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[[File:Long_tail.svg|500px|缩略图|右|一个幂律图示例，展示了人气排名的规律。右侧是数量庞大但人气微弱的[https://en.wikipedia.org/wiki/Long_tail “长尾”]，左侧则是少数“主宰”。 (又称80–20法则)。]]
[[File:Long_tail.svg|500px|缩略图|右|一个幂律图示例，展示了人气排名的规律。右侧是数量庞大但人气微弱的[https://en.wikipedia.org/wiki/Long_tail “长尾”]，左侧则是少数“主宰”。 (又称80–20法则)。]]
在统计学中，'''幂律'''表示的是两个量之间的函数关系，其中一个量的相对变化会导致另一个量的相应幂次比例的变化，且与初值无关：表现为一个量是另一个量的'''幂次方'''。例如，正方形面积与边长的关系，如果长度加倍，那么面积扩大四倍。<ref name="a">Bar-Yam, Yaneer (2015) [https://pattern.swarma.org/paper?id=3f7cfb86-6a75-11ea-8988-0242ac1a0005 "Concepts: Power Law"].New England Complex Systems Institute. Retrieved 18 August.</ref>
在统计学中，'''幂律'''表示的是两个量之间的函数关系，其中一个量的相对变化会导致另一个量的相应幂次比例的变化，且与初值无关：表现为一个量是另一个量的'''幂次方'''。例如，正方形面积与边长的关系，如果长度扩大到两倍，那么面积扩大到四倍。<ref name="a">Bar-Yam, Yaneer (2015) [https://pattern.swarma.org/paper?id=3f7cfb86-6a75-11ea-8988-0242ac1a0005 "Concepts: Power Law"].New England Complex Systems Institute. Retrieved 18 August.</ref>
==实例==
==实例==
幂律的涉及范围极其广泛，各种各样的物理、生物和人为现象的分布在大致遵循着幂律，包括月球表面月坑的大小，太阳耀斑的强度<ref name="Newman">Newman， M. E. J. (2005). "Power laws, Pareto distributions and Zipf's law". [https://en.wikipedia.org/wiki/Contemporary_Physics Contemporary Physics.] 46 (5): 323–351. [https://arxiv.org/abs/cond-mat/0412004 arXiv:cond-mat/0412004 ]Freely accessible. [http://adsabs.harvard.edu/abs/2005ConPh..46..323N Bibcode:2005ConPh..46..323N.] [https://doi.org/10.1080/00107510500052444 doi:10.1080/00107510500052444]</ref>
，各物种的觅食模式<ref name="Humphries">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 (2010) [https://pattern.swarma.org/paper?id=cc7c08d0-6cd5-11ea-8bce-0242ac1a0005 "Environmental context explains Lévy and Brownian movement patterns of marine predators" (PDF)].Nature, Bibcode:2010Natur.465.1066H. doi:10.1038/nature09116. PMID 20531470..465.(1066--1069)</ref>，神经元集群活动模式的规模<ref name="Klaus">Klaus, A, Yu, S, Plenz, D (2011) [https://pattern.swarma.org/paper?id=53ec16fc-6cd6-11ea-80b8-0242ac1a0005 Zochowski M (ed, "Statistical Analyses Support Power Law Distributions Found in Neuronal Avalanches"].PLoS ONE, Bibcode:2011PLoSO...619779K. doi:10.1371/journal.pone.0019779. PMC 3102672. PMID 21720544..6.(e19779)</ref>，大多数语言的用词频率，各姓氏的频率，生物支系的物种丰富度<ref name="Albert">Albert, J. S., H. J. Bart, & R. E. Reis (2011) [https://pattern.swarma.org/paper?id=8d094202-6cd6-11ea-91d5-0242ac1a0005 Historical Biogeography of Neotropical Freshwater Fishes].University of California Press.</ref>，停电的影响范围，每个罪犯的刑事罪名指控，火山喷发<ref name="Can">Cannavò, Giuseppe (2016) [https://pattern.swarma.org/paper?id=91d35726-6cd8-11ea-8ed0-0242ac1a0005 "On a Possible Unified Scaling Law for Volcanic Eruption Durations"].Scientific Reports, 622289C. doi:10.1038/srep22289. ISSN 2045-2322. PMC 4772095. PMID 26926425..6.(22289, 2016NatSR)</ref>，人类对刺激的敏感度<ref name = "s">Stevens, S. S. (1957). On the psychophysical law. Psychological Review, 64, 153-181</ref><ref name = "ts" >Staddon, J. E. R. (1978). Theory of behavioral power functions. Psychological Review, 85, 305-320.</ref> ，等等<ref name="N10">Clauset, A.; Shalizi, C. R.; Newman, M. E. J. (2009). "Power-Law Distributions in Empirical Data". SIAM Review. 51 (4): 661–703. [https://arxiv.org/abs/0706.1062 arXiv:0706.1062 Freely accessible.] [http://adsabs.harvard.edu/abs/2009SIAMR..51..661C Bibcode:2009SIAMR..51..661C.] [https://doi.org/10.1137/070710111 doi:10.1137/070710111]</ref>。
，各物种的觅食模式<ref name="Humphries">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 (2010) [https://pattern.swarma.org/paper?id=cc7c08d0-6cd5-11ea-8bce-0242ac1a0005 "Environmental context explains Lévy and Brownian movement patterns of marine predators" (PDF)].Nature, Bibcode:2010Natur.465.1066H. doi:10.1038/nature09116. PMID 20531470..465.(1066--1069)</ref>，神经元集群活动模式的规模<ref name="Klaus">Klaus, A, Yu, S, Plenz, D (2011) [https://pattern.swarma.org/paper?id=53ec16fc-6cd6-11ea-80b8-0242ac1a0005 Zochowski M (ed, "Statistical Analyses Support Power Law Distributions Found in Neuronal Avalanches"].PLoS ONE, Bibcode:2011PLoSO...619779K. doi:10.1371/journal.pone.0019779. PMC 3102672. PMID 21720544..6.(e19779)</ref>，大多数语言的用词频率，一个国家各姓氏出现的频率，生物支系的物种丰富度<ref name="Albert">Albert, J. S., H. J. Bart, & R. E. Reis (2011) [https://pattern.swarma.org/paper?id=8d094202-6cd6-11ea-91d5-0242ac1a0005 Historical Biogeography of Neotropical Freshwater Fishes].University of California Press.</ref>，停电的影响范围，每个罪犯的刑事罪名指控，火山喷发<ref name="Can">Cannavò, Giuseppe (2016) [https://pattern.swarma.org/paper?id=91d35726-6cd8-11ea-8ed0-0242ac1a0005 "On a Possible Unified Scaling Law for Volcanic Eruption Durations"].Scientific Reports, 622289C. doi:10.1038/srep22289. ISSN 2045-2322. PMC 4772095. PMID 26926425..6.(22289, 2016NatSR)</ref>，人类对刺激的敏感度<ref name = "s">Stevens, S. S. (1957). On the psychophysical law. Psychological Review, 64, 153-181</ref><ref name = "ts" >Staddon, J. E. R. (1978). Theory of behavioral power functions. Psychological Review, 85, 305-320.</ref> ，等等<ref name="N10">Clauset, A.; Shalizi, C. R.; Newman, M. E. J. (2009). "Power-Law Distributions in Empirical Data". SIAM Review. 51 (4): 661–703. [https://arxiv.org/abs/0706.1062 arXiv:0706.1062 Freely accessible.] [http://adsabs.harvard.edu/abs/2009SIAMR..51..661C Bibcode:2009SIAMR..51..661C.] [https://doi.org/10.1137/070710111 doi:10.1137/070710111]</ref>。
不过很少有经验分布完全符合幂律分布，更确切地说，他们通常是遵循幂律末尾的规律。在许多复杂介质中，[https://en.wikipedia.org/wiki/Acoustic_attenuation 声衰减 Acoustic attenuation]遵循宽频带内的频率幂律；揭示生物变量之间关系的[[异速标度律 Allometric scaling]]是自然界中已知的最著名的幂律函数之一。
==性质==
==性质==