Research ArticleSOCIAL SCIENCES

Urban growth and the emergent statistics of cities

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Science Advances  19 Aug 2020:
Vol. 6, no. 34, eaat8812
DOI: 10.1126/sciadv.aat8812
  • Fig. 1 Scheme of statistical theory, assumptions, and derived consequences.

    Basic assumptions are shown as blue boxes, and derived results are shown as red boxes; arrows indicate outcomes, while dashed lines represent alternative scenarios. The budget condition, y – c, is the common basic assumption for urban agents, generalizing energy conservation in simpler systems. Recognizing its dynamical, stochastic nature leads to the central assumption of the manuscript that agents must actively control its associated volatility; the simplest way to do this is through the time averaging of expenditures (consumption smoothing). Then, the resource growth rate volatility, σ2, becomes finite and small, both at the individual and group levels. This leads to emergent stochastic geometric dynamics of resources both at the individual and population levels with exponential growth and lognormal statistics observable at long times (right). Averaging over populations derives the growth rate statistics for cities, which determines when dynamics become self-similar across scales and preserve urban scaling (left): First, if under group averaging variations of the growth rate (γ) are correlated to those in agent resources (r) inequality will change within the population. Second, if effective growth rates are independent of population size, the dynamics becomes self-similar and urban scaling is preserved over time. Alternatively, if the growth rate volatility(σ2) is population size dependent, corrections to mean-field exponents result: They are calculable via B ≠ 0 and are controlled by the volatility’s magnitude. For large σ2(N), the statistics become dominated by fluctuations, and urban scaling breaks down. The existence of strong group volatilities contradicts the assumption of effective control at lower levels. This regime would be unstable, signaling the loss of control over resource flows for most of the population and entailing wide-spread crises and eventual collapse. See text for detailed notation.

  • Fig. 2 Urban scaling and the dynamics of growth and deviations.

    (A) Total wages for U.S. metropolitan areas 1969–2016. Each circle is a city in a given year from blue (1969) to brown (2015). Yellow squares show the urban system’s centers (〈 ln N〉, 〈 ln Y〉), which account for collective economic and population growth (movement, upward and to the right). Urban scaling relations for each year (black lines) are derived through the consideration of a short-term spatial equilibrium (inset), which changes on a very slow time scale. (B) Centered data, obtained from (A) by removing the center’s motion (inset). This allows the decomposition of temporal change into two separate processes: collective growth (center’s motion) and deviations from scaling, ξi(t), characteristic of each city i. We see that scaling with a common exponent (global fit β = 1.114, 95% confidence interval = [1.111, 1.117], R2 = 0.935) is preserved over time, and net growth is a property of the urban system and not of individual cities. (C) The statistics of deviations, obtained from the residuals of the centered scaling fit of (B). While the ξ distribution is well localized and symmetrical, it is not very well fit by a normal distribution (blue line). Instead, the red dashed line, which follows from theory developed in the paper, produces a much better account of the data. (D) The deviations ξi(t) of a few selected cities: Silicon Valley (San Jose–Santa Clara MSA) and Boulder, CO show two of the more exceptional trajectories in wage gains for their city sizes, whereas Las Vegas, NV and Havasu, AZ illustrate wage losses. New York City, Los Angeles, and the exceptionally poor McAllen, TX show no relative change in their positions over nearly 50 years.

  • Fig. 3 General properties of stochastic geometric growth and their consequences for cities.

    (A) Example of growth trajectories for a simple process of geometric Brownian motion (Eq. 4). The blue trajectory shows typical growth with small fluctuations and positive effective growth rate, the orange line shows a similar situation with larger fluctuations, and the green line shows a trajectory with critical γr = 0. The purple and red lines illustrate negative effective growth rate trajectories. The critical growth time, t*, is shown for growing trajectories. (B) An ensemble of trajectories with stochastic growth rates similar to those of U.S. MSAs, starting with the same initial conditions. The yellow line shows the temporal trajectory of the ensemble average, and the black lines show the 95% confidence interval. Note that both the mean and the SD are time dependent (see text). The inset shows the resource distribution at a later time, which becomes asymptotically lognormal (red line). (C) The general properties of stochastic growth imply that a positive growth rate is necessary to overcome temporal decay due to rate fluctuations. If volatility increases, growth will ultimately stop, and decay will ensue. The critical point γr=η¯rσr22=0 is characterized by large fluctuations with a diverging t* so that agents will not be able to tell whether they are experiencing growth and may be unable to exert effective control (see text). (D) Under general conditions, multiplicative random growth can be self-similar across group sizes, providing a simple theory that applies at all scales, from individual agents to populations and cities (see section S3) (29). However, the key parameters of the theory “run” across scales and are in general sensitive to both group size, temporal averaging, and inequality. These dependences define urban scaling as a dynamical statistical theory beyond the mean-field approximation.

  • Fig. 4 Dynamically balancing income and costs via feedback control leads to simple statistics for resource growth rates.

    (A) Example trajectories for the income-to-resources and costs-to-resources ratios, b(t) (red) and a(t) (blue), respectively. Note that when income is larger than costs, there can be growth, but fluctuations need to be controlled. (B) Control scheme to deliver average growth rate and tame fluctuations εr(t). Costs a(t) become a control variable that, in part, adapts to environmental fluctuations to generate εr(t) with small, known variance. (C) The dynamics of the resulting error εr(t) is now centered around zero and (D) displays a Gaussian distribution (red line) with variance given by the ratio of the environmental variance to control parameters (see text). In this way, adaptive agents’ behavior can lead to predictable growth in stochastic environments with a chosen variance.

  • Fig. 5 Effective diffusive growth of deviations ξi (t) and the emerging statistics of cities.

    (A) On the average over cities, the displacement from their initial deviations in 1969 grows linearly (red line) (gradient = 0.00108, 95% confidence interval = [0.00102, 0.00115]; intercept = −2.13279, 95% confidence interval = [−2.25885, −2.00672], R2 = 0.93), as expected from pure random diffusion of the growth rates. Note that this is a mean temporal behavior and that there are periods when deviations grow faster or slower. Periods of economic recession are shown in gray. (B) The trajectory of deviations for all cities (different colors) but having set all deviations in 1969 to zero so that all trajectories depart from a common origin. The red line indicates the diffusive behavior, same as in (A), clearly showing that deviations tend to increase in magnitude over time. (C) The prediction of the wage growth volatility for U.S. MSAs by three methods: the fit of (A) and (B) and the averages over time and sets of cities, demonstrating the ergodic character of the statistical dynamics. Shaded areas show the overlapping 95% intervals in these estimates. (D) The distribution of deviations, year by year, using the same color scheme as in Fig. 2 (A and B). We see that, unlike our first approach in Fig. 2C, the width of the distributions is increasing slowly over time (brown most recent) and that the data for wages (a flow) should be fit by a distribution that is well described as the sum of two Gaussians: a universal broad distribution due to resource compounding and a contingent short-term narrow distribution (Eq. 20), which depends on most recent environmental shocks.

Supplementary Materials

  • Supplementary Materials

    Urban growth and the emergent statistics of cities

    Luis M. A. Bettencourt

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