Abstract
Density scaling laws complement traditional population scaling laws by enabling the analysis of the full range of human settlements and revealing rural to urban transitions with breakpoints at consistent population densities. However, previous studies covering all areas, not just cities, have been constrained by the granularity of available rural and urban units, as well as limitations in the quantity and diversity of indicators. This study addresses these gaps by examining Middle Layer Super Output Areas (MSOAs) in England and Wales, incorporating an extensive set of 117 indicators for the year 2021, spanning age, ethnicity, educational attainment, religion, disability, economic activity, mortality, crime, property transactions, and road accidents. Results indicate that the relationship between indicator density and population density is best described by a segmented power law model with a consistent breakpoint (33 ± 5 persons per hectare) for 92 of the 117 indicators. Additionally, increasing granularity reveals further rural to urban transitions not observed at coarser spatial resolutions. Our findings also highlight the influence of population characteristics on scaling exponents, where stratifying dementia and ischaemic heart disease by older age groups (aged 70 and above) significantly affects these exponents, illustrating a protective urban effect.
Data availability
All data generated or analysed during this study are included in this published article (and its supplementary information files). This data was compiled from a range of publicly available sources as noted in the manuscript. These are provided as the Following files: S1Dataset.csv, S2Dataset.csv, S3Dataset.csv and S4Dataset.csv.
Code Availability
We have also provided a set of R-scripts as supplementary information. This has been provided as S1Code.R.
References
Kühnert, C., Helbing, D. & West, G. B. Scaling laws in urban supply networks. Phys. A: Stat. Mech. Its Appl. 363, 96–103 (2006).
Bettencourt, L. M. A., Lobo, J., Helbing, D., Kühnert, C. & West, G. B. Growth, innovation, scaling, and the pace of life in cities. Proc. Natl. Acad. Sci. 104, 7301–7306 (2007).
Bettencourt, L. M. A. The origins of scaling in cities. Sci. (1979). 340, 1438–1441 (2013).
Anabalón, A., Willison, S. & Zanelli, J. General relativity from a gauged Wess-Zumino-Witten term. Phys. Rev. D. 75, 024009 (2007).
Bettencourt, L. M. A., Lobo, J., Strumsky, D. & West, G. B. Urban scaling and its deviations: revealing the structure of Wealth, innovation and crime across cities. PLoS One. 5, e13541 (2010).
Bettencourt, L. M. A. & Lobo, J. Urban scaling in Europe. J. R Soc. Interface. 13, 20160005 (2016).
Bettencourt, L. M. A., Lobo, J. & Youn, H. The hypothesis of urban scaling: formalization, implications and challenges. arXiv preprint https://doi.org/10.48550/arXiv.1301.5919 (2013).
Nordbeck, S. Urban allometric growth. Geogr. Ann. Ser. B. 53, 54–67 (1971).
Antonio, F. J., de Picoli, S., Teixeira, J. J. V. & Mendes, R. dos S. Growth Patterns and Scaling Laws Governing AIDS Epidemic in Brazilian Cities. PLoS One. 9, e111015 (2014).
Bilal, U. et al. Scaling of mortality in 742 metropolitan areas of the Americas. Sci Adv 7, 7–114 (2021).
Francisco Cardoso, B. H. & Gonçalves, S. Universal scaling law for human-to-human transmission diseases. Europhys. Lett. 133, 58001 (2021).
Patterson-Lomba, O. & Gomez-Lievano, A. On the scaling patterns of infectious disease incidence in cities. Epi-SCIENCE 1, 116 (2023).
Ribeiro, H. V., Sunahara, A. S., Sutton, J., Perc, M. & Hanley, Q. S. City size and the spreading of COVID-19 in Brazil. PLoS One. 15, e0239699 (2020).
Ribeiro, H. V., Oehlers, M., Moreno-Monroy, A. I., Kropp, J. P. & Rybski, D. Association between population distribution and urban GDP scaling. PLoS One. 16, e0245771 (2021).
Batty, M. & Longley, P. Fractal Cities: A Geometry of Form and Function (Academic, 1994).
Arcaute, E. et al. Cities and regions in Britain through hierarchical percolation. R Soc. Open. Sci 3(4), 150691.https://doi.org/10.1098/rsos.150691 (2016).
Cobb, C. W. & Douglas, P. H. A theory of production. Am. Econ. Rev. 18, 139–165 (1928).
Christensen, L. R., Jorgenson, D. W. & Lau, L. J. Transcendental logarithmic production frontiers. Rev. Econ. Stat. 55, 28 (1973).
Loureiro, N. A., Neto, C. R., Sutton, J., Perc, M. & Ribeiro, H. V. Impact of inter-city interactions on disease scaling. Sci. Rep. 15, 498 (2025).
Ribeiro, H. V., Rybski, D. & Kropp, J. P. Effects of changing population or density on urban carbon dioxide emissions. Nat. Commun. 10, 3204 (2019).
Arcaute, E. et al. Constructing cities, deconstructing scaling laws. J. R Soc. Interface. 12 (102), 2015 (2015).
Leitão, J. C., Miotto, J. M., Gerlach, M. & Altmann, E. G. Is this scaling nonlinear? R Soc. Open. Sci. 3, 150649 (2016).
Altmann, E. G. Statistical Laws in Complex Systems: Combining Mechanistic Models and Data Analysis (Springer Nature Switzerland, 2024). https://doi.org/10.1007/978-3-031-73164-8
Sutton, J., Shahtahmassebi, G., Hanley, Q. S. & Ribeiro, H. V. A heteroscedastic bayesian generalized logistic regression model with application to scaling problems. Chaos Solitons Fractals. 182, 114787 (2024).
Sutton, J., Shahtahmassebi, G., Ribeiro, H. V. & Hanley, Q. S. Population density and spreading of COVID-19 in England and Wales. PLoS One. 17, e0261725 (2022).
Hanley, Q. S., Lewis, D. & Ribeiro, H. V. Rural to urban population density scaling of crime and property transactions in english and Welsh parliamentary constituencies. PLoS One. 11, e0149546 (2016).
Sutton, J., Shahtahmassebi, G., Ribeiro, H. V. & Hanley, Q. S. Rural–urban scaling of age, mortality, crime and property reveals a loss of expected self-similar behaviour. Sci. Rep. 10, 16863 (2020).
Ribeiro, H. V., Hanley, Q. S. & Lewis, D. Unveiling relationships between crime and property in England and Wales via density scale-adjusted metrics and network tools. PLoS One. 13, e0192931 (2018).
Batty, M. The New Science of Cities (MIT Press, 2013).
Muggeo, V. M. R. Estimating regression models with unknown break-points. Stat. Med. 22, 3055–3071 (2003).
Muggeo, V. M. R. Interval Estimation for the breakpoint in segmented regression: a smoothed score-based approach. Aust N Z. J. Stat. 59, 311–322 (2017).
Muggeo, V. M., Atkins, D. C., Gallop, R. J. & Dimidjian, S. Segmented mixed models with random changepoints: a maximum likelihood approach with application to treatment for depression study. Stat. Modelling. 14, 293–313 (2014).
Burnham, K. P. & Anderson, D. R. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (Springer New York, 2002). https://doi.org/10.1007/b97636
R Core Team. R: A Language and Environment for Statistical Computing. Preprint at https://www.r-project.org/ (2013).
Dragulescu, A. A. & Arendt, C. Read, Write, Format Excel 2007 and Excel 97/2000/XP/2003. CRAN: Contributed Packages Preprint at (2018). https://cran.r-project.org/package=xlsx
Chan, C., Chan, G. C. H., Leeper, T. J., Becker, J. & rio A Swiss-army knife for data file I/O. CRAN: Contributed Packages Preprint at https://rdrr.io/cran/rio/man/rio.html (2021).
Muggeo, V. M. R. Testing with a nuisance parameter present only under the alternative: a score-based approach with application to segmented modelling. J. Stat. Comput. Simul. 86, 3059–3067 (2016).
Fasola, S., Muggeo, V. M. R. & Küchenhoff, H. A heuristic, iterative algorithm for change-point detection in abrupt change models. Comput. Stat. 33, 997–1015 (2018).
Warnes, G. R. et al. Gplots: various R programming tools for plotting data. CRAN: Contributed Packages Preprint at. https://doi.org/10.32614/CRAN.package.gplots (2022).
BBC News. Greater Manchester Police ‘failed to record 80,000 crimes in a year’. https://www.bbc.co.uk/news/uk-england-manchester-55251366 (2020).
Neanidis, K. C. & Rana, M. P. Crime in the era of COVID-19: evidence from England. SSRN Electron. J. https://doi.org/10.2139/ssrn.3784821 (2022).
Cabrera-Arnau, C., Curiel, P., Bishop, S. R. & R. & Uncovering the behaviour of road accidents in urban areas. R Soc. Open. Sci. 7, 191739 (2020).
Finance, O. & Cottineau, C. Are the absent always wrong? Dealing with zero values in urban scaling. Environ. Plan. B Urban Anal. City Sci. 46, 1663–1677 (2019).
UK Government. What qualification levels mean. https://www.gov.uk/what-different-qualification-levels-mean/list-of-qualification-levels
Devanand, D. P. et al. Computerized games versus crosswords training in mild cognitive impairment. NEJM Evidence 1, EVIDoa2200121.https://doi.org/10.1056/EVIDoa2200121(2022).
OECD. Fiscal Sustainability of Health Systems: How To Finance More Resilient Health Systems when Money Is Tight? https://doi.org/10.1787/880f3195-en (OECD, 2024).
Funding
This research was funded by East Midland’s Secure Data Environment as part of the NHS England initiative. Additional support was provided by ‘1 Decembrie 1918’ University of Alba Iulia through scientific research funds. Thomas Peron acknowledges support from FAPESP (Gran No. 2023/07481-6) and CNPq/Brazil (Grant No. 310248/2023-0).
Author information
Authors and Affiliations
Contributions
J.S., Q.S.H., G.M., O.B., H.V.R., T.P., G.S., and P.S. designed research, performed research, analysed data, and wrote the paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
About this article
Cite this article
Sutton, J., Hanley, Q.S., Mortimore, G. et al. Comprehensive indicators and fine granularity refine density scaling laws in rural-urban systems. Sci Rep (2026). https://doi.org/10.1038/s41598-026-40238-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41598-026-40238-7
