Power law: Difference between revisions
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A '''power law''' is a mathematical relationship between two quantities of the form, | {{subpages}} | ||
A '''power law''' is a mathematical relationship between two quantities where one is proportional to a [[exponentiation|power]] of the other: that is, of the form, | |||
:<math>y(x) = ax^k\!</math> | :<math>y(x) = ax^k\!</math> | ||
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:<math>\log\left(y\right) = k \log x + \log a</math> | :<math>\log\left(y\right) = k \log x + \log a</math> | ||
which has the same form, <math>Y = mX + b</math>, as a [[line (mathematics)|straight line]]. Equations that do not follow the above formula strictly may display '''power law tails''', meaning that the ratio <math>y(x)/ax^k</math> tends towards one as <math>x \to \infty</math>. | |||
Strictly speaking the term "power law" includes many well-known formulas, such as those for calculating areas or volumes (e.g. <math>\pi r^2</math> for the area of a circle), [[Isaac Newton|Newton]]'s [[inverse-square law]] of gravity, and so on. However, the term is typically used in the context of power-law [[probability distribution]]s such as the [[Gutenberg-Richter law]] for earthquake sizes, or scaling relationships such as those observed in [[fractal]]s, [[1/f noise]] and [[allometric law|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 system]]s. Explanations for these findings remain a topic of considerable debate in the scientific literature. | Strictly speaking the term "power law" includes many well-known formulas, such as those for calculating areas or volumes (e.g. <math>\pi r^2</math> for the area of a circle), [[Isaac Newton|Newton]]'s [[inverse-square law]] of gravity, and so on. However, the term is typically used in the context of power-law [[probability distribution]]s such as the [[Gutenberg-Richter law]] for earthquake sizes, or scaling relationships such as those observed in [[fractal]]s, [[1/f noise]] and [[allometric law|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 system]]s. Explanations for these findings remain a topic of considerable debate in the scientific literature. | ||
==Properties of power laws== | ==Properties of power laws== | ||
===Exponents, scale invariance and universality=== | ===Exponents, scale invariance and universality=== | ||
One of the key properties of power laws is their [[scale invariance#Scale invariance of functions and self-similarity|scale invariance]]. Suppose that for a given power law, <math>y(x) = ax^k</math>, we change the length scale of our observation from <math>x</math> to <math>Ax</math>, where <math>A</math> is a constant. Then, | One of the key properties of power laws is their [[scale invariance#Scale invariance of functions and self-similarity|scale invariance]]. Suppose that for a given power law, <math>y(x) = ax^k</math>, we change the length scale of our observation from <math>x</math> to <math>Ax</math>, where <math>A</math> is a constant. Then, | ||
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===Measuring the exponent from empirical data=== | ===Measuring the exponent from empirical data=== | ||
Since a [[log-log graph|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 this method can produce wildly inaccurate estimates, as can be demonstrated by testing a randomly-generated data set from a known power law distribution<ref name="GMY 2004"> | Since a [[log-log graph|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 this method can produce wildly inaccurate estimates, as can be demonstrated by testing a randomly-generated data set from a known power law distribution<ref name="GMY 2004"> | ||
{{cite journal | {{cite journal | ||
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==Power law probability distributions (Pareto distributions)== | ==Power law probability distributions (Pareto distributions)== | ||
Power law probability distributions, frequently referred to as '''Pareto distributions''' in honour of the economist [[Vilfredo Pareto]] who introduced them in the late 19th century<ref name="Pareto"> | Power law probability distributions, frequently referred to as '''Pareto distributions''' in honour of the economist [[Vilfredo Pareto]] who introduced them in the late 19th century<ref name="Pareto"> | ||
{{cite book | {{cite book | ||
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==Power laws in nature== | ==Power laws in nature== | ||
== See Also == | |||
== | * [[logarithm]] (which is a different function) | ||
* [[Exponential growth]] (different again) | |||
===Notes=== | ===Notes=== | ||
{{reflist}}[[Category:Suggestion Bot Tag]] | |||
[[Category: | |||
Latest revision as of 11:01, 6 October 2024
A power law is a mathematical relationship between two quantities where one is proportional to a power of the other: that is, of the form,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x) = ax^k\!}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} are constants, with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} being referred to as the exponent. Plotted on a log-log graph, this appears as a linear relationship with a slope of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} , since
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log\left(y\right) = k \log x + \log a}
which has the same form, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y = mX + b} , as a straight line. Equations that do not follow the above formula strictly may display power law tails, meaning that the ratio Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x)/ax^k} tends towards one as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \to \infty} .
Strictly speaking the term "power law" includes many well-known formulas, such as those for calculating areas or volumes (e.g. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi 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.
Properties of power laws
Exponents, scale invariance and universality
One of the key properties of power laws is their scale invariance. Suppose that for a given power law, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x) = ax^k} , we change the length scale of our observation from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ax} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is a constant. Then,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(Ax) = a(Ax)^k = aA^{k}x^{k} = A^{k}ax^{k} = A^{k}y(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.
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 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x_{i}\}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 1, \dots , N} ,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x_{i}\}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 1, \dots , N} , the maximum likelihood exponent is the solution to the transcendental equation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{\mathrm{min}}} in the equation, so the power law is assumed to range from 1 to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} . Further, these two equations are not equivalent, and the continuous version should not be applied to discrete data, nor vice versa.
Power law probability distributions (Pareto distributions)
Power law probability distributions, frequently referred to as Pareto distributions in honour of the economist Vilfredo Pareto who introduced them in the late 19th century[3][4], describe many phenomena in nature, for example the Gutenberg-Richter law for the distribution of earthquake sizes. If we suppose a distribution to be of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = ax^{-k}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x \rangle = \int_{x_{\mathrm{min}}}^{\infty} x p(x) \mathrm{d}x = a \int_{x_{\mathrm{min}}}^{\infty} x^{-k+1} \mathrm{d}x}
In the special case k = 2 this is of course the integral of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/x} , which yields,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x \rangle = a [\log x]_{x_{\mathrm{min}}}^{\infty}}
while for k ≠ 2 we have,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle x \rangle = \frac{a}{2 - k}[x^{-k+2}]_{x_{\mathrm{min}}}^{\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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x) = Pr(x' > x)} ,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} accurately requires an appropriate choice of bin width for the data, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.
Power laws in nature
See Also
- logarithm (which is a different function)
- Exponential growth (different again)
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. Research Blogging.
- ↑ 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.
- ↑ Pareto, V. (1897). Cours d'Économie Politique (in French). Lausanne: Rouge.
- ↑ Pareto used a power law distribution to describe the distribution of income in society. Ironically, this is now recognised as being better described by a Lévy distribution, which has a power law tail for large values of the quantity described but is closer to an exponential distribution for smaller values.