Demography and the emergence of universal patterns in urban systems.

Urban areas exist in a wide variety of population sizes, from small towns to huge megacities. No proposed form for the statistical distribution of city sizes has received more attention than Zipf's law, a Pareto distribution with power law exponent equal to one. However, this distribution is typically violated by empirical evidence for small and large cities. Moreover, no theory presently exists to derive city size distributions from fundamental demographic choices while also explaining consistent variations. Here we develop a comprehensive framework based on demography to show how the structure of migration flows between cities, together with the differential magnitude of their vital rates, determine a variety of city size distributions. This approach provides a powerful mathematical methodology for deriving Zipf's law as well as other size distributions under specific conditions, and to resolve puzzles associated with their deviations in terms of concepts of choice, symmetry, information, and selection.