The Mathematical Law that Predicts City Sizes

By Wesley Fenlon

Originally applied to linguistics, Zipf's law now, strangely, tells us how big our biggest cities will be.

Math is weird. Take, for example, Zipf's law, which io9 wrote about on Monday. Linguist George Zipf discovered, back in the 1940s, that if he ranked words by their popular usage, a surprising pattern appeared. The most popular word was used twice as frequently as the next most popular, and that word was used twice as frequently as the next most popular, and so on. Zipf called this the rank vs. frequency rule, though it's now known as Zipf's law. And Zipf's law doesn't just apply to words. It applies to the sizes of cities, too.

Strangely, the population distribution among cities in many countries follows the pattern of Zipf's law. It doesn't work 100 percent of the time, but Zipf's law is surprisingly accurate when applied to cities over 100,000 population.

"Just take a look at the top ranked cities in the United States by population," io9 writes. "In the 2010 census, the biggest city in the U.S., New York, had a population of 8,175,133. Los Angeles, ranked number 2, had a population of 3,792,621. And the cities in the next three ranks, Chicago, Houston and Philadelphia, clock in at 2,695,598, 2,100,263 and 1,526,006 respectively. You can see that obviously the numbers aren't exact, but looked at statistically, they are remarkably consistent with Zipf's predictions."

Photo credit: Flickr user c1ssou via Creative Commons.

Economist Xavier Gabaix wrote a paper titled "Zipf's Law for Cities: An Explanation" that showed cities graphed very closely to a line representing Zipf's law. Gabaix writes that Zipf's law applies to countries like the US and China and India, even though their backgrounds differ enormously. He concludes that "cities in the upper tail follow similar growth processes," referencing Gibrat's law.

Perhaps the most interesting thing about Zipf's law, though, is how it applies to what, exactly, constitutes a city. The urban sprawl surrounding a city may not technically be considered part of the city itself, but colloquially, it certainly would. New York city, for example, has a population of about 8.3 million. But the New York metropolitan area has a population of 23.3 million.

Turns out, Zipf's law still applies.

The paper "Zipf’s Law for All the Natural Cities in the United States: A Geospatial Perspective" looked at the United States' large cities and attempted to define them as "natural" cities, including the appropriate metropolitan area. The paper concludes "Our investigation leads to the finding that Zipf’s law holds remarkably well for the entire country. This implies that cities are power law distributed, and that the Zipf value is around 1.0."

The paper did note, however, that if you divide the cities up by individual states, Zipf's law breaks down. Scale seems like an important component in Zipf's law, and as io9 points out, it relates to how cities are connected economically. That's why it doesn't work for cities on a global scale, but works within countries. And, of course, it's not perfect. You can always find exceptions. But for the most part, we don't know exactly why our cities grow at the pace they do, but we know Zipf's law is strangely good at predicting that growth.