Power law
In
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
Properties
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
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
On the one hand, this makes it incorrect to apply traditional statistics that are based on variance and standard deviation (such as regression analysis).[12] On the other hand, this also allows for cost-efficient interventions.[11] 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.[13]
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.[2]
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.[citation needed][clarification needed] 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.
However, much of the recent interest in power laws comes from the study of
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
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).[16] Among them are:
Artificial Intelligence
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
- allometric lawsin general
- The two-thirds power law, relating speed to curvature in the human motor system.[17]
- The Taylor's law relating mean population size and variance of populations sizes in ecology
- Neuronal avalanches[5]
- The species richness (number of species) in clades of freshwater fishes[18]
- The Harlow Knapp effect, where a subset of the published research[19]
- The size of forest patches globally follows a power law [20]
- 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[3]
- The size of rain-shower cells[21]
- Energy dissipation in cyclones[22]
- Diameters of dust devils on Earth and Mars [23]
General science
- Exponential growth and random observation (or killing)[24]
- Progress through exponential growth and exponential diffusion of innovations[25]
- 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)[26]
- Populations of cities (Gibrat's law)[27]
- Bibliograms, and frequencies of words in a text (Zipf's law)[28]
- Richardson's Law for the severity of violent conflicts (wars and terrorism)[31][32]
- The relationship between a CPU's cache size and the number of cache misses follows the power law of cache misses.
- The deep neural networks[33]
Economics
- urban network, Zipf's law.
- Distribution of artists by the average price of their artworks.[34]
- Income distribution in a market economy.
- Distribution of degrees in banking networks.[35]
- Firm-size distributions.[36]
Finance
- Returns for high-risk venture capital investments[37]
- The mean absolute change of the logarithmic mid-prices[38]
- Large price changes, volatility, and transaction volume on stock exchanges[39]
- Average waiting time of a directional change[40]
- 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
- The input-voltage–output-current curves of 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 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
- The Curie–von Schweidler law in dielectric responses to step DC voltage input.
- The damping force over speed relation in antiseismic dampers calculus
- Folded solvent-exposed surface areas of centered amino acids in protein structure segments[42]
Political Science
- Cube root law of assembly sizes
Psychology
Variants
Broken power law
A broken power law is a
- for
- .
Smoothly broken power law
The pieces of a broken power law can be smoothly spliced together to construct a smoothly broken power law.
There are different possible ways to splice together power laws. One example is the following:[47]
When the function is plotted as a log-log plot with horizontal axis being and vertical axis being , the plot is composed of linear segments with slopes , separated at , smoothly spliced together. The size of determines the sharpness of splicing between segments .
Power law with exponential cutoff
A power law with an exponential cutoff is simply a power law multiplied by an exponential function:[10]
Curved power law
Power-law probability distributions
In a looser sense, a power-law probability distribution is a distribution whose density function (or mass function in the discrete case) has the form, for large values of ,[49]
where , and is a slowly varying function, which is any function that satisfies for any positive factor . 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 of the Pareto distribution
where the pre-factor to is the normalizing constant. We can now consider several properties of this distribution. For instance, its moments 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 , 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. These power-law probability distributions are also called Pareto-type distributions, distributions with Pareto tails, or distributions with regularly varying tails.
A modification, which does not satisfy the general form above, with an exponential cutoff,[10] is
In this distribution, the exponential decay term eventually overwhelms the power-law behavior at very large values of . This distribution does not scale[further explanation needed] and is thus not asymptotically as a power law; however, it does approximately scale over a finite region before the cutoff. The pure form above is a subset of this family, with . This distribution is a common alternative to the asymptotic power-law distribution because it naturally captures finite-size effects.
The
Graphical methods for identification
Although more sophisticated and robust methods have been proposed, the most frequently used graphical methods of identifying power-law probability distributions using random samples are Pareto quantile-quantile plots (or Pareto Q–Q plots),[citation needed] mean residual life plots[52][53] and log–log plots. Another, more robust graphical method uses bundles of residual quantile functions.[54] (Please keep in mind that power-law distributions are also called Pareto-type distributions.) It is assumed here that a random sample is obtained from a probability distribution, and that we want to know if the tail of the distribution follows a power law (in other words, we want to know if the distribution has a "Pareto tail"). Here, the random sample is called "the data".
Pareto Q–Q plots compare the quantiles of the log-transformed data to the corresponding quantiles of an exponential distribution with mean 1 (or to the quantiles of a standard Pareto distribution) by plotting the former versus the latter. If the resultant scatterplot suggests that the plotted points " asymptotically converge" to a straight line, then a power-law distribution should be suspected. A limitation of Pareto Q–Q plots is that they behave poorly when the tail index (also called Pareto index) is close to 0, because Pareto Q–Q plots are not designed to identify distributions with slowly varying tails.[54]
On the other hand, in its version for identifying power-law probability distributions, the mean residual life plot consists of first log-transforming the data, and then plotting the average of those log-transformed data that are higher than the i-th order statistic versus the i-th order statistic, for i = 1, ..., n, where n is the size of the random sample. If the resultant scatterplot suggests that the plotted points tend to "stabilize" about a horizontal straight line, then a power-law distribution should be suspected. Since the mean residual life plot is very sensitive to outliers (it is not robust), it usually produces plots that are difficult to interpret; for this reason, such plots are usually called Hill horror plots [55]
Log–log plots are an alternative way of graphically examining the tail of a distribution using a random sample. Caution has to be exercised however as a log–log plot is necessary but insufficient evidence for a power law relationship, as many non power-law distributions will appear as straight lines on a log–log plot.[10][56] This method consists of plotting the logarithm of an estimator of the probability that a particular number of the distribution occurs versus the logarithm of that particular number. Usually, this estimator is the proportion of times that the number occurs in the data set. If the points in the plot tend to "converge" to a straight line for large numbers in the x axis, then the researcher concludes that the distribution has a power-law tail. Examples of the application of these types of plot have been published.[57] A disadvantage of these plots is that, in order for them to provide reliable results, they require huge amounts of data. In addition, they are appropriate only for discrete (or grouped) data.
Another graphical method for the identification of power-law probability distributions using random samples has been proposed.[54] This methodology consists of plotting a bundle for the log-transformed sample. Originally proposed as a tool to explore the existence of moments and the moment generation function using random samples, the bundle methodology is based on residual quantile functions (RQFs), also called residual percentile functions,[58][59][60][61][62][63][64] which provide a full characterization of the tail behavior of many well-known probability distributions, including power-law distributions, distributions with other types of heavy tails, and even non-heavy-tailed distributions. Bundle plots do not have the disadvantages of Pareto Q–Q plots, mean residual life plots and log–log plots mentioned above (they are robust to outliers, allow visually identifying power laws with small values of , and do not demand the collection of much data).[citation needed] In addition, other types of tail behavior can be identified using bundle plots.
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 (ccdf) that is, the survival function, ,
The cdf is also a power-law function, 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 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.[10][65] The survival function, on the other hand, is more robust to (but not without) such biases in the data and preserves the linear signature on doubly logarithmic axes. Though a survival function representation is favored over that of the pdf while fitting a power law to the data with the linear least square method, it is not devoid of mathematical inaccuracy. Thus, while estimating exponents of a power law distribution, maximum likelihood estimator is recommended.
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. Some of 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.[10]
Maximum likelihood
For real-valued,
to the data , where the coefficient is included to ensure that the distribution is normalized. Given a choice for , the log likelihood function becomes:
The maximum of this likelihood is found by differentiating with respect to parameter , setting the result equal to zero. Upon rearrangement, this yields the estimator equation:
where are the data points .[2][66] This estimator exhibits a small finite sample-size bias of order , which is small when n > 100. Further, the standard error of the estimate is . This estimator is equivalent to the popular[
For a set of n integer-valued data points , again where each , the maximum likelihood exponent is the solution to the transcendental equation
where 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.
Further, both of these estimators require the choice of . For functions with a non-trivial function, choosing too small produces a significant bias in , while choosing it too large increases the uncertainty in , and reduces the
More about these methods, and the conditions under which they can be used, can be found in .[10] Further, this comprehensive review article provides usable code (Matlab, Python, R and C++) for estimation and testing routines for power-law distributions.
Kolmogorov–Smirnov estimation
Another method for the estimation of the power-law exponent, which does not assume
with
where and denote the cdfs of the data and the power law with exponent , respectively. As this method does not assume iid data, it provides an alternative way to determine the power-law exponent for data sets in which the temporal correlation can not be ignored.[5]
Two-point fitting method
This criterion[67] can be applied for the estimation of power-law exponent in the case of scale-free distributions and provides a more convergent estimate than the maximum likelihood method. It has been applied to study probability distributions of fracture apertures. In some contexts the probability distribution is described, not by the cumulative distribution function, by the cumulative frequency of a property X, defined as the number of elements per meter (or area unit, second etc.) for which X > x applies, where x is a variable real number. As an example,[citation needed] the cumulative distribution of the fracture aperture, X, for a sample of N elements is defined as 'the number of fractures per meter having aperture greater than x . Use of cumulative frequency has some advantages, e.g. it allows one to put on the same diagram data gathered from sample lines of different lengths at different scales (e.g. from outcrop and from microscope).
Validating power laws
Although power-law relations are attractive for many theoretical reasons, demonstrating that data does indeed follow a power-law relation requires more than simply fitting a particular model to the data.[25] This is important for understanding the mechanism that gives rise to the distribution: superficially similar distributions may arise for significantly different reasons, and different models yield different predictions, such as extrapolation.
For example, log-normal distributions are often mistaken for power-law distributions:[68] a data set drawn from a lognormal distribution will be approximately linear for large values (corresponding to the upper tail of the lognormal being close to a power law)[clarification needed], but for small values the lognormal will drop off significantly (bowing down), corresponding to the lower tail of the lognormal being small (there are very few small values, rather than many small values in a power law).[citation needed]
For example, Gibrat's law about proportional growth processes produce distributions that are lognormal, although their log–log plots look linear over a limited range. An explanation of this is that although the logarithm of the lognormal density function is quadratic in log(x), yielding a "bowed" shape in a log–log plot, if the quadratic term is small relative to the linear term then the result can appear almost linear, and the lognormal behavior is only visible when the quadratic term dominates, which may require significantly more data. Therefore, a log–log plot that is slightly "bowed" downwards can reflect a log-normal distribution – not a power law.
In general, many alternative functional forms can appear to follow a power-law form for some extent.[69] Stumpf & Porter (2012) proposed plotting the empirical cumulative distribution function in the log-log domain and claimed that a candidate power-law should cover at least two orders of magnitude.[70] Also, researchers usually have to face the problem of deciding whether or not a real-world probability distribution follows a power law. As a solution to this problem, Diaz[54] proposed a graphical methodology based on random samples that allow visually discerning between different types of tail behavior. This methodology uses bundles of residual quantile functions, also called percentile residual life functions, which characterize many different types of distribution tails, including both heavy and non-heavy tails. However, Stumpf & Porter (2012) claimed the need for both a statistical and a theoretical background in order to support a power-law in the underlying mechanism driving the data generating process.[70]
One method to validate a power-law relation tests many orthogonal predictions of a particular generative mechanism against data. Simply fitting a power-law relation to a particular kind of data is not considered a rational approach. As such, the validation of power-law claims remains a very active field of research in many areas of modern science.[10]
See also
References
Notes
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- Buchanan, Mark (2000). Ubiquity. Weidenfeld & Nicolson. ISBN 0-297-64376-2.
- Clauset, A.; Shalizi, C. R.; Newman, M. E. J. (2009). "Power-Law Distributions in Empirical Data". SIAM Review. 51 (4): 661–703. S2CID 9155618.
- Laherrère, J.; Sornette, D. (1998). "Stretched exponential distributions in nature and economy: "fat tails" with characteristic scales". European Physical Journal B. 2 (4): 525–539. S2CID 119467988.
- Mitzenmacher, M. (2004). "A Brief History of Generative Models for Power Law and Lognormal Distributions" (PDF). Internet Mathematics. 1 (2): 226–251. S2CID 1671059.
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External links
- Zipf, Power-laws, and Pareto – a ranking tutorial Archived 2007-10-26 at the Wayback Machine
- Stream Morphometry and Horton's Laws
- "How the Finance Gurus Get Risk All Wrong" by Benoit Mandelbrot & Nassim Nicholas Taleb. Fortune, July 11, 2005.
- "Million-dollar Murray": power-law distributions in homelessness and other social problems; by Malcolm Gladwell. The New Yorker, February 13, 2006.
- Benoit Mandelbrot & Richard Hudson: The Misbehaviour of Markets (2004)
- Philip Ball: Critical Mass: How one thing leads to another (2005)
- Tyranny of the Power Law from The Econophysics Blog
- So You Think You Have a Power Law – Well Isn't That Special? from Three-Toed Sloth, the blog of Cosma Shalizi, Professor of Statistics at Carnegie-Mellon University.
- Simple MATLAB script which bins data to illustrate power-law distributions (if any) in the data.
- The Erdős Webgraph Server Archived 2021-03-01 at the Wayback Machine visualizes the distribution of the degrees of the webgraph on the download page.