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Conceived and designed the experiments: LMB JL GBW. Analyzed the data: LMB JL DS. Contributed reagents/materials/analysis tools: LMB. Wrote the paper: LMB JL GBW.

The authors have declared that no competing interests exist.

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 (

How rich, creative or safe can we expect a city to be? How can we establish which cities are the most creative, the most violent, or most effective at generating wealth? The conventional answer is to use the rank order of per capita measures of performance

Per capita measures of urban performance are ubiquitous in official statistics, policy documents and in the scientific literature. For example, official statistics on wages, income or gross domestic product (GDP) compiled by governmental agencies and international bodies worldwide

The use of per capita indicators assumes implicitly that, on average, specific urban characteristics,

These empirical regularities strongly suggest that underlying these apparently diverse phenomena there is a universal socioeconomic dynamic reflecting average organizational behavior of human interactions in cities

Thus, scaling laws provide the average baseline behavior and, by extension, the null model necessary for addressing the long-standing problem of how to rank specific cities meaningfully and assess the effects of local events, historical contingency, and policy, independently of population size. These agglomeration laws provide the

Agglomeration effects in cities are typically manifested as (

Non-linear agglomeration effects are manifested as simple scaling laws. Recent studies

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,

Eq. (1) is motivated by the more general observation that diverse characteristics of many complex adaptive systems, and especially those of biological organisms

For a given value of

To illustrate this methodology and its potential impact we analyze data from U.S. Metropolitan Statistical Areas (MSAs) (see Materials for data sources and city definitions). These are socioeconomic units defined via commuting flows, in contrast to more arbitrary political divisions such as counties or administrative cities. This definition emphasizes social interactions as the defining feature of cities. It attempts to circumscribe the city geographically as a mixing population where all residents can come into contact with each other, a familiar concept in epidemiology and ecology

We find that the variation in local quantities corresponding to different cities in the same year is well characterized statistically by a Laplace (exponential) distribution density

Interestingly, this Laplace distribution for SAMIs implies that the statistics of the urban indicators themselves also follow a power-law distribution density. Substituting, the definition of SAMIs, Eq. (2), into the Laplace distribution (3), and accounting for the change in measure in the probability density

The first use of SAMIs is to provide a meaningful way to rank cities.

The probability distribution of SAMIs, Eq. (3), might suggest that they behave much like random fluctuations. However, as illustrated in

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,

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 = 0.38

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

In general, higher rates of violent crime positively correlate with higher average incomes. However, this is primarily because both quantities scale similarly with city population size. SAMIs allow us to factor out these dominant general size effects and identify local relationships.

Place and geography are important in the development of cities

This lack of greater spatial similarity in socioeconomic SAMIs raises the question of whether the local dynamics of different cities are idiosyncratic and unique (random spatial fluctuations), or whether there are common patterns across the urban system. To investigate this question we ask more specifically if the SAMI histories of different cities, see

In this paper, we have proposed a systematic procedure for solving the long-standing problem of constructing meaningful, science-based metrics for ranking and assessing local features of cities

Population size plays a fundamental role in this approach. In the spirit of the successful application of scaling analysis to many other system - from collective physical phenomena

From this viewpoint, the general statistically stable properties of cities emerge as a hierarchy of interrelated fundamental quantities. First, it has been known for some time that the population size distribution of cities has remained relatively stable over time and across many different nations and is well-described by a Zipf power law distribution

Secondly, perhaps the most conspicuous property of SAMIs is that they do not randomly fluctuate over time but, instead, show long temporal persistence. This indicates that, even though the size and structure of a city's population may change considerably over time, any initial advantage or disadvantage that it has relative to its scaling expectation tends to be preserved over decades. In this sense, either for good or for bad, cities are remarkably robust. Examples are Phoenix, which has remained a mild economic under-performer over the last four decades maintaining a similar value of

Our analyses show that average spatial correlations between cities in the US are relatively short ranged (

Despite the lack of greater similarity due to geographical proximity, we find that most cities in the US show strong similarity with groups of other cities so that all US MSAs fall into a small number of classes of kindred cities sharing common historical paths. The same is true in terms of dissimilarity (or negative correlation) among cities, indicating that beneficial periods in specific sectors of the urban system coincide with negative developments in others, as

Finally, it is important to emphasize that the average properties of most socioeconomic quantities such as wealth creation, crime and innovation are strongly predicted by the scaling laws expressed in Eq. (1), which are non-linear functions of population size and account for 65–97% of the variance in the data (see

In summary, we have used the empirical manifestations of the underlying principles of agglomeration and the implicit network structures and dynamics responsible for the formation of cities to account systematically for urban dynamics at different scales. This paradigm allows us to separate measures of true local dynamics and organization in cities from their generic universal behavior. We have shown that these local indicators (SAMIs) have well defined statistics and that the consideration of their temporal and spatial properties is an essential element of models and theory of urban evolution and a new tool for the formulation of improved urban policy.

Our spatial unit of analysis is the metropolitan statistical area (MSA). MSAs are defined by the U.S. Office of Management and Budget and are standardized county-based areas having at least one urbanized area of 50,000 or more population, plus adjacent territory that has a high degree of social and economic integration with the core, as measured by commuting ties. Data on Gross Metropolitan Product (GMP) was recently made available by the US Department of Commerces Bureau of Economic Analysis and is a measure - in 2001 chain-weighted dollars - of the market value of final goods and services produced within a metropolitan area in a particular period of time. Data on the number of violent crimes is provided by the US Federal Bureau of Investigation (Uniform Crime Reports). Metropolitan patent counts were constructed using data provided by the U.S. Patent and Trademark Office, see

Data for GMP, personal income, violent crime and patents for each MSA corresponding to the same year were transformed logarithmically and fitted using Ordinary Least Squares to the logarithm of population, according to (1). Residuals from these fits,

The magnitude of the SAMIs corresponding to a given quantity and year for each city were used to rank cities. Two examples are shown in

The temporal autocorrelation is defined as

In

Interactive maps and tables of SAMIs for each quantity and year were produced using Exhibit (

Spatial similarity between cities was computed in terms of the equal-time cross-correlation of their SAMI time-series

Heatmaps were created by clustering the SAMI

Fit of cumulative exponential (Laplace) and Gaussian distributions to residuals for personal income in 2005. Both distributions give an excellent fit, but the exponential (Laplace) distribution is better, especially for residues around zero.

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Dendrogram of U.S. metropolitan areas grouped by incidence of violent crime, for cities with population above 1 million. Only cities reported by the FBI every year between 2001–06 are shown.

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Heatmap of U.S. metropolitan areas grouped by incidence of violent crime for cities with population above 1 million. Only cities reported by the FBI every year from 2001 to 2006 are shown.

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Dendrogram of U.S. metropolitan areas grouped by patenting rates for cities with population above 1 million. Data covers the period of 1975–2005.

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Heatmap of U.S. metropolitan areas grouped by patenting rates for cities with population above 1 million. Data covers the period of 1975–2005.

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Dendrogram of U.S. metropolitan areas grouped by Gross Metropolitan Product (GMP) for cities with population above 1 million. Data covers the period of 2001–2006.

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Heatmap of U.S. metropolitan areas grouped by Gross Metropolitan Product (GMP) for cities with population above 1 million. Data covers the period of 2001–2006.

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Summary statistics for 2005. Scaling exponent with 95% confidence interval and R-squared for log-log fits of total urban indicator versus total population. Two fits to the residual distribution using an exponential (Laplace) and Gaussian distributions. The parameter s measures the width of the Laplace distribution. Similarly, σ is the standard deviation of the Gaussian. Values of R-squared shown for these parameters indicate goodness of fit of the cumulative residual distributions to the data (see

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This describes in greater detail our methodology for assigning patents to metropolitan statistical areas.

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This contains a summary of statistical analysis of correlations between SAMIs and population growth rates of Metropolitan Statistical Areas.

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