Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life, in Organisms, Cities, Economies, and Companies

by Geoffrey West

There was, however, one major exception to this and that is a famous scaling law known as Zipf’s law for the ranking of cities in terms of their population size. This is shown graphically in Figure 39. It’s an intriguing observation: in its simplest form, it states that the rank order of a city is inversely proportional to its population size. Thus, the largest city in an urban system should be about twice the size of the second largest, three times the size of the third largest, four times the size of the fourth largest, and so on. So, for example, in the 2010 census, the biggest city in the United States was New York with a population of 8,491,079. According to Zipf’s law, the second largest city, Los Angeles, should have a population of approximately one half of this, namely 4,245,539, the third largest, Chicago, should have a population of about one third, or 2,830,359, and Houston, the fourth largest, should have a population of about one quarter, namely 2,122,769, and so on. The actual numbers were: Los Angeles, 3,928,864; Chicago, 2,722,389; and Houston, 2,239,558, all in reasonably good agreement with Zipf’s law to within less than 7 percent. Zipf’s law owes its name to the Harvard linguist George Kingsley Zipf, who popularized it through his fascinating book Human Behavior and the Principle of Least Effort, published in 1949.17 He first enunciated his law in 1935, not for cities but for the frequency of word use in languages.