  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 speaks 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
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 first 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.
  Keywords: Night-time imagery, city-size distributions, head/tail division
rule, head/tail breaks, big data
