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'''Urban scaling'''<ref name=":0">{{Cite journal |last1=Bettencourt |first1=Luis |last2=West |first2=Geoffrey |date=2010 |title=A unified theory of urban living |url=https://www.nature.com/articles/467912a |journal=Nature |language=en |volume=467 |issue=7318 |pages=912–913 |doi=10.1038/467912a |pmid=20962823 |bibcode=2010Natur.467..912B |issn=1476-4687}}</ref> is an area of research within the study of cities as [[Complex system|complex systems]]. It examines how various urban indicators change systematically with city size.
The literature on urban scaling was motivated by the success of scaling theory in biology, itself motivated in turn by the success of scaling in physics.<ref name=":5">{{Cite book |last=Whitfield |first=John |url=https://www.worldcat.org/title/ocm67346041 |title=In the beat of a heart: life, energy, and the unity of nature |date=2006 |publisher=Joseph Henry Press |isbn=978-0-309-09681-2 |location=Washington, D.C |oclc=ocm67346041}}</ref> Crucial insights from scaling analysis applied to a system are followed from finding [[Power law|power-law function]] relationships between variables of interest and the size of the system (as opposed to finding [[Power-law tail|power-law]] [[Probability distribution|probability distributions]]). Power-laws have an implicit self-similarity which suggests universal mechanisms at work, which in turn support the search for fundamental laws.<ref>{{Cite book |last=Schroeder |first=Manfred Robert |title=Fractals, chaos, power laws: minutes from an infinite paradise |date=2009 |publisher=Dover Publications |isbn=978-0-486-47204-1 |edition=Dover |location=Mineola, N.Y}}</ref>
The phenomenon of scaling in biology is often referred to as [[allometric scaling]]. These relationships were originally mentioned by Galileo (e.g., in terms of the area width of animals as a function of their mass) and then studied a century ago by [[Max Kleiber]] (see [[Kleiber's law]]). A theoretical explanation of allometric scaling laws in biology was provided by the [[Metabolic Scaling Theory]].<ref name=":5" />
[https://www.santafe.edu/people/profile/luis-bettencourt Luis Bettencourt], [[Geoffrey West]], and [https://sustainability-innovation.asu.edu/person/jose-lobo/ Jose Lobo]'s seminal work<ref name=":1" /> demonstrated that many urban indicators are associated with population size through a [[Power law|power-law]] relationship, in which socio-economic quantities tend to scale superlinearly<ref name=":2" />, while measures of infrastructure (such as the number of gas stations) scale sublinearly with population size<ref name=":3" />. They argue for a quantitative, predictive framework to understand cities as collective wholes, guiding urban policy, improving sustainability, and managing urban growth.<ref name=":0" />▼
In the context of cities, the application of scaling was inspired by the idea that cities are [[emergent phenomena]] arising from the interactions of many individuals. The expectation is that collective effects should result in the form of large-scale quantitative regularities that ought to hold across cultures, countries and history. If such regularities are observed, then it would support the search for a general mathematical theory of cities.
▲Indeed, [https://www.santafe.edu/people/profile/luis-bettencourt Luis Bettencourt], [[Geoffrey West]], and [https://sustainability-innovation.asu.edu/person/jose-lobo/ Jose Lobo]'s seminal work<ref name=":1" /> demonstrated that many urban indicators are associated with population size through a [[Power law|power-law]] relationship, in which socio-economic quantities tend to scale superlinearly<ref name=":2" />, while measures of infrastructure (such as the number of gas stations) scale sublinearly with population size<ref name=":3" />. They argue for a quantitative, predictive framework to understand cities as collective wholes, guiding urban policy, improving sustainability, and managing urban growth.<ref name=":0" />
The literature has grown, with many theoretical explanations for these emergent power-laws. Ribeiro and Rybski summarized these in their paper "[https://www.sciencedirect.com/science/article/abs/pii/S0370157323000650 Mathematical models to explain the origin of urban scaling laws]"<ref name=":4">{{Cite journal |last1=Ribeiro |first1=Fabiano L. |last2=Rybski |first2=Diego |date=2023 |title=Mathematical models to explain the origin of urban scaling laws |url=https://doi.org/10.1016/j.physrep.2023.02.002 |journal=Physics Reports |volume=1012 |pages=1–39 |doi=10.1016/j.physrep.2023.02.002 |bibcode=2023PhR..1012....1R |issn=0370-1573}}</ref>. Examples include Arbesman et al.'s 2009 model<ref>{{Cite journal |last1=Arbesman |first1=Samuel |last2=Kleinberg |first2=Jon M. |last3=Strogatz |first3=Steven H. |date=2009-01-30 |title=Superlinear scaling for innovation in cities |url=https://link.aps.org/doi/10.1103/PhysRevE.79.016115 |journal=Physical Review E |volume=79 |issue=1 |pages=016115 |doi=10.1103/PhysRevE.79.016115|pmid=19257115 |arxiv=0809.4994 |bibcode=2009PhRvE..79a6115A }}</ref>, Bettencourt's 2013 model<ref>{{Cite journal |last=Bettencourt |first=Luís M. A. |date=2013-06-21 |title=The Origins of Scaling in Cities |url=https://www.science.org/doi/10.1126/science.1235823 |journal=Science |language=en |volume=340 |issue=6139 |pages=1438–1441 |doi=10.1126/science.1235823 |pmid=23788793 |bibcode=2013Sci...340.1438B |issn=0036-8075}}</ref>, Gomez-Lievano et al.'s 2017 model<ref>{{Cite journal |last1=Gomez-Lievano |first1=Andres |last2=Patterson-Lomba |first2=Oscar |last3=Hausmann |first3=Ricardo |date=2016-12-22 |title=Explaining the prevalence, scaling and variance of urban phenomena |url=https://www.nature.com/articles/s41562-016-0012 |journal=Nature Human Behaviour |language=en |volume=1 |issue=1 |pages=1–6 |doi=10.1038/s41562-016-0012 |issn=2397-3374}}</ref>, and Yang et al.'s 2019 model<ref>{{Cite journal |last1=Yang |first1=V. Chuqiao |last2=Papachristos |first2=Andrew V. |last3=Abrams |first3=Daniel M. |date=2019-09-16 |title=Modeling the origin of urban-output scaling laws |url=https://link.aps.org/doi/10.1103/PhysRevE.100.032306 |journal=Physical Review E |volume=100 |issue=3 |pages=032306 |doi=10.1103/PhysRevE.100.032306|pmid=31639910 |arxiv=1712.00476 |bibcode=2019PhRvE.100c2306Y }}</ref>, among others (see <ref name=":4" />).
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