Exploiting the cascade structure of cities and based on a dataset for U.S. cities between 1840 and 2016, the aim of this short paper is to answer three important questions: First, do we observe that the U.S. city size distribution exhibits a smooth transition to Zipf's law from the beginning or are there periods showing a pronounced departure from Zipf's law? Second, if we observe periods of departure, which alternative laws instead should be used to accurately describe the city size distribution? Third, employing information from the cascade structure of cities, do we always find evidence for primate cities for a specific period of time? Inter alia, we find that the exact Zipf's law has evolved over time from the more general, so-called three-parameter Zipf's law which can be traced back to Mandelbrot (1982).
city size distributions Zipf’s law hierachical scaling law urban systems