Size distribution of cities, Zipf’s law revisited

Abstract

Several authors (Berry 1970, Krugman 1996 or Eaton and Eckstein 1997, among many others) have experienced amazement how in most places the law of "least effort" established by Zipf (1949) is met very clearly. Cities, ranked by population, seem to follow almost exactly a function log/log, in which the logarithm of the "mass" (population, density, number of employees, etc..) correlates almost perfectly with the logarithm of the order of that mass. This function log/log, advanced by Pareto in the nineteenth century, has attracted quite a number of researchers, to occur in scenarios, both natural phenomena (earthquakes, meteorites, living species, ...) as derivatives of society (language, or distribution of cities), which has led to investigate its theoretical basis (Simon 1955, Brakmar et al. 1999, Gabaix 1999). While some authors (Rosen and Resnick 1980, Fan and Casetti 1994) have discussed the linear validity of Zipf's Law, introducing nonlinear models, technical literature has focused on the "upper tail" of the urban hierarchy, cities or large metropolitan areas, tending to silence the fact that the function log/log at all seems to be a general model. This paper attempts to show that when taking into account all the cases (ie, all populated localities in a particular territory), the log / log model seems to be only a special case of "the big." In fact it shows that the log/lin model tends to be more efficient, even with "folded tails." This has led to the hypothesis that tries to be tested in this study, that the logarithm of the urban mass tends to have a "normal distribution", leading its cumulative distribution (and ordered by rank) to be distributed according to a logistical structure, called "S". In this sense, the observation repeated of fulfillment of the Law of Zipf in the size of the cities would be just "the tip of the iceberg", in which cities of small and medium size also take its part, and where a "law" of a higher level appears. The presented research questions if this "normal" emergency of the logarithm of the mass could be shaped in a simple and elegant form, and tries an experiment in this regard.