Generative Historical Model of City Size Hierarchies : 430 BCE – 2005 ∗

Despite the enormous changes in city size and differentiation of functions over time and space, and discontinuities in growth processes, key spatio-temporal and distributional processes shaping city sizes are often assumed to remain invariant. We demonstrate in this article that most of the facile assumptions about such invariance, that is, constancy over long historical periods, are unsupported when comparisons are made concerning city size distributions. We examine four aspects of the problem: the portion of the size distributions that are thought to be largely invariant (with exceptions for abnormal growth); how the city samples are constructed; the shapes of the mathematical functions that fit these distributions; and the historical variation in fitted constants of these functions. First, many geographers dispute Zipf’s rank-size ”law” empirically because the Zipfian is only satisfied in some cases for the tails of urban distributions, and there is a notorious deviation from an urban size power law when cities with smaller populations are considered (Malacarne, Mendes, and Lenzi 2002:2). Cities in centrally planned polities such as Russia and China also fail to follow the Zipfian (Marsili and Zang 2004:1). Second, these distributions vary as the sampling boundaries are changed, so that some cities might appear unusually large simply because the sampling region is drawn too narrowly. Third, others dispute Zipf’s and other scale-free laws of the distributions because they are not explanations but merely mathematical descriptions, and Rapoport (1978:847) cautions that just about any group of objects arranged according to size will fit some monotonically decreasing curve. Carroll (1979) notes that in addition to the power-law distribution, the lognormal, Pareto and Yule distributions all have long tails, like the Zipfian. Simon (1995) believes that the rank-size portion of a skew distribution results from any growth process at equilibrium in which size is initially random and growth is proportional to current size (see also Gabaix 1999), and Carroll also notes that this result also requires an assumption of a constant and steady rate of new entrants into the urban system. None of these conditions can be assumed historically. In order to address the fourth aspect of questions of distributional invariance, we examine all of the most