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The Statistics of Urban Scaling and Their Connection to Zipf’s Law

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  • Andres Gomez-Lievano
  • HyeJin Youn
  • Luís M A Bettencourt
Abstract
Urban scaling relations characterizing how diverse properties of cities vary on average with their population size have recently been shown to be a general quantitative property of many urban systems around the world. However, in previous studies the statistics of urban indicators were not analyzed in detail, raising important questions about the full characterization of urban properties and how scaling relations may emerge in these larger contexts. Here, we build a self-consistent statistical framework that characterizes the joint probability distributions of urban indicators and city population sizes across an urban system. To develop this framework empirically we use one of the most granular and stochastic urban indicators available, specifically measuring homicides in cities of Brazil, Colombia and Mexico, three nations with high and fast changing rates of violent crime. We use these data to derive the conditional probability of the number of homicides per year given the population size of a city. To do this we use Bayes’ rule together with the estimated conditional probability of city size given their number of homicides and the distribution of total homicides. We then show that scaling laws emerge as expectation values of these conditional statistics. Knowledge of these distributions implies, in turn, a relationship between scaling and population size distribution exponents that can be used to predict Zipf’s exponent from urban indicator statistics. Our results also suggest how a general statistical theory of urban indicators may be constructed from the stochastic dynamics of social interaction processes in cities.

Suggested Citation

  • Andres Gomez-Lievano & HyeJin Youn & Luís M A Bettencourt, 2012. "The Statistics of Urban Scaling and Their Connection to Zipf’s Law," PLOS ONE, Public Library of Science, vol. 7(7), pages 1-11, July.
    • Handle: RePEc:plo:pone00:0040393
    DOI: 10.1371/journal.pone.0040393
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    References listed on IDEAS

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    1. Masahisa Fujita & Paul Krugman & Anthony J. Venables, 2001. "The Spatial Economy: Cities, Regions, and International Trade," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262561476, April.
    2. Luis Bettencourt & Geoffrey West, 2010. "A unified theory of urban living," Nature, Nature, vol. 467(7318), pages 912-913, October.
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    Cited by:

    1. Fabiano L Ribeiro & Joao Meirelles & Vinicius M Netto & Camilo Rodrigues Neto & Andrea Baronchelli, 2020. "On the relation between transversal and longitudinal scaling in cities," PLOS ONE, Public Library of Science, vol. 15(5), pages 1-20, May.
    2. Matias Nehuen Iglesias, 2021. "The Overlooked Insights from Correlation Structures in Economic Geography," Papers in Evolutionary Economic Geography (PEEG) 2105, Utrecht University, Department of Human Geography and Spatial Planning, Group Economic Geography, revised Jan 2021.
    3. Hadi Arbabi & Martin Mayfield & Philip McCann, 2020. "Productivity, infrastructure and urban density—an allometric comparison of three European city regions across scales," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 183(1), pages 211-228, January.
    4. Rudolf Cesaretti & José Lobo & Luís M A Bettencourt & Scott G Ortman & Michael E Smith, 2016. "Population-Area Relationship for Medieval European Cities," PLOS ONE, Public Library of Science, vol. 11(10), pages 1-27, October.
    5. Luiz G A Alves & Haroldo V Ribeiro & Ervin K Lenzi & Renio S Mendes, 2013. "Distance to the Scaling Law: A Useful Approach for Unveiling Relationships between Crime and Urban Metrics," PLOS ONE, Public Library of Science, vol. 8(8), pages 1-8, August.
    6. Lei, Weiqian & Jiao, Limin & Xu, Zhibang & Zhu, Xinhua, 2024. "Evolution of urban land and population system coupling micro–dynamics and macro-stability: Trends and paths," Land Use Policy, Elsevier, vol. 141(C).
    7. Joao Meirelles & Fabiano L. Ribeiro & Gabriel Cury & Claudia R. Binder & Vinicius M. Netto, 2021. "More from Less? Environmental Rebound Effects of City Size," Sustainability, MDPI, vol. 13(7), pages 1-20, April.
    8. José Lobo & Luís M A Bettencourt & Deborah Strumsky & Geoffrey B West, 2013. "Urban Scaling and the Production Function for Cities," PLOS ONE, Public Library of Science, vol. 8(3), pages 1-10, March.
    9. Deborah Strumsky & Jose Lobo & Charlotta Mellander, 2021. "As different as night and day: Scaling analysis of Swedish urban areas and regional labor markets," Environment and Planning B, , vol. 48(2), pages 231-247, February.
    10. Cardoso, M. & Souza, J.T.G. & Neli, R.R. & Souza, W.E., 2023. "Scaling laws from Brazilian state election results point out that, the candidate’s chance to win increases by investing more campaign efforts in smaller electorates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 619(C).
    11. Sutton, Jack & Shahtahmassebi, Golnaz & Hanley, Quentin S. & Ribeiro, Haroldo V., 2024. "A heteroscedastic Bayesian generalized logistic regression model with application to scaling problems," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    12. Caio Porto de Castro & Gervásio Ferreira dos Santos & Anderson Dias de Freitas & Maria Isabel dos Santos & Roberto Fernandes Silva Andrade & Maurício Lima Barreto, 2020. "Socio-economic urban scaling properties: Influence of regional geographic heterogeneities in Brazil," PLOS ONE, Public Library of Science, vol. 15(12), pages 1-17, December.
    13. Yunfei Li & Diego Rybski & Jürgen P. Kropp, 2021. "Singularity cities," Environment and Planning B, , vol. 48(1), pages 43-59, January.
    14. Clémentine Cottineau, 2017. "MetaZipf. A dynamic meta-analysis of city size distributions," PLOS ONE, Public Library of Science, vol. 12(8), pages 1-22, August.
    15. Cardoso, M. & Afonso, L.H.D. & Neli, R.R. & Souza, W.E., 2024. "Simplified model relating blank and null votes to voter turnout from Brazilian state elections results," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 645(C).
    16. Diego Rybski & Dominik E Reusser & Anna-Lena Winz & Christina Fichtner & Till Sterzel & Jürgen P Kropp, 2017. "Cities as nuclei of sustainability?," Environment and Planning B, , vol. 44(3), pages 425-440, May.
    17. Andres Gomez-Lievano & Michail Fragkias, 2024. "The benefits and costs of agglomeration: insights from economics and complexity," Papers 2404.13178, arXiv.org.

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