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Draft:Urban Scaling


Urban scaling[1] is an area of research within the study of cities as 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.[2] Crucial insights from scaling analysis applied to a system can emerge from finding power-law function relationships between variables of interest and the size of the system (as opposed to finding power-law probability distributions). Power-laws have an implicit self-similarity which suggests universal mechanisms at work, which in turn support the search for fundamental laws.[3]

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' legs as a function of their mass) and then studied a century ago by Max Kleiber (see Kleiber's law) in terms of the relationship between basal metabolic rate and mass. A theoretical explanation of allometric scaling laws in biology was provided by the Metabolic Scaling Theory.[2]

The application of scaling in the context of cities is inspired by the idea that cities are emergent phenomena arising from the interactions of many individuals in close physical proximity. This is in contrast to applying scaling to countries or other social group delineations, which are more ad-hoc sociological constructions. The expectation is that collective effects in cities should result in the form of large-scale quantitative urban 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.[4]

Indeed, Luis Bettencourt, Geoffrey West, and Jose Lobo's seminal work[5] demonstrated that many urban indicators are associated with population size through a power-law relationship, in which socio-economic quantities tend to scale superlinearly[6], while measures of infrastructure (such as the number of gas stations) scale sublinearly with population size[7]. They argue for a quantitative, predictive framework to understand cities as collective wholes, guiding urban policy, improving sustainability, and managing urban growth.[1]

The literature has grown, with many theoretical explanations for these emergent power-laws. Ribeiro and Rybski summarized these in their paper "Mathematical models to explain the origin of urban scaling laws"[8]. Examples include Arbesman et al.'s 2009 model[9], Bettencourt's 2013 model[10], Gomez-Lievano et al.'s 2017 model[11], and Yang et al.'s 2019 model[12], among others (see for a more thorough review of the models [8]). The ultimate explanation of scaling laws observed in cities is still debated.[13][14]

  1. ^ a b Bettencourt, Luis; West, Geoffrey (2010). "A unified theory of urban living". Nature. 467 (7318): 912–913. Bibcode:2010Natur.467..912B. doi:10.1038/467912a. ISSN 1476-4687. PMID 20962823.
  2. ^ a b Whitfield, John (2006). In the beat of a heart: life, energy, and the unity of nature. Washington, D.C: Joseph Henry Press. ISBN 978-0-309-09681-2. OCLC 67346041.
  3. ^ Schroeder, Manfred Robert (2009). Fractals, chaos, power laws: minutes from an infinite paradise (Dover ed.). Mineola, N.Y: Dover Publications. ISBN 978-0-486-47204-1.
  4. ^ Batty, Michael (2017). The new science of cities (First paperback ed.). Cambridge, Massachusetts London, England: The MIT Press. ISBN 978-0-262-53456-7.
  5. ^ Cite error: The named reference :1 was invoked but never defined (see the help page).
  6. ^ Cite error: The named reference :2 was invoked but never defined (see the help page).
  7. ^ Cite error: The named reference :3 was invoked but never defined (see the help page).
  8. ^ a b Ribeiro, Fabiano L.; Rybski, Diego (2023). "Mathematical models to explain the origin of urban scaling laws". Physics Reports. 1012: 1–39. Bibcode:2023PhR..1012....1R. doi:10.1016/j.physrep.2023.02.002. ISSN 0370-1573.
  9. ^ Arbesman, Samuel; Kleinberg, Jon M.; Strogatz, Steven H. (2009-01-30). "Superlinear scaling for innovation in cities". Physical Review E. 79 (1): 016115. arXiv:0809.4994. Bibcode:2009PhRvE..79a6115A. doi:10.1103/PhysRevE.79.016115. PMID 19257115.
  10. ^ Bettencourt, Luís M. A. (2013-06-21). "The Origins of Scaling in Cities". Science. 340 (6139): 1438–1441. Bibcode:2013Sci...340.1438B. doi:10.1126/science.1235823. ISSN 0036-8075. PMID 23788793.
  11. ^ Gomez-Lievano, Andres; Patterson-Lomba, Oscar; Hausmann, Ricardo (2016-12-22). "Explaining the prevalence, scaling and variance of urban phenomena". Nature Human Behaviour. 1 (1): 1–6. doi:10.1038/s41562-016-0012. ISSN 2397-3374.
  12. ^ Yang, V. Chuqiao; Papachristos, Andrew V.; Abrams, Daniel M. (2019-09-16). "Modeling the origin of urban-output scaling laws". Physical Review E. 100 (3): 032306. arXiv:1712.00476. Bibcode:2019PhRvE.100c2306Y. doi:10.1103/PhysRevE.100.032306. PMID 31639910.
  13. ^ Gomez-Lievano, Andres; Fragkias, Michail (2024). "The benefits and costs of agglomeration: insights from economics and complexity". arXiv:2404.13178 [physics.soc-ph].
  14. ^ Ribeiro, Fabiano L.; Netto, Vinicius M. (2024). "Urban Scaling Laws". arXiv:2404.02642 [physics.soc-ph].

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