Clever Geek Handbook
📜 ⬆️ ⬇️

Power law

An example of a power function graph used to demonstrate ranking in popularity. On the right there is a long tail , and on the left those that dominate (see principle 80-20 ).

In statistics, a power law is a functional relationship between two quantities, in which a relative change in one quantity leads to a proportional relative change in another quantity, regardless of the initial values ​​of these quantities: the dependence of one quantity on another is a power function . For example, consider the dependence of the square of a square on the length of its side. If the length is doubled, the area will increase fourfold. [one]

Content

Case Studies

In many physical, biological and artificial phenomena, distributions are observed that approximately correspond to a power law on various scales: for example, the sizes of lunar craters and solar flares , [2] patterns of nutrition for different species, [3] the activity of neuron populations, [4] the frequency of use of words in most languages, the prevalence of the names , the number of species in hoards organisms [5] scale accidents in electric power systems , the number of criminal charges per offender number eruptions, [6], human otse ki stimulus intensity [7], [8] and many other quantities. [9] Empirical distributions can correspond to a power law over the whole range of their values, or, for example, in the tail. Attenuation of sound vibrations follows a power law in wide frequency bands in many complex environments. Allometric patterns for the relationship between biological variables are among the most well-known examples of power laws in nature.

Properties

Scale Invariance

For power law, scale invariance is characteristic. If performedf(x)=ax-k {\ displaystyle f (x) = ax ^ {- k}}   , then the scaling argumentx {\ displaystyle x}   by constant factorc {\ displaystyle c}   will lead to proportional scaling of the function itself. I.e:

f(cx)=a(cx)-k=c-kf(x)αf(x),{\ displaystyle f (cx) = a (cx) ^ {- k} = c ^ {- k} f (x) \ propto f (x), \!}  

Whereα {\ displaystyle \ propto}   denotes direct proportionality . In other words, multiplying an argument by a constantc {\ displaystyle c}   simply results in multiplying the value of a function by a constantc-k {\ displaystyle c ^ {- k}}   . Thus, all power laws with a given exponent are equivalent up to multiplication by a constant, since all of them are only scaled versions of each other. This generates a linear relationship between the logarithms off(x) {\ displaystyle f (x)}   andx {\ displaystyle x}   , and a straight line on the graph on a double logarithmic scale (log-log), which is often considered a characteristic feature of a power law. In real data, this feature is necessary, but not sufficient, to conclude that there is a power law. There are many ways to generate finite data volumes that mimic the correspondence to a power law, but deviate from it in the asymptotic limit (for example, if the data generation process follows a lognormal distribution ). Testing models for compliance with a power law is an actual area of ​​research in statistics, see below.

Absence of a strictly defined mean value

Power lawx-k {\ displaystyle x ^ {- k}}   has a well-defined mean value forx∈[one,∞) {\ displaystyle x \ in [1, \ infty)}   , only ifk>2 {\ displaystyle k> 2}   , and has a finite variance only ifk>3 {\ displaystyle k> 3}   . For most of the known power laws in nature, the values ​​of the exponent are such that the mean value is strictly defined, but the variance is not, therefore, there is the possibility of occurrence of black swan events. [10] This can be illustrated by the following thought experiment: [11] Imagine yourself in a room with friends and estimate the average monthly income in this room. Now imagine that the richest man in the world entered this room with a monthly income of about US $ 1 billion . How to change the value of the average monthly income in the room? The distribution of income follows a power law known as the Pareto distribution (for example, the capital of the Americans is distributed according to a power law with a power of 2).

On the one hand, this does not allow for the correct application of traditional statistics based on variance and standard deviation (for example, regression analysis ). On the other hand, it allows cost effective intervention. [11] For example, let automobile exhaust gases be distributed according to a power law among automobiles (that is, most of the pollution is carried out by a very small number of automobiles). Then it will be enough to remove this small number of cars from the roads in order to significantly reduce the total amount of emissions. [12]

The median exists: for the power law x - k with exponentk>one {\ displaystyle k> 1}   it takes the value 2 1 / ( k - 1) x min , where x min is the minimum value for which the power law is fulfilled [13]

Check for compliance with power law

Although the power law is attractive for many theoretical reasons, the proof that the data actually follow the power law requires more than the simple selection of model parameters. [14] It is important to understand the mechanism of occurrence of a distribution: externally similar distributions may arise for significantly different reasons, and different models give different predictions, for example, when extrapolating. [15] [16]

See also

  • Sound attenuation
  • Allometry
  • Fat tail distribution
  • Singularity for a finite time
  • Fractional Calculus
  • Fractional dynamics
  • Heavy-tailed distribution
  • Hyperbolic growth
  • Levi's Flight
  • A long tail
  • Power law of viscosity of liquids
  • Simon's model
  • Sustainable distribution
  • Stevens law
  • Concentration of wealth
  • Web graph

Links

Footnotes

  1. ↑ Yaneer Bar-Yam. Concepts: Power Law (Unsolved) . New England Complex Systems Institute. The appeal date is August 18, 2015.
  2. ↑ Newman, MEJ Power laws, Pareto distributions and Zipf's law (Eng.) // Contemporary Physics : journal. - 2005. - Vol. 46 , no. 5 - P. 323-351 . - DOI : 10.1080 / 00107510500052444 . - . - arXiv : cond-mat / 0412004 .
  3. Ump Humphries NE, Queiroz N., Dyer JR, Pade NG, Musyl MK, Schaefer KM, Fuller DW, Brunnschweiler JM, Doyle TK, Houghton JD, Hays GC, Jones CS, Noble LR, Wearmouth VJ, Southall EJ, Sims DW Environmental context explains Lévy and the Brownian movement patterns of marine predators (Eng.) // Nature: journal. - 2010. - Vol. 465 , no. 7301 . - P. 1066-1069 . - DOI : 10.1038 / nature09116 . - . - PMID 20531470 .
  4. ↑ Klaus A., Yu. S., Plenz D. Statistical Analyzes for Supporting Power Laws (Eng.) // PLoS ONE : journal / Zochowski, Michal. - 2011. - Vol. 6 , no. 5 - P. e19779 . - DOI : 10.1371 / journal.pone.0019779 . - . - PMID 21720544 .
  5. ↑ Historical Biogeography of Neotropical Freshwater Fishes . - Berkeley: University of California Press, 2011.
  6. ↑ Cannavò, Flavio; Nunnari, Giuseppe. Scaling Law for Volcanic Eruption Durations (English) // Scientific Reports : journal. - 2016. - 1 March ( vol. 6 ). - P. 22289 . - ISSN 2045-2322 . - DOI : 10.1038 / srep22289 . - . - PMID 26926425 .
  7. ↑ Stevens, SS (1957). On the psychophysical law. Psychological Review, 64, 153-181
  8. ↑ Staddon, JER (1978). Theory of behavioral power functions. Psychological Review, 85, 305–320.
  9. ↑ Clauset, Shalizi .
  10. ↑ Newman, MEJ; Reggiani, Aura; Nijkamp, ​​Peter. Power laws, Pareto distributions and Zipf's law (Eng.) // Cities . - Elsevier , 2005. - Vol. 30 , no. 2005 . - P. 323-351 . - DOI : 10.1016 / j.cities.2012 2012.03.001 . - arXiv : cond-mat / 0412004 .
  11. ↑ 1 2 9na CEPAL Charlas Sobre Sistemas Complejos Sociales (CCSSCS): Leyes de potencias, https://www.youtube.com/watch?v=4uDSE86xCI
  12. ↑ Malcolm Gladwell (2006), Million-Dollar Murray; Archived copy (Unsolved) . The appeal date is June 14, 2015. Archived March 18, 2015.
  13. ↑ Newman, Mark EJ. "Power laws, Pareto distributions and Zipf's law." Contemporary physics 46.5 (2005): 323-351.
  14. ↑ Hilbert, Martin. Scale-free power-laws as interaction between progress and diffusion (English) // Complexity: journal. - 2013. - Vol. 19 , no. 4 - p . 56-65 . - DOI : 10.1002 / cplx.21485 . - .
  15. ↑ Hall, P. On Some Simple Estimates Of An Exponent Of Regular Variation (Eng.) // Journal Of The Royal Statistical Society, Series B : journal. - 1982. - Vol. 44 , no. 1 . - P. 37—42 .
  16. ↑ Stumpf, MPH Critical Truths about Power Laws (eng.) // Science : journal. - 2012. - Vol. 335 , no. 6069 . - P. 665-666 . - DOI : 10.1126 / science.1216142 . - . - PMID 22323807 .

Bibliography

  • Bak, Per (1997) How nature works , Oxford University Press
  • Clauset, A .; Shalizi, CR; Newman, MEJ Power-Law Distributions in Empirical Data (Unreferenced) // SIAM Review. - 2009. - V. 51 , № 4 . - p . 661-703 . - DOI : 10.1137 / 070710111 . - . - arXiv : 0706.1062 .
  • Laherrère, J .; Sornette, D. Stretched exponential distributions in nature and the economy: "The fat tails" with characteristic scales (Eng.) // The European Physical Journal B : journal. - 1998. - Vol. 2 , no. 4 - P. 525-539 . - DOI : 10.1007 / s100510050276 . - . - arXiv : cond-mat / 9801293 .
  • Mitzenmacher, M.A. The Law and Lognormal Distributions ( Internet ) // Internet Mathematics: journal. - 2004. - Vol. 1 , no. 2 - P. 226-251 . - DOI : 10.1080 / 15427951.2004.10129088 .
  • Alexander Saichev, Yannick Malevergne and Didier Sornette (2009), Springer (November 2009),
  • Simon, HA On a Class of Skew Distribution Functions (Neopr.) // Biometrika . - 1955. - V. 42 , No. 3/4 . - p . 425-440 . - DOI : 10.2307 / 2333389 .
  • Sornette, Didier. Critical Phenomena in Natural Sciences: Chaos, Fractals, Self-organization and Disorder: Concepts and Tools. - 2nd. - Heidelberg: Springer, 2006. - ISBN 978-3-540-30882-9 .
  • Mark Buchanan (2000) Ubiquity , Weidenfeld & Nicolson
  • Stumpf, MPH; Porter, MA Critical Truths about Power Laws (eng.) // Science. - 2012. - Vol. 335 , no. 6069 . - P. 665-666 . - DOI : 10.1126 / science.1216142 . - . - PMID 22323807 .

Links

  • Zipf's law
  • Zipf, Power-laws, and Pareto - a ranking tutorial
  • Stream Morphometry and Horton's Laws
  • Clay Shirky on Institutions & Collaboration:
  • Clay Shirky on Power Laws, Weblogs, and Inequality
  • "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 Misbehavior of Markets (2004)
  • Philip Ball: Critical Mass: How one thing leads to another (2005)
  • Tyranny of the Econophysics Blog
  • Do You Have That Special? from the Cosma Shalizi, The Carnigie-Mellon University.
  • Simple MATLAB script is a simple MATLAB script that builds a histogram of data in a logarithmic scale.
  • The Erdős Webgraph Server - visualize the distribution of web graph degrees on the download page.
Source - https://ru.wikipedia.org/w/index.php?title=Stepnoy_akon&oldid=101121309


More articles:

  • Parvitsky, Nikolai Ivanovich
  • Zotikov, Vladimir Ivanovich
  • Coremacera
  • Round-the-clock party-goers
  • San Martin de los Andes
  • Parliamentary Elections in Senegal (1957)
  • Meshcheryakova, Olga Grigorievna
  • Norwegian Air UK
  • Strictly chordal graph
  • Kleinhenchen

All articles

Clever Geek | 2019