The Rank-Frequency Form of Zipf's Law

Abstract

Abstract Suppose that there are K regions in a country, with Ni people and Mi cities in the ith region. Let Ni be large, Mi random, given Ni , and such that the distribution of M i N i –1, given Ni , converges to a limiting distribution F, with F(x) ∼ Cx γ as x → 0, γ > 0. Let L (r) be the size of the rth largest city in the country. If, given Mi and Ni , there is a Bose-Einstein allocation of the Ni people to the Mi cities in region i, independently for the various regions, then a plot of L (r) against r will be approximately proportional to r –(1+α), for 1 + α = γ–1.