Urban scaling and its deviations: revealing the structure of wealth, innovation and crime across cities

Abstract

With urban population increasing dramatically worldwide, cities are playing an increasingly critical role in human societies and the sustainability of the planet. An obstacle to effective policy is the lack of meaningful urban metrics based on a quantitative understanding of cities. Typically, linear per capita indicators are used to characterize and rank cities. However, these implicitly ignore the fundamental role of nonlinear agglomeration integral to the life history of cities. As such, per capita indicators conflate general nonlinear effects, common to all cities, with local dynamics, specific to each city, failing to provide direct measures of the impact of local events and policy. Agglomeration nonlinearities are explicitly manifested by the superlinear power law scaling of most urban socioeconomic indicators with population size, all with similar exponents (1.15). As a result larger cities are disproportionally the centers of innovation, wealth and crime, all to approximately the same degree. We use these general urban laws to develop new urban metrics that disentangle dynamics at different scales and provide true measures of local urban performance. New rankings of cities and a novel and simpler perspective on urban systems emerge. We find that local urban dynamics display long-term memory, so cities under or outperforming their size expectation maintain such (dis)advantage for decades. Spatiotemporal correlation analyses reveal a novel functional taxonomy of U.S. metropolitan areas that is generally not organized geographically but based instead on common local economic models, innovation strategies and patterns of crime.

Figure 1. Urban Agglomeration effects result in per capita nonlinear scaling of urban metrics.

Subtracting these effects produces a truly local measure of urban dynamics and a reference scale for ranking cities. a) A typical superlinear scaling law (solid line): Gross Metropolitan Product of US MSAs in 2006 (red dots) vs. population; the slope of the solid line has exponent,  = 1.126 (95% CI [1.101,1.149]). b) Histogram showing frequency of residuals, (SAMIs, see Eq. (2)); the statistics of residuals is well described by a Laplace distribution (red line). Scale independent ranking (SAMIs) for US MSAs by c) personal income and d) patenting (red denotes above average performance, blue below). For more details see Text S1, Table S1 and Figure S1.

Figure 2. The temporal evolution of scale independent indicators (SAMIs) displays long-term memory.

The value of SAMIs as functions of time for a) income (1969–2006) and b) patents (1975–2006) for selected MAs. Shaded grey areas indicate periods of national economic recession. The temporal autocorrelation c) for patents and personal income and exponential fits, , with characteristic decay times of  = 18.9 and 34.9 years, respectively and d) temporal Fourier power spectrum for the same quantities shows that their dynamics is dominated by long timescales.

Figure 3. Relationships between local urban performance measured by personal income, patents and violent crime and their spatial distributions.

A) normalized SAMIs for income versus patents are shown in polar coordinates, see SI, together with best-fit linear relation capturing overall average correlation (solid line, gradient = 0.380.04,  = 0.20). The color and size of circles both denote the magnitude of the combined SAMIs for each city; b) similar representation for income versus violent crime with best-fit linear relation (gradient = −0.190.07,  = 0.05), and c) similar representation for patents versus violent crime with best-fit linear relation (gradient = −0.340.05,  = 0.12). Note that B) and C) show a small amount of anti-correlation between SAMIs, which contrasts with the positive correlations for the per capita quantities due to their size dependence. d) Spatial distribution of income residuals (SAMIs) in 2006 (created with Google maps, see online (http://www.santafe.edu/urban_observatory/).). Red (blue) dots correspond to deviations above (below) expectation for city size. The size of the circle denotes the magnitude of the SAMIs. e) Average cross-correlation between SAMIs versus spatial separation distance, showing short-range spatial correlation. Averages shown are subject to large variation for distances 200 km (124 miles) with standard deviation 0.6.

Figure 4. Families of kindred cities.

The cross-correlation between SAMI time-series gives a measure of similarity, which can be used to group cities into clusters with similar characteristics; A) sorted correlation matrix (heatmap) for personal income in US MSAs with population over 1 million. Red (blue) denotes greatest (dis)similarity; B)Dendrogram showing detailed urban taxonomy of USMAs according to personal income. This clearly manifests clustering among cities with similar time trajectories. Here we used a decorrelation measure as distance between any two cities, where is the cross-correlation of Figure 4A. When the decorrelation , , indicating no correlation(dashed line), revealing five families of kindred cities. See Figures S2, S3, S4, S5, S6, and S7 for other quantities.