Urban Scaling Functions: Emission, Pollution and Health.

Urban scaling is widely used to characterize the population dependence of city indicators including greenhouse gas emission. Here we analyze the population dependence of CO ${}_2$ and PM2.5 emissions and concentrations across all European cities. Our analysis reveals considerable variations in emissions among cities of comparable population size which are not captured by conventional urban scaling. We thus characterize these fluctuations by multi-parameter scaling functions and multifractal spectral analysis. We find that the distribution of emissions and population is multifractal while that of air pollution is not, leading to non-trivial relations between emission and pollution in some large cities. We also analyze the impact of forests in curbing emission and the impact of air pollution on health. Our work provides a detailed picture of the fluctuations in the scaling of urban metabolism in Europe and suggests a general strategy that goes beyond conventional urban scaling laws.