Talk:Power law/Revised article

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A power law is a mathematical equation of the form,

$f(x) = ax^{-k}\!$

where $a$ and $k$ are constants, with $k$ being referred to as the exponent (the $-$ sign is placed there for convenience since in many power laws the exponent is negative). Plotted on a log-log graph, this appears as a linear relationship with a slope of − $k$ , since

$\log\left(ax^{-k}\right) = -k \log x + \log a$

which has the same form, $Y = m X + b$ , as a straight line. Equations that do not follow the above formula strictly may display power law tails, meaning that the ratio $f (x) / a x - k$ tends towards one as as $x \to \infty$ .

Strictly speaking the term "power law" includes many well-known formulas, such as those for calculating areas or volumes (e.g. $π r 2$ for the area of a circle), Newton's inverse-square law of gravity, and so on. However, the term is typically used in the context of power-law probability distributions such as the Gutenberg-Richter law for earthquake sizes, or scaling relationships such as those observed in fractals, 1/f noise and allometric scaling laws in living organisms. Much of the interest springs from the great variety of natural situations in which such power laws are observed, and their occurrence as a common feature of diverse complex systems. Explanations for these findings remain a topic of considerable debate in the scientific literature.

1 Properties of power laws
2 Power laws in nature
3 What follows is text from the original article. I don't intend to copy-and-paste much: some things will be removed, most substantially rewritten and restructured.
4 See also
5 External links
6 References
- 6.1 Notes
- 6.2 Bibliography

[edit] Properties of power laws

[edit] Exponents, scale invariance and universality

One of the key properties of power laws is their scale invariance. Suppose that for a given power law, $f (x) = a x - k$ , we change the length scale of our observation from $x$ to $A x$ , where $A$ is a constant. Then,

$f(Ax) = a(Ax)^{-k} = aA^{-k}x^{-k} = A^{-k}ax^{-k} = A^{-k}f(x)\!$

which leaves the power law intact, changing only the constant of proportionality. It follows that power laws with the same exponent are to some extent equivalent, since each is simply a rescaling of the other.

In some cases this equivalence is reflected in the dynamical origins of power laws. 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. 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 the same critical exponents are said to belong to the same universality class.

[edit] Power law probability distributions

Many of the power laws observed in nature are found in the probability distributions of various quantities, for example the Gutenberg-Richter law for the distribution of earthquake sizes. If we suppose a distribution to be of the form $p (x) = a x - k$ , where $x$ is a continuous variable, then aside from the above-mentioned scale invariance, a number of other features are observed.

To begin with, if we attempt to calculate the mean of x, we find,

$\langle x \rangle = \int_{x \in (0,\infty)} x p(x) \mathrm{d}x = a \int_{x \in (0,\infty)} x^{-k+1} \mathrm{d}x$

In the special case k = 2 this is of course the integral of $1 / x$ , which yields,

$\langle x \rangle = a [\log x]_{x \in (0,\infty)}$

while for k ≠ 2 we have,

$\langle x \rangle = \frac{a}{2 - k}[x^{-k+2}]_{x \in (0,\infty)}$

It follows that the mean is finite only if k > 2, since for k ≤ 2 the above integral diverges.

If now we try instead to calculate the (complementary) cumulative distribution, $P (x) = P r (x' > x)$ ,

$P(x) = \int_{x}^{\infty} p(x')\mathrm{d}x' = a\int_{x}^{\infty} x'^{-k} \mathrm{d}x'= \frac{a}{k-1} x^{-(k-1)}$

Thus, $P (x)$ also follows a power law, with exponent (k – 1). This observation can be particularly useful when giving a graphical representation of a power law: whereas plotting $p (x)$ accurately requires an appropriate choice of bin width for the data, $P (x)$ is well defined for every value of x, and so avoids the possibility that a wrong choice of binning skews the value of k displayed on a graphical plot.

[edit] Measuring the exponent from empirical data

Since a log-log plot of a power law yields a straight line, one simple way to estimate the exponent would be to perform linear regression on the log-values of the data. Unfortunately, however, this method can produce wildly inaccurate estimates, as can be demonstrated by testing a randomly-generated data set from a known power law distribution^[1].

An unbiased method, based on maximum likelihood estimation, chooses the maximally probable value for the exponent based on a given set of data points^[2].

Given a set of real-valued data points ${x i}$ , $i = 1, \dots , N$ ,

$k = 1 + N \left[ \sum_{i=1}^{N} \ln \frac{x_{i}}{x_{\mathrm{min}}} \right]^{-1}$

For a set of integer-valued data points ${x i}$ , $i = 1, \dots , N$ , the maximum likelihood exponent is the solution to the transcendental equation

$\frac{\zeta'(k)}{\zeta(k)} = -\frac{1}{N} \sum_{i=1}^{N} \ln x_{i}$

Note first that in this case, there is no value of $x min$ in the equation, so the power law is assumed to range from 1 to $\infty$ . Further, these two equations are not equivalent, and the continuous version should not be applied to discrete data, nor vice versa.

[edit] Power laws in nature

[edit] What follows is text from the original article. I don't intend to copy-and-paste much: some things will be removed, most substantially rewritten and restructured.

Power laws are observed in many fields, including physics, biology, geography, sociology, economics, linguistics, and even in war and terrorism^[3]^[4]. Power laws are among the most frequent scaling laws that describe the scale invariance found in many natural phenomena.

A power law relationship between two scalar quantities x and y is one where the relationship can be written as

$y = ax^k\,\!$

where a (the constant of proportionality) and k (the exponent of the power law) are constants.

Power laws can be seen as a straight line on a log-log graph since, taking logs of both sides, the above equation becomes

$\log(y) = k\log(x) + \log(a)\,\!$

which has the same form as the equation for a line

$y = mx+c\,\!$

Because both the power law and the log-normal distribution are asymptotic distributions, they can be easy to confuse without using robust statistical methods such as Bayesian model selection or statistical hypothesis testing. Indeed, a log-log plot of a log-normal distribution can often look nearly straight for certain ranges of x and y. One rule of thumb is the distribution conforms to a power law if it is straight on a log-log graph over 3 or more orders of magnitude.

Examples of power law relationships:

The Stefan-Boltzmann law
The Gompertz Law of Mortality
The Ramberg-Osgood stress-strain relationship
The inverse-square law of Newtonian gravity
Gamma correction relating light intensity with voltage
Kleiber's law relating animal metabolism to size
Behaviour near second-order phase transitions involving critical exponents
Frequency of events or effects of varying size in self-organized critical systems, e.g. Gutenberg-Richter Law of earthquake magnitudes and Horton's laws describing river systems
Proposed form of experience curve effects
Scale-free networks, where the distribution of links is given by a power law (in particular, the World Wide Web)
The differential energy spectrum of cosmic-ray nuclei

Examples of power law probability distributions:

The Pareto distribution
Zipf's law

These appear to fit such disparate phenomena as the popularity of websites, the wealth of individuals, the popularity of given names, and the frequency of words in documents. Benoît Mandelbrot and Nassim Taleb have recently popularised the analysis of financial market volatility in terms of a power law distribution (as opposed to the traditional Gaussian distribution), and Aventis science prize-winning author Philip Ball has argued that the same power law relationships that are evident in phase transitions also apply to various manifestations of collective human behaviour.

[edit] See also

[edit] External links

Zipf's law
Power laws, Pareto distributions and Zipf's law
Zipf, Power-laws, and Pareto - a ranking tutorial
Zipf Law, Zipf Distribution: An Introduction
Gutenberg-Richter Law
Stream Morphometry and Horton's Laws
A claim that the blogosphere obeys a powerlaw distribution
"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, Feb 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

[edit] References

The references of this article are divided into two groups. Notes are references or comments regarding specific details discussed in the article. The Bibliography contains general reference works that give an overview of power laws and their properties.

[edit] Notes

^ Goldstein, M. L., Morris, S. A. and Yen, G. G. (2004). "Problems with fitting to the power-law distribution". European Physical Journal B 41: 255–258. DOI:10.1140/epjb/e2004-00316-5.
^ This article gives only a basic, methodical description. For a more detailed exposition of the technique and how to derive the equations given here, see Newman (2005) in the bibliography, and Goldstein, Morris and Yen (2004), op. cit.
^ Clauset, Aaron; Young, Maxwell (2005-05-01). Scale Invariance in Global Terrorism. Retrieved on 2006-07-30.
^ Johnson, N; Spagat, M.; Restrepo, J.; Bohorquez, J.; Suarez, N.; Restrepo, E.; Zarama, R. (2005-05-29). From old wars to new wars and global terrorism. Retrieved on 2006-07-30.