Modeling the origin of urban-output scaling laws

Superlinear urban scaling by functional organization: A metabolic interpretation of sectoral water consumption

Prevailing view asserts that the disproportionately greater productivities of larger cities, or superlinear urban scaling, are the result of human interactions channeled by urban networks. But this view was established by considering the spatial organization of urban infrastructure and social networks-the urban "arteries" effects-but neglecting the functional organization of urban production and consumption entities-the urban "organs" effects. Here, adopting a metabolic view and using water consumption as a proxy for metabolism, we empirically quantify the scaling of entity number, size, and metabolic rate for the functionally specific urban residential, commercial, public or institutional, and industrial sectors. Sectoral urban metabolic scaling is highlighted by a disproportionate coordination between residential and enterprise metabolic rates, attributable to the functional mechanisms of mutualism, specialization, and entity size effect. The resultant whole-city metabolic scaling exhibits a constant superlinear exponent for water-abundant regions in numerical agreement with superlinear urban productivity, with varying exponent deviations for water-deficient regions explainable as adaptations to climate-driven resource constraints. These results present a functional organizational, non-social-network explanation of superlinear urban scaling.

Scaling of urban income inequality in the USA

Urban scaling analysis, the study of how aggregated urban features vary with the population of an urban area, provides a promising framework for discovering commonalities across cities and uncovering dynamics shared by cities across time and space. Here, we use the urban scaling framework to study an important, but under-explored feature in this community-income inequality. We propose a new method to study the scaling of income distributions by analysing total income scaling in population percentiles. We show that income in the least wealthy decile (10%) scales close to linearly with city population, while income in the most wealthy decile scale with a significantly superlinear exponent. In contrast to the superlinear scaling of total income with city population, this decile scaling illustrates that the benefits of larger cities are increasingly unequally distributed. For the poorest income deciles, cities have no positive effect over the null expectation of a linear increase. We repeat our analysis after adjusting income by housing cost, and find similar results. We then further analyse the shapes of income distributions. First, we find that mean, variance, skewness and kurtosis of income distributions all increase with city size. Second, the Kullback-Leibler divergence between a city's income distribution and that of the largest city decreases with city population, suggesting the overall shape of income distribution shifts with city population. As most urban scaling theories consider densifying interactions within cities as the fundamental process leading to the superlinear increase of many features, our results suggest this effect is only seen in the upper deciles of the cities. Our finding encourages future work to consider heterogeneous models of interactions to form a more coherent understanding of urban scaling.

The interpretation of urban scaling analysis in time

Scaling is a general analytical framework used by many disciplines-from physics to biology and the social sciences-to characterize how population-averaged properties of a collective vary with its size. The observation of scale invariance over some range identifies general system types, be they ideal gases, ecosystems or cities. The use of scaling in the analysis of cities quantifies many of their arguably fundamental general characteristics, especially their capacity to create interrelated economies of scale in infrastructure and increasing returns to scale in socio-economic activities. However, the measurement of these effects, and the relationship of observable parameters to theory, hinge on how scaling analysis is used empirically. Here, we show how two equivalent approaches to urban scaling-cross-sectional and temporal-lead to the measurement of different mixtures of the same fundamental parameters describing pure scale and pure temporal phenomena. Specifically, temporal exponents are sensitive to the intensive growth of urban quantities and to circumstances when population growth vanishes, leading to instabilities and infinite divergences. These spurious effects are avoided in cross-sectional scaling, which is more common and closer to theory in terms of quantitative testable expectations for its parameters.