Does Zipf's Law Hold for Primate Cities? Some Evidence from a Discriminant Analysis of World Countries

According Zipf's Law, city sizes follow a Pareto distribution, with the rank (R) of a city i being proportional to its size (S): R(i)=A*S-ß or ln(R) = ln(A)-ß*ln(S), where ß is a slope gradient or Pareto parameter, varying around 1. However, several empirical studies, carried out to date, indicate that the sizes of the first largest cities in many countries (with ranks of 1 and 2) are not exactly given to Zip's Law, but with relatively large errors. In our study, we consider the ratio between the size of the first largest city and the size of the second largest city (B-ratio) for a very large ensemble of 177 countries across the world. A surprising result of this work is that only a small number of countries (about 35%) have their B-ratios within the limits expected under Zipf's Law. As we also learn from the discriminant analysis of our country-wide data, high urbanization levels are likely to reduce the gap in population sizes between the first and the second city, while the first city being the national capital is likely to widen the gap between it and its "nearest neighbor" in the national city-size distribution.