemingly unrelated systems (for instance, see both 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 and robustness. A few notable examples of power laws are the Gutenberg-Richter law for earthquake sizes, or 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 extremely active in many fields of modern science, including physics, computer science, linguistics, geophysics, sociology, economics and more. Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ... Complex systems have a number of properties, some of which are listed below. ... In seismology, the Gutenberg-Richter law states that the number of earthquakes per year of Richter magnitude M statistically has the form Number of earthquakes of size M per year ~ exp(a - bM) where exp is the exponential function. ... A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. ... Allometric law (or power-law) are relationships between living organisms body parts or process, usually expressed in power-law form: or in a logarithmic form: Examples allometric law of cruising speed vs body mass Some examples of allometric laws: Kleibers law, the proportionality between metabolic rate and body... This is a discussion of a present category of science. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... Linguistics is the scientific study of language, which can be theoretical or applied. ... â€¹ The template below has been proposed for deletion. ... Sociology (from Latin: socius, companion; and the suffix -ology, the study of, from Greek Î»ÏŒÎ³Î¿Ï‚, lÃ³gos, knowledge) is an academic and applied discipline that studies society and human social interaction. ... â€¹ The template below is being considered for deletion. ...

Properties of power laws

Scale invariance

The main property of power laws that makes them interesting is their scale invariance. Given a relation

f (x) = a x k

, or, indeed any homogeneous polynomial, scaling the argument

x

by a constant factor causes only a proportionate scaling of the function itself. That is, In physics, scale invariance is the feature of physical objects of laws that do not change if the space is magnified, i. ... In mathematics, a homogeneous polynomial is a polynomial whose terms are monomials all having the same total degree; or are elements of the same dimension. ...

That is, scaling by a constant simply multiplies the original power-law relation by the constant

c k

. 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

f (x)

and

x

, 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. This article is about the field of statistics. ...

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. In physics, a phase transition, (or phase change) is the transformation of a thermodynamic system from one phase to another. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... In physical chemistry, thermodynamics, chemistry and condensed matter physics, a critical point, also called a critical state, specifies the conditions (temperature, pressure) at which the liquid state of the matter ceases to exist. ... In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ... The theory of self-organized criticality (SOC) claims that whenever a self-organizing dynamical system is open or dissipative, it exhibits critical (scale-invariant) behaviour similar to that displayed by static systems undergoing a second-order phase transition. ... In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ... In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. ... In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. ...

Power-law functions

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

o (x k)

term is most typically replaced by a deviation term

ε

, 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): Allometric law (or power-law) are relationships between living organisms body parts or process, usually expressed in power-law form: or in a logarithmic form: Examples allometric law of cruising speed vs body mass Some examples of allometric laws: Kleibers law, the proportionality between metabolic rate and body... In the mathematics of probability, a stochastic process is a random function. ...

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

x

and

y

measurements. If the residuals are log normally distributed, e.g. if the spread in

y

is multiplicative (increasing proportionally with

x

), 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. dataset with approximating polynomials Nonlinear regression in statistics is the problem of fitting a model to multidimensional x,y data, where f is a nonlinear function of x with parameters Î¸. In general, there is no algebraic expression for the best-fitting parameters, as there is in linear regression. ... Allometry is the science studying the differential growth rates of the parts of a living organisms body part or process. ... In statistics, linear regression is a regression method that models the relationship between a dependent variable Y, independent variables Xp, and a random term Îµ. The model can be written as where Î²1 is the intercept (constant term), the Î²is are the respective parameters of independent variables, and p is the... In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed (the base of the logarithmic function is immaterial in that loga X is normally distributed if and only if logb X is normally distributed). ... In geometry, the semi-major axis (also semimajor axis) a applies to ellipses and hyperbolas. ...

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 In statistics, principal components analysis (PCA) is a technique that can be used to simplify a dataset; more formally it is a linear transformation that chooses a new coordinate system for the data set such that the greatest variance by any projection of the data set comes to lie on... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ...

where

μ x

and

μ y

are the sample means of the

x

and

y

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 usable code (C++, R, and Matlab) for estimation and testing routines for power-law functions.

Examples of power law functions

The Stefan-Boltzmann law, also known as Stefans law, states that the total energy radiated per unit surface area of a black body in unit time (known variously as the black-body irradiance, energy flux density, radiant flux, or the emissive power), j*, is directly proportional to the fourth... The Gompertz-Makeham law states that death rate is a sum of age-independent component (Makeham term) and age-dependent component (Gompertz function), which increases exponentially with age. ... The Ramberg-Osgood equation was created to describe the non linear relationship between stress and strainâ€”that is, the stress-strain curveâ€”in materials near their yield points. ... This diagram shows how the law works. ... The law of universal gravitation states that gravitational force between masses decreases with the distance between them, according to an inverse-square law. ... Example of CRT gamma correction Plot of the sRGB standard gamma-expansion nonlinearity (red), and its local gamma value, slope in logâ€“log space (blue). ... Kleibers law, named after Max Kleibers biological work in the early 1930s, is the observation that, for the vast majority of animals, an animals metabolic rate scales to the 3/4 power of the animals mass. ... Allometric law (or power-law) are relationships between living organisms body parts or process, usually expressed in power-law form: or in a logarithmic form: Examples allometric law of cruising speed vs body mass Some examples of allometric laws: Kleibers law, the proportionality between metabolic rate and body... In physics, a phase transition, (or phase change) is the transformation of a thermodynamic system from one phase to another. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... The learning curve effect and the closely related experience curve effect express the relationship between experience and efficiency. ... The energy spectrum for cosmic rays The Moons cosmic ray shadow, as seen in secondary muons detected 700m below ground, at the Soudan 2 detector Cosmic rays are energetic particles originating from space that impinge on Earths atmosphere. ... This diagram shows how the law works. ... The square-cube law is a principle, drawn from the mathematics of proportion, that is applied in engineering and biomechanics. ... Constructal design of a cooling system The constructal theory of global optimization under local constraints explains in a simple manner the shapes that arise in nature. ... The boundary of the Mandelbrot set is a famous example of a fractal. ...

Power-law distributions

where

α > 1

, and

L (x)

is a slowly varying function, which is any function that satisfies with

t

constant. This property of

L (x)

follows directly from the requirement that

p (x)

be asymptotically scale invariant; thus, the form of

L (x)

only controls the shape and finite extent of the lower tail. For instance, if

L (x)

is the constant function, then we have a power-law that holds for all values of

x

. In many cases, it is convenient to assume a lower bound

x min

from which the law holds. Combining these two cases, and where

x

is a continuous variable, the power law has the form

where the constant is necessary to guarantee that the distributions is properly normalized. Briefly, we can consider several properties of this distribution.

which is only well defined for

m < α - 1

. That is, all moments diverge: when

α < 2

, the average and all higher-order moments are infinite; when

2 < α < 3

, the mean exists, but the variance and higher-order moments are infinite, etc. For finite-size samples drawn from such distribution, this behavior implies that the central moment estimators (like the mean and the variance) for diverging moments will never converge - as more data is accumulated, they continue to grow.

Another kind of power-law distribution, which does not satisfy the general form above, is the power law with an exponential cutoff

where we introduce an exponential decay term

e - λ x

that overwhelms the power-law behavior at large values of

x

. This distribution does not scale and is thus not asymptotically a power law; however, it does approximately scale over a finite region before the cutoff. (Note that the pure form above is a subset of this family, with

λ = 0

.) This distribution is a common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects. For instance, although the Gutenberg-Richter Law is commonly cited as an example of a power-law distribution, the distribution of earthquake magnitudes cannot scale as a power law in the limit because there is a finite amount of energy in the Earth's crust. Thus, there must be some maximum size earthquake, and the scaling behavior must taper off beyond a slightly smaller value. Note that finiteness is even the deeper cause for many power-laws. In seismology, the Gutenberg-Richter law states that the number of earthquakes per year of Richter magnitude M statistically has the form Number of earthquakes of size M per year ~ exp(a - bM) where exp is the exponential function. ...

Plotting power-law distributions

In general, power-law distributions are plotted on doubly logarithmic axes, which emphasizes the upper tail region. The most convenient way to do this is via the (complementary) cumulative distribution (cdf),

P (x) = Pr(X > x)

, A log-log plot of y=x (green), y=x^2 (blue), and y=x^3 (red). ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

Note that the cdf is also a power-law distribution, but with a smaller scaling exponent. For data, an equivalent form of the cdf is the rank-frequency approach, in which we first sort the

n

observed values in ascending order, and plot them against the vector .

Although it can be convenient to log-bin the data, or otherwise smooth the probability density (mass) function directly, these methods introduce an implicit bias in the representation of the data, and thus should be avoided. The cdf, on the other hand, introduces no bias in the data and preserves the linear signature on doubly logarithmic axes.

Estimating the exponent from empirical data

There are many ways of estimating the value of the scaling exponent for a power-law tail, however not all of them yield unbiased and consistent answers. The most reliable techniques are often based on the method of maximum likelihood. Alternative methods are often based on making a linear regression on either the log-log probability, the log-log cumulative distribution function, or on log-binned data, but these approaches should be avoided as they can all lead to highly biased estimates of the scaling exponent (see the Goldstein et al. reference below). Maximum likelihood estimation (MLE) is a popular statistical method used to make inferences about parameters of the underlying probability distribution from a given data set. ... In statistics, the method of maximum likelihood, pioneered by geneticist and statistician Sir Ronald A. Fisher, is a method of point estimation, that uses as an estimate of an unobservable population parameter the member of the parameter space that maximizes the likelihood function. ...

to the data . Given a choice for

x min

, a simple derivation by this method yields the estimator equation

where

{x i}

are the

n

data points . (For a more detailed derivation, see Hall or Newman below.) This estimator exhibits a small finite sample-size bias of order

O (n - 1)

, which is small when

n > 100

. Further, the uncertainty in the estimation can be derived from the maximum likelihood argument, and has the form . This estimator is equivalent to the popular Hill estimator from quantitative finance and extreme value theory. Financial mathematics is the branch of applied mathematics concerned with the financial markets. ... Extreme value theory is a branch of statistics dealing with the extreme deviations from the median of probability distributions. ...

For a set of

n

integer-valued data points

{x i}

, again where each , the maximum likelihood exponent is the solution to the transcendental equation

where

ζ(α, x min)

is the incomplete zeta function. The uncertainty in this estimate follows the same formula as for the continuous equation. However, the two equations for are not equivalent, and the continuous version should not be applied to discrete data, nor vice versa. In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ...

Further, both of these estimators require the choice of

x min

. For functions with a non-trivial

L (x)

function, choosing

x min

too small produces a significant bias in , while choosing it too small increases the uncertainty in , and reduces the statistical power of our model. In general, the optimum choice of

x min

depends strongly on the particular form of the lower tail, represented by

L (x)

above. The power of a statistical test is the probability that the test will reject a false null hypothesis (that it will not make a Type II error). ...

Examples of power-law distributions

A great many power-law distributions have been conjectured in recent years. For instance, power laws are thought to characterize the behavior of the upper tails for the popularity of websites, number of species per genus, the popularity of given names, the size of financial returns, and many others. However, much debate remains as to which of these tails are actually power-law distributed and which are not. For instance, it is commonly accepted now that the famous Gutenberg-Richter Law decays more rapidly than a pure power-law tail because of a finite exponential cutoff in the upper tail. The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ... In probability theory and statistics, the zeta distribution is a discrete probability distribution. ... In probability and statistics, the Yule-Simon distribution is a discrete probability distribution. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and Î³ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... Originally, Zipfs law stated that, in a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table. ... In probability theory and statistics, the Zipf-Mandelbrot law is a discrete probability distribution. ... A scale-free network is a noteworthy kind of complex network because many real-world networks fall into this category. ... A bibliogram is a verbal construct made when noun phrases from extended stretches of text are ranked high to low by their frequency of co-occurrence with one or more user-supplied seed terms. ... In seismology, the Gutenberg-Richter law states that the number of earthquakes per year of Richter magnitude M statistically has the form Number of earthquakes of size M per year ~ exp(a - bM) where exp is the exponential function. ... An earthquake is the result of a sudden release of stored energy in the Earths crust that creates seismic waves. ... Robert Elmer Horton (May 18, 1875 - April 22, 1945) was an American ecologist and soil scientist, considered by many to be the father of modern hydrology. ... Look up Appendix:Most popular given names by country in Wiktionary, the free dictionary. ... In seismology, the Gutenberg-Richter law states that the number of earthquakes per year of Richter magnitude M statistically has the form Number of earthquakes of size M per year ~ exp(a - bM) where exp is the exponential function. ...

Validating power laws

Although power-law relations are attractive for many theoretical reasons, demonstrating that data do indeed follow a power-law relation requires more than simply fitting such a model to the data. In general, many alternative functional forms can appear to follow a power-law form for some extent. Thus, the preferred method for validation of power-law relations is by testing many orthogonal predictions of a particular generative mechanism against data, and not simply fitting a power-law relation to a particular kind of data. As such, the validation of power-law claims remains a very active field of research in many areas of modern science.

Contents

Properties of power laws

Scale invariance

Universality

Power-law functions

Estimating the exponent from empirical data

Examples of power law functions

Power-law distributions

Plotting power-law distributions

Estimating the exponent from empirical data

Examples of power-law distributions

Validating power laws

See also

External links

Bibliography

Contents

Properties of power laws GA_googleFillSlot("encyclopedia_square");

Scale invariance

Universality

Power-law functions

Estimating the exponent from empirical data

Examples of power law functions

Power-law distributions

Plotting power-law distributions

Estimating the exponent from empirical data

Examples of power-law distributions

Validating power laws

See also

External links

Bibliography

Properties of power laws