Urban hierarchy

The Urban hierarchy ranks each city based on the size of population residing within the nationally defined statistical urban area. Because urban population depends on how governments define their metropolitan areas, urban hierarchies are conventionally ranked at the national level; however, the ranking can be extended globally to include all cities. Urban hierarchies tell us about the general organization of cities and yield some important insights. First, it tells us that within a system of cities, some cities will grow to be very large, but that number will be small relative to the universe of cities. Second, it refutes the expectation of an optimally sized city. Lastly, it establishes cities as belonging to an inter-related network where one city's growth affects others'.

The hierarchy is usually related to the empirical regularity with which cities are distributed. The pattern has been formulated in a number of ways, but usually as a variation of the power law. Formally, it is a frequency distribution of rank data where frequency is inversely proportional to rank such that cities with population larger than S are approximately proportional to S^−a, where a is normally close to 1. There are no good explanations for the exponent consistently being close to 1. This is problematic because an exponent of 1 in the power law implies an infinite population. Paul Krugman proposes that, in the case of cities, the power law operates according to the percolation theory. This relaxes the condition on the exponent approaching the value of 1 and breaking down the model. Importantly, the application of a percolation model leads to one of the key insights regarding city sizes: geography and economic conditions give cities advantages that allow them to grow more than cities with a relative scarcity of these benefits.

A simpler formulation of the relationship between rank and frequency is expressed with reference to Zipf's Law. The law applied to cities states that "if cities are ranked in decreasing population size, then the rank of a given city will be inversely proportional to its population." According to this intuitive formulation, in a country where the largest city has a population of 10 million, the second largest will have population size of 5 million, the third largest 3 .33 million, etc.

The urban hierarchy has been described in detail in the United States where the power law has held consistently for over a century. In 1991, there were 40 U.S. Metropolitan Areas with population above 1 million, 20 above 2 million, and 9 with more than 4 million.

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