Zipf's law for all the natural cities around the world

Two fundamental issues surrounding research on Zipf's law regarding city sizes are whether and why this law holds. This paper does not deal with the latter issue with respect to why, and instead investigates whether Zipf's law holds in a global setting, thus involving all cities around the world. Unlike previous studies, which have mainly relied on conventional census data such as populations and census-bureau-imposed definitions of cities, we adopt naturally (in terms of data speak for itself) delineated cities, or natural cities, to be more precise, in order to examine Zipf's law. We find that Zipf's law holds remarkably well for all natural cities at the global level, and it remains almost valid at the continental level except for Africa at certain time instants. We further examine the law at the country level, and note that Zipf's law is violated from country to country or from time to time. This violation is mainly due to our limitations; we are limited to individual countries, or to a static view on city-size distributions. The central argument of this paper is that Zipf's law is universal, and we therefore must use the correct scope in order to observe it. We further find Zipf's law applied to city numbers; the number of cities in the largest country is twice as many as that in the second largest country, three times as many as that in the third largest country, and so on. These findings have profound implications for big data and the science of cities.