Pith. sign in

REVIEW 4 major objections 4 minor 52 references

Impact of inter-city interactions on disease scaling

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Commuting flows change how disease cases scale with city population.

desk verdict A useful, transparent application of production-function models to disease scaling, but the central claim that commuters add explanatory power beyond population is undercut by a missing nonlinear population-only baseline. read the letter →

arxiv 2501.01395 v1 pith:5NMDUYF6 submitted 2025-01-02 physics.soc-ph q-bio.PE

Nathalia A. Loureiro , Camilo R. Neto , Jack Sutton , Matjaz Perc , Haroldo V. Ribeiro This is my paper · ORCID
classification physics.soc-phq-bio.PE
keywords urbanscalingdiseaseincidencecommutingnetworkinter-cityinteractionstranslogproductionfunctionelasticityofscaleBraziliancitiesregimes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the standard urban scaling law, in which disease cases depend only on population, misses a major driver: interactions between cities. Using the Brazilian commuting network as a proxy for those interactions, it adds a second variable, a city's total number of commuters $S$, and fits disease cases with a translog production function. Across seven infectious diseases, this two-input model fits observed cases better than population-only scaling and removes most of the underestimation in large cities. The model's elasticity of scale is not constant but depends on the product of population and commuters, so each city has its own scaling regime: most are sublinear, some are superlinear, and a few small isolated cities show negative elasticity, where growth and connectivity accompany fewer reported cases. If the claim holds, disease burden in cities cannot be summarized by a single scaling exponent.

What carries the argument

The load-bearing object is the translog production function, $\log Y \sim \beta_N \log N + \beta_S \log S + \beta_C \log N \log S$, borrowed from production economics and applied to disease counts as output, population $N$ as one input, and the weighted total degree $S$ of the commuting network as the other. The weighted total degree $S$ is the total number of commuters moving between a city and all other cities, regardless of direction; it is intended as a proxy for the strength of inter-city interactions. The interaction term $\beta_C \log N \log S$ makes the elasticity $\epsilon = \beta_N + \beta_S + \beta_C \log(NS)$ depend on the initial values of $N$ and $S$, which is what generates the city-specific scaling regimes and the negative-elasticity pockets in small cities. Ridge regression is used to estimate the parameters because $\log N$ and $\log S$ are strongly collinear.

What would settle it

Fit the same translog model with $S$ replaced by a randomized commuting count or by an unrelated variable such as city area, and check whether the improvement over urban scaling persists; if it does, the claimed effect is not specific to inter-city interaction. Alternatively, hold out the largest cities and refit: if the model's advantage over urban scaling vanishes, the result is carried by the same high-leverage points the model was introduced to fix.

Watch

Extended reading notes

Core claim

The central claim is that inter-city commuting, measured as the weighted total degree $S$ of each city in the commuting network, is a genuine second input in the scaling of infectious disease incidence. Formally, the paper replaces $\log Y \sim \beta_N \log N$ with $\log Y \sim \beta_N \log N + \beta_S \log S + \beta_C \log N \log S$, the translog production function, and estimates its parameters with ridge regression. For all seven diseases the interaction coefficient $\beta_C$ is positive and the commuter coefficient $\beta_S$ is negative, while $\beta_N$ changes sign across diseases. The resulting city-level elasticity $\epsilon = \beta_N + \beta_S + \beta_C \log(NS)$ crosses thresholds: when the product $NS$ is small, $\epsilon$ can be negative; for most cities $0 < \epsilon < 1$; and for large connected cities $\epsilon > 1$. The paper reads these regimes as evidence that small, isolated cities can benefit from growth and connectivity through better healthcare access and vaccination, while large cities experience conditions that amplify transmission. It also reports that changes in population matter more than changes in commuters for almost all cities and diseases, with pertussis as a partial exception.

Load-bearing premise

The whole argument rests on treating a city's total number of commuters $S$ as a faithful proxy for all inter-city interactions that matter for disease transmission; if commuter counts mainly track some other city characteristic, the model's better fit could come from that confounder rather than from genuine inter-city effects.

Editorial extensions

If this is right

  • Disease scaling exponents estimated from population alone will be systematically biased in large, well-connected cities, since the population-only model underestimates cases there.
  • A city's scaling regime is set by whether its product of population and commuters, $NS$, lies below, between, or above the thresholds defined by the fitted parameters, so regime predictions are testable city by city.
  • For most Brazilian cities, a 1% simultaneous rise in population and commuters is associated with less than a 1% rise in cases; for a minority it is more than 1%.
  • Population remains the dominant driver: proportional changes in population affect cases more than proportional changes in commuters in the large majority of cities for all diseases except pertussis.
  • The negative-elasticity cities, especially for pertussis, point to a window in which growth and connectivity are associated with fewer cases, implying that the health benefits of connectivity can be outgrown.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the commuting proxy is right, the same translog machinery should transfer to other interaction channels such as air travel, trade, or migration; a direct test would replace $S$ with those flows and see whether the interaction term survives.
  • The functional form suggests a policy-relevant threshold: interventions that raise connectivity in small isolated cities may initially lower disease burden, but the same policy in larger, denser cities could increase transmission; this crossover is an empirical prediction that could be checked with time series.
  • Because the model is cross-sectional, the negative elasticity for small cities is a correlation, not proof of causation; a panel version following individual cities over time as they grow and gain commuters would test whether the same city traverses negative, sublinear, and superlinear regimes.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper extends urban scaling analyses of infectious disease incidence by adding an inter-city interaction term, measured as the weighted total degree S of each Brazilian city in the 2010 commuting network. It compares three models for the number of reported cases Y as a function of population N and S: standard urban scaling (log Y ~ beta_N log N), Cobb-Douglas (log Y ~ beta_N log N + beta_S log S), and a translog production function (log Y ~ beta_N log N + beta_S log S + beta_C log N log S). The translog model yields the lowest AIC/BIC and highest R2 for all seven diseases, and the associated elasticity epsilon = beta_N + beta_S + beta_C log(NS) is used to classify cities into negative, sublinear, and superlinear scaling regimes. The authors conclude that inter-city interactions, approximated by commuting flows, improve disease-incidence modeling and that population size remains the dominant factor.

Significance. The production-function translog framework is a promising extension of urban scaling, and the paper is methodologically transparent in several respects: it discusses multicollinearity, applies ridge regression, and compares models with three criteria across seven diseases. The thresholds separating scaling regimes are explicit and falsifiable in principle. If the main inference is supported, the work would provide a relatively simple way to incorporate inter-city connectivity into disease scaling and could inform public-health planning. However, the central claim that commuting interactions improve the description over and above population size is not yet established, because the model comparison lacks a flexible population-only baseline and the dataset conditions on cities with at least one reported case. These gaps are fixable and should be addressed before the results are interpreted as evidence for the role of inter-city interactions.

major comments (4)
  1. [Results, Fig. 3 and Methods] The central claim that inter-city interactions matter is inferred from the translog model (Eq. 3) outperforming the urban scaling model (Eq. 1) and Cobb-Douglas model (Eq. 2). But since log S is strongly collinear with log N (acknowledged in the Methods), the interaction term beta_C log N log S is approximately a quadratic function of log N under log S ≈ a + b log N. A population-only model with a quadratic term, log Y ~ beta_N log N + beta_2 (log N)^2, has one fewer parameter and is never reported. Without this baseline, the lower AIC/BIC and higher R2 for the translog can be explained by extra flexibility in N rather than by an independent contribution of S. Please add such a baseline (or a spline in log N) and show that including S still improves fit and, ideally, out-of-sample prediction.
  2. [Data, Fig. 5 and Discussion] The analysis is restricted to cities with at least one reported case, but the number and population distribution of excluded cities are not given. Because small cities are more likely to report zero cases, truncating on the outcome can induce artificial curvature in the N-Y relation and may contribute to the negative-elasticity regime found for small, isolated cities. The Discussion lists zero disease counts as a limitation, but the manuscript should provide a quantitative robustness check (e.g., a zero-inflated or hurdle count model, or an analysis that includes zeros) and report how many cities are excluded per disease.
  3. [Methods, ridge regression and Fig. 4] The manuscript reports standard errors and statistical significance for the ridge estimates (Fig. 4), but it does not state how these standard errors are computed. Ridge coefficients are biased, and their usual OLS-based standard errors are not directly valid; bootstrap or a stated closed-form covariance under the chosen regularization is needed. Since the thresholds Omega*, S*, N* and the elasticity epsilon are nonlinear functions of these coefficients, the absence of uncertainty propagation for these derived quantities is a gap. Please provide bootstrap intervals or an equivalent treatment for the elasticity values and thresholds.
  4. [Discussion and limitations] The commuting weighted degree S is the sole proxy for inter-city interactions, and no alternative or additional proxies (e.g., air travel, trade flows, social-network ties) are tested. The paper acknowledges this in the Discussion, but the title and abstract make a stronger claim: that inter-city interactions are 'critical' for disease transmission. To make the inference more convincing, at least one alternative proxy or a falsifiable comparison should be reported, or the conclusions should be reworded to reflect that the evidence is specific to the commuting-network operationalization.
minor comments (4)
  1. [Fig. 4 caption] The caption for Fig. 4C says 'Parameter beta_S' but it should read 'Parameter beta_C'.
  2. [Abstract and Results] The text lists 'seven infectious diseases' but then enumerates only six (HIV/AIDS, influenza, pertussis, syphilis, tuberculosis, viral hepatitis); meningitis is missing.
  3. [Eqs. 1–5] The logarithm base is not specified in the equations and thresholds; the figures use base-10 logarithms, so please state this explicitly and keep it consistent throughout.
  4. [Fig. 4] The statement that all parameters are statistically significantly different from zero should be accompanied by the significance criterion and a multiple-comparison correction, given the several diseases and parameters tested.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the translog fit is an empirical model comparison, and the scaling regimes are algebraic consequences of fitted coefficients, not recycled inputs.

full rationale

The derivation chain is not circular. The urban scaling, Cobb-Douglas, and translog models (Eqs. 1-3) are fit to the observed disease-case data and compared by R2, AIC, and BIC, so the claim that the translog model improves the description is an empirical model-selection result rather than a consequence of how the inputs are defined. The elasticity epsilon = beta_N + beta_S + beta_C log(NS) in Eq. 4, and the thresholds Omega*, Omega-tilde*, S*, and N*, are indeed algebraic consequences of the fitted translog parameters, and the paper itself says 'we focus on interpreting its adjusted behavior for each disease type,' acknowledging that the regimes come from the adjusted model rather than from an independent out-of-sample prediction. That is standard post-fit interpretation, not a fitted input renamed as a prediction. The commuting strength S is an independently measured census quantity, not defined through disease cases, so there is no self-definitional step. The self-citations (refs. 27, 30, and 41) supply the Cobb-Douglas/translog ansatz and the ridge-regression recipe, but the disease-specific fits and model comparisons are computed in this paper, so the cited prior work is not load-bearing evidence for the central disease claim. The Discussion's limitation statement that 'the strong correlations between population size and the number of commuters may constrain the ability to disentangle their individual effects' is a genuine identification caveat, and the absence of a population-only quadratic baseline is a model-competition concern, but both bear on robustness and correctness rather than on circularity. Overall, the central claim is self-contained against the data and does not reduce to its inputs by construction.

Assumptions & free parameters 28 free parameters · 5 assumptions · 0 invented entities

The central claim rests on 28 fitted values (21 translog coefficients and 7 ridge regularization parameters), all estimated from the same data used to evaluate the model. No new theoretical entities are introduced, but the commuting network proxy and the production-function functional forms are domain assumptions imported without independent validation.

free parameters (28)
  • beta_N, tuberculosis = not given numerically (Fig. 4)
    Translog coefficient for log population in tuberculosis model.
  • beta_S, tuberculosis = not given numerically (Fig. 4)
    Translog coefficient for log commuters in tuberculosis model.
  • beta_C, tuberculosis = not given numerically (Fig. 4)
    Translog interaction coefficient in tuberculosis model.
  • beta_N, HIV/AIDS = not given numerically (Fig. 4)
    Translog coefficient for log population in HIV/AIDS model.
  • beta_S, HIV/AIDS = not given numerically (Fig. 4)
    Translog coefficient for log commuters in HIV/AIDS model.
  • beta_C, HIV/AIDS = not given numerically (Fig. 4)
    Translog interaction coefficient in HIV/AIDS model.
  • beta_N, viral hepatitis = not given numerically (Fig. 4)
    Translog coefficient for log population in viral hepatitis model.
  • beta_S, viral hepatitis = not given numerically (Fig. 4)
    Translog coefficient for log commuters in viral hepatitis model.
  • beta_C, viral hepatitis = not given numerically (Fig. 4)
    Translog interaction coefficient in viral hepatitis model.
  • beta_N, meningitis = not given numerically (Fig. 4)
    Translog coefficient for log population in meningitis model.
  • beta_S, meningitis = not given numerically (Fig. 4)
    Translog coefficient for log commuters in meningitis model.
  • beta_C, meningitis = not given numerically (Fig. 4)
    Translog interaction coefficient in meningitis model.
  • beta_N, syphilis = not given numerically (Fig. 4)
    Translog coefficient for log population in syphilis model.
  • beta_S, syphilis = not given numerically (Fig. 4)
    Translog coefficient for log commuters in syphilis model.
  • beta_C, syphilis = not given numerically (Fig. 4)
    Translog interaction coefficient in syphilis model.
  • beta_N, influenza = not given numerically (Fig. 4)
    Translog coefficient for log population in influenza model.
  • beta_S, influenza = not given numerically (Fig. 4)
    Translog coefficient for log commuters in influenza model.
  • beta_C, influenza = not given numerically (Fig. 4)
    Translog interaction coefficient in influenza model.
  • beta_N, pertussis = not given numerically (Fig. 4)
    Translog coefficient for log population in pertussis model.
  • beta_S, pertussis = not given numerically (Fig. 4)
    Translog coefficient for log commuters in pertussis model.
  • beta_C, pertussis = not given numerically (Fig. 4)
    Translog interaction coefficient in pertussis model.
  • lambda, tuberculosis = not reported
    Ridge regularization hyperparameter selected by leave-one-out cross-validation.
  • lambda, HIV/AIDS = not reported
    Ridge regularization hyperparameter selected by leave-one-out cross-validation.
  • lambda, viral hepatitis = not reported
    Ridge regularization hyperparameter selected by leave-one-out cross-validation.
  • lambda, meningitis = not reported
    Ridge regularization hyperparameter selected by leave-one-out cross-validation.
  • lambda, syphilis = not reported
    Ridge regularization hyperparameter selected by leave-one-out cross-validation.
  • lambda, influenza = not reported
    Ridge regularization hyperparameter selected by leave-one-out cross-validation.
  • lambda, pertussis = not reported
    Ridge regularization hyperparameter selected by leave-one-out cross-validation.
assumptions (5)
  • domain assumption Commuting network weighted degree S is a valid proxy for inter-city interactions.
    Used throughout to define the second input to production functions; if interactions relevant to disease spread are not captured by commuter flows, the model's improvement is confounded.
  • domain assumption Production function forms (Cobb-Douglas, translog) are appropriate for modeling disease case counts.
    The paper adopts these from economic theory without mechanistic derivation, and acknowledges this limitation.
  • domain assumption Reported disease cases from DATASUS reflect true incidence and are comparable across cities.
    Underreporting and surveillance heterogeneity would bias scaling exponents; the paper does not correct for this.
  • domain assumption Cross-sectional 2010 data can identify scaling relationships.
    All variables are from 2010; no temporal dynamics are modeled.
  • standard math Ridge regression with standardized predictors yields meaningful parameter estimates despite collinearity.
    Needed to stabilize inversion of the Gram matrix; standard practice but introduces bias.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Impact of inter-city interactions on disease scaling." pith.science (2026). https://pith.science/paper/5NMDUYF6

@misc{pith2026250101395,
  author       = {Pith},
  title        = {Pith review of: Impact of inter-city interactions on disease scaling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5NMDUYF6}},
  note         = {Machine review of arXiv:2501.01395}
}
read the original abstract

Inter-city interactions are critical for the transmission of infectious diseases, yet their effects on the scaling of disease cases remain largely underexplored. Here, we use the commuting network as a proxy for inter-city interactions, integrating it with a general scaling framework to describe the incidence of seven infectious diseases across Brazilian cities as a function of population size and the number of commuters. Our models significantly outperform traditional urban scaling approaches, revealing that the relationship between disease cases and a combination of population and commuters varies across diseases and is influenced by both factors. Although most cities exhibit a less-than-proportional increase in disease cases with changes in population and commuters, more-than-proportional responses are also observed across all diseases. Notably, in some small and isolated cities, proportional rises in population and commuters correlate with a reduction in disease cases. These findings suggest that such towns may experience improved health outcomes and socioeconomic conditions as they grow and become more connected. However, as growth and connectivity continue, these gains diminish, eventually giving way to challenges typical of larger urban areas - such as socioeconomic inequality and overcrowding - that facilitate the spread of infectious diseases. Our study underscores the interconnected roles of population size and commuter dynamics in disease incidence while highlighting that changes in population size exert a greater influence on disease cases than variations in the number of commuters.

Figures

Figures reproduced from arXiv: 2501.01395 by the authors.

Figure 1
Figure 1. Commuting network among Brazilian cities. The map displays the locations of Brazilian cities, which correspond to the network nodes. Connections represent the flow of commuters between city pairs, irrespective of direction. Node sizes are proportional to the weighted total degree of the cities and are also color-coded accordingly. Edge widths indicate the number of commuters between city pairs. In this visualization… view at source ↗
Figure 2
Figure 2. Urban scaling predictions of disease cases compared with the enhanced descriptions of the Cobb-Douglas and translog models. (A) Relationship between the predictions from the urban scaling model (Eq. 1) and the observed cases of HIV/AIDS, meningitis, and influenza. (B) Improved predictions considering the number of commuters in the Cobb-Douglas model (Eq. 2). (C) Improved predictions considering the number of commute… view at source ↗
Figure 3
Figure 3. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Parameters of the translog model estimated for each disease type. (A) Parameter βN quantifying the association between population size and disease cases. (B) Parameter βS quantifying the association between the number of commuters and disease cases. (C) Parameter βS qu…
Figure 5
Figure 5. Figure 5: Elasticity of scale for each city and transitions in the scaling regime. (A) Dependence of the elasticity of scale ε on population and commuter numbers for every Brazilian city reporting cases of tuberculosis, HIV/AIDS, viral hepatitis, meningitis, syphilis, influenza,…
Figure 6
Figure 6. Figure 6: Marginal products of population and total number of commuters. Probability distributions of the marginal products for (A) population ∆N and (B) commuters ∆S, calculated for all Brazilian cities reporting cases of tuberculosis, HIV/AIDS, viral hepatitis, meningitis, syp…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 39 canonical work pages

  1. [1]

    Available: http://data.worldbank.org/ indicator/SP.URB.TOTL.IN.ZS

    United Nations, World Urbanization Prospects: Urban population (% of total). Available: http://data.worldbank.org/ indicator/SP.URB.TOTL.IN.ZS. Accessed: 27 May 2020

    work page 2020

  2. [2]

    & O’Neill, B

    Jiang, L. & O’Neill, B. C. Global urbanization projections for the Shared Socioeconomic Pathways. Glob. Environ. Chang. 42, 193–199, DOI: 10.1016/j.gloenvcha.2015.03.008 (2017)

    work page doi:10.1016/j.gloenvcha.2015.03.008 2017

  3. [3]

    The New Science of Cities (MIT Press, Cambridge, MA, 2013)

    Batty, M. The New Science of Cities (MIT Press, Cambridge, MA, 2013)

    work page 2013

  4. [4]

    West, G. B. Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies (Penguin, New York, 2017)

    work page 2017

  5. [5]

    D’Acci, L. S. (ed.) Urban Scaling: Allometry in Urban Studies and Spatial Science. Advances in Regional Economics, Science and Policy (Routledge, Abingdon, 2025)

    work page 2025

  6. [6]

    Woldenberg, M. Part II. Harvard Papers in Theoretical Geography: Geography and the Properties of Surfaces Series (1967)

    work page 1967

  7. [7]

    Urban allometric growth

    Nordbeck, S. Urban allometric growth. Geografiska Annaler: Ser. B, Hum. Geogr. 53, 54–67, DOI: 10.2307/490887 (1971)

    work page doi:10.2307/490887 1971

  8. [8]

    & West, G

    Kühnert, C., Helbing, D. & West, G. B. Scaling laws in urban supply networks. Phys. A 363, 96–103, DOI: 10.1016/j. physa.2006.01.058 (2006)

    doi:10.1016/j 2006

Show all 52 references
  1. [9]

    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, DOI: 10.1073/pnas.0610172104 (2007)

    2007 doi

  2. [10]

    Bettencourt, L. M. A. The origins of scaling in cities. Science 340, 1438–1441, DOI: 10.1126/science.1235823 (2013)

    2013 doi

  3. [11]

    B., Brown, J

    West, G. B., Brown, J. H. & Enquist, B. J. A general model for the origin of allometric scaling laws in biology. Science 276, 122–126, DOI: 10.1126/science.276.5309.122 (1997)

    1997 doi

  4. [12]

    J., de Picoli Jr, S., Teixeira, J

    Antonio, F. J., de Picoli Jr, S., Teixeira, J. J. V . & Mendes, R. D. S. Growth patterns and scaling laws governing AIDS epidemic in Brazilian cities. PLOS ONE 9, e111015, DOI: 10.1371/journal.pone.0111015 (2014)

    2014 doi

  5. [13]

    E., Thorson, A

    Rocha, L. E., Thorson, A. E. & Lambiotte, R. The non-linear health consequences of living in larger cities. J. Urban Heal. 92, 785–799, DOI: 10.1007/s11524-015-9976-x (2015)

    2015 doi

  6. [14]

    Schläpfer, M. et al. The scaling of human interactions with city size. J. Royal Soc. Interface 11, 20130789, DOI: 10.1098/rsif.2013.0789 (2014)

    2014

  7. [15]

    & Towers, S

    Patterson-Lomba, O., Goldstein, E., Gómez-Liévano, A., Castillo-Chavez, C. & Towers, S. Per capita incidence of sexually transmitted infections increases systematically with urban population size: A cross-sectional study. Sex. Transm. Infect. 91, 610–614, DOI: 10.1136/sextrans...

    2015 doi

  8. [16]

    Bilal, U. et al. Scaling of mortality in 742 metropolitan areas of the Americas. Sci. Adv. 7, eabl6325, DOI: 10.1126/sciadv. abl6325 (2021)

    2021 doi

  9. [17]

    & Gómez-Lievano, A

    Patterson-Lomba, O. & Gómez-Lievano, A. On the scaling patterns of infectious disease incidence in cities. Epi-SCIENCE 1, 60–86, DOI: 10.15517/es.2023.55397 (2023). 11/13

    2023

  10. [18]

    J., Berman, M

    Stier, A. J., Berman, M. G. & Bettencourt, L. M. Early pandemic COVID-19 case growth rates increase with city size. npj Urban Sustain. 1, 31, DOI: 10.1209/0295-5075/133/58001 (2021)

    2021 doi

  11. [19]

    V ., Sunahara, A

    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, DOI: 10.1371/journal.pone.0239699 (2020)

    2020 doi

  12. [20]

    Ribeiro, F. L. & Rybski, D. Mathematical models to explain the origin of urban scaling laws. Phys. Reports 1012, 1–39, DOI: 10.1016/j.physrep.2023.02.002 (2023)

    2023 doi

  13. [21]

    & Amaral, L

    Guimera, R., Uzzi, B., Spiro, J. & Amaral, L. A. N. Team assembly mechanisms determine collaboration network structure and team performance. Science 308, 697–702, DOI: 10.1126/science.1106340 (2005)

    2005 doi

  14. [22]

    S., Beckfield, J

    Alderson, A. S., Beckfield, J. & Sprague-Jones, J. Intercity relations and globalisation: The evolution of the global urban hierarchy, 1981—2007. Urban Stud. 47, 1899–1923, DOI: 10.1177/0042098010372679 (2010)

    2010 doi

  15. [23]

    & Bishop, S

    Prieto Curiel, R., Pappalardo, L., Gabrielli, L. & Bishop, S. R. Gravity and scaling laws of city to city migration. PLOS ONE 13, e0199892, DOI: 10.1371/journal.pone.0199892 (2018)

    2018 doi

  16. [24]

    Nelson, G. D. & Rae, A. An economic geography of the United States: From commutes to megaregions. PLOS ONE 11, e0166083, DOI: 10.1371/journal.pone.0166083 (2016)

    2016 doi

  17. [25]

    Altmann, E. G. Spatial interactions in urban scaling laws. PLOS ONE 15, e0243390, DOI: 10.1371/journal.pone.0243390 (2020)

    2020 doi

  18. [26]

    V ., Oehlers, M., Moreno-Monroy, A

    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, DOI: 10.1371/journal.pone.0245771 (2021)

    2021 doi

  19. [27]

    Alves, L. G. A., Rybski, D. & Ribeiro, H. V . Commuting network effect on urban wealth scaling.Sci. Reports 11, 22918, DOI: 10.1038/s41598-021-02327-7 (2021)

    2021 doi

  20. [28]

    A., Balland, P.-A., Zheng, S

    Liang, X., Hidalgo, C. A., Balland, P.-A., Zheng, S. & Wang, J. Intercity connectivity and urban innovation. Comput. Environ. Urban Syst. 109, 102092, DOI: 10.1016/j.compenvurbsys.2024.102092 (2024)

    2024

  21. [29]

    Heathfield, D. F. & Wibe, S. An Introduction to Cost and Production Functions (Macmillan: London, 1987)

    1987

  22. [30]

    V ., Rybski, D

    Ribeiro, H. V ., Rybski, D. & Kropp, J. P. Effects of changing population or density on urban carbon dioxide emissions. Nat. Commun. 10, 3204, DOI: 10.1038/s41467-019-11184-y (2019)

    2019 doi

  23. [31]

    Census micro-data 2010

    Brazilian Institute of Geography and Statistics (IBGE). Census micro-data 2010. ftp://ftp.ibge.gov.br/Censos/Censo_ Demografico_2010/Resultados_Gerais_da_Amostra/Microdados (2010). Accessed: 18 September 2024

    2010

  24. [32]

    Available: http://datasus.saude.gov

    Brazil’s Public Healthcare System (SUS), Department of Data Processing (DATASUS). Available: http://datasus.saude.gov. br. Accessed: 18 September 2024

    2024

  25. [33]

    Moura, D. C. 3D density histograms for criteria-driven edge bundling. arXiv preprint arXiv:1504.02687 (2015)

    2015 arXiv

  26. [34]

    Hunter, J. D. Matplotlib: A 2D graphics environment. Comput. Sci. & Eng. 9, 90–95, DOI: 10.1109/MCSE.2007.55 (2007)

    2007 doi

  27. [35]

    Jordahl, K. et al. GeoPandas, DOI: 10.5281/zenodo.3946761

    doi

  28. [36]

    A., Schult, D

    Hagberg, A. A., Schult, D. A. & Swart, P. J. Exploring network structure, dynamics, and function using NetworkX. In Varoquaux, G., Vaught, T. & Millman, J. (eds.)Proceedings of the 7th Python in Science Conference, 11–15 (Pasadena, CA USA, 2008)

    2008

  29. [37]

    Cobb, C. W. & Douglas, P. H. A theory of production. The Am. Econ. Rev. 18, 139–165 (1928)

    1928

  30. [38]

    R., Jorgenson, D

    Christensen, L. R., Jorgenson, D. W. & Lau, L. J. Transcendental logarithmic production frontiers. The Rev. Econ. Stat. 55, 28–45, DOI: 10.2307/1927992 (1973)

    1973 doi

  31. [39]

    & Friedman, J

    Hastie, T., Tibshirani, R. & Friedman, J. The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer, New York, 2009), 2nd edn

    2009

  32. [40]

    Hoerl, A. E. & Kennard, R. W. Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 12, 55–67, DOI: 10.2307/1271436 (1970)

    1970 doi

  33. [41]

    Ribeiro, H. V . & Rybski, D.Urban Scaling: Allometry in Urban Studies and Spatial Science, chap. Urban carbon dioxide emissions and the role of population, area, and density, 219–233. Advances in Regional Economics, Science and Policy (Routledge, Abingdon, 2025)

    2025

  34. [42]

    Pedregosa, F. et al. Scikit-learn: Machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830 (2011). 12/13

    2011

  35. [43]

    Burnham, K. P. & Anderson, D. R. Model Selection and Multimodel Inference (Springer, New York, 2002)

    2002

  36. [44]

    K., Barttfeld, P., Havlin, S., Sigman, M

    Gallos, L. K., Barttfeld, P., Havlin, S., Sigman, M. & Makse, H. A. Collective behavior in the spatial spreading of obesity. Sci. Reports 2, 454, DOI: 10.1038/srep00454 (2012)

    2012 doi

  37. [45]

    J., Itami, A

    Antonio, F. J., Itami, A. S., de Picoli, S., Teixeira, J. J. V . & Mendes, R. d. S. Spatial patterns of dengue cases in Brazil. PLOS ONE 12, e0180715, DOI: 10.1371/journal.pone.0180715 (2017)

    2017 doi

  38. [46]

    Alves, L. G. A., Lenzi, E. K., Mendes, R. S. & Ribeiro, H. V . Spatial correlations, clustering and percolation-like transitions in homicide crimes. Europhys. Lett. 111, 18002, DOI: 10.1209/0295-5075/111/18002 (2015)

    2015 doi

  39. [47]

    G., Andrade Jr, J

    Alves, L. G., Andrade Jr, J. S., Hanley, Q. S. & Ribeiro, H. V . The hidden traits of endemic illiteracy in cities.Phys. A 515, 566–574, DOI: 10.1016/j.physa.2018.09.153 (2019)

    2019 doi

  40. [48]

    S., Lewis, D

    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, DOI: 10.1371/journal.pone.0149546 (2016)

    2016 doi

  41. [49]

    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. Reports 10, 16863, DOI: 10.1038/s41598-020-74015-x (2020)

    2020 doi

  42. [50]

    C., Miotto, J

    Leitao, J. C., Miotto, J. M., Gerlach, M. & Altmann, E. G. Is this scaling nonlinear? Royal Soc. Open Sci. 3, 150649, DOI: 10.1098/rsos.150649 (2016)

    2016 doi

  43. [51]

    & Altmann, E

    Gerlach, M. & Altmann, E. G. Testing statistical laws in complex systems. Phys. Rev. Lett. 122, 168301, DOI: 10.1103/PhysRevLett.122.168301 (2019)

    2019 doi

  44. [52]

    Altmann, E. G. Statistical Laws in Complex Systems: Combining Mechanistic Models and Data Analysis. Understanding Complex Systems (Springer, New York, 2024). 13/13 Impact of inter-city interactions on disease scaling Nathalia A. Loureiro1, Camilo R. Neto1, Jack Sutton2, Matjaˇ...

    2024

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.