Testing Heaps’ law for cities using administrative and gridded population data sets

Abstract Since 2008 the number of individuals living in urban areas has surpassed that of rural areas and in the next decades urbanisation is expected to further increase, especially in developing countries. A country’s urbanisation depends both on the distribution of city sizes, describing the fraction of cities with a given population (or area), and the overall number of cities in the country. Here we present empirical evidence suggesting the validity of Heaps’ law for cities: the expected number of cities in a country is only a function of the country’s total population (or built-up area) and the distribution of city sizes. This implies the absence of correlations in the spatial distribution of cities. We show that this result holds at the country scale using the official administrative definition of cities provided by the Geonames dataset, as well as at the local scale, for areas of 128 × 128 $128 \times 128$  km2 in the United States, using a morphological definition of urban clusters obtained from the Global Rural-Urban Mapping Project (GRUMP) dataset. We also derive a general theoretical result applicable to all systems characterised by a Zipf distribution of group sizes, which describes the relationship between the expected number of groups (cities) and the total number of elements in all groups (population), providing further insights on the relationship between Zipf’s law and Heaps’ law for finite-size systems.