Zipf’s Law

Sizing up cities:

Math and the City, by Steven Strogatz: ...The mathematics of cities was launched in 1949 when George Zipf, a linguist working at Harvard,... noticed that if you tabulate the biggest cities in a given country and rank them according to their populations, the largest city is always about twice as big as the second largest, and three times as big as the third largest, and so on. In other words, the population of a city is, to a good approximation, inversely proportional to its rank. Why this should be true, no one knows. ...

Given the different social conditions from country to country, the different patterns of migration a century ago and many other variables that you’d think would make a difference, the generality of Zipf’s law is astonishing.

Keep in mind that this pattern emerged on its own. ... Many inventive theorists working in disciplines ranging from economics to physics have taken a whack at explaining Zipf’s law, but no one has completely solved it. Paul Krugman ... wryly noted that “the usual complaint about economic theory is that our models are oversimplified — that they offer excessively neat views of complex, messy reality. [In the case of Zipf’s law] the reverse is true: we have complex, messy models, yet reality is startlingly neat and simple.” ...

Around 2006, scientists started discovering new mathematical laws about cities that are nearly as stunning as Zipf’s. ... For instance,... populous ... cities have more gas stations than smaller ones (of course), but not nearly in direct proportion to their size. The number of gas stations grows only in proportion to the 0.77 power of population. The crucial thing is that 0.77 is less than 1. This implies that ... bigger cities enjoy economies of scale. In this sense, bigger is greener.

The same pattern holds for other measures of infrastructure. Whether you measure miles of roadway or length of electrical cables, you find that all ... show an exponent between 0.7 and 0.9. Now comes the spooky part. The same law is true for living things. That is, if you mentally replace cities by organisms and city size by body weight, the mathematical pattern remains the same.

For example, suppose you measure how many calories a mouse burns per day, compared to an elephant. ... The relevant law of metabolism, called Kleiber’s law, states that the metabolic needs of a mammal grow in proportion to its body weight raised to the 0.74 power.

This 0.74 power is uncannily close to the 0.77 observed for the law governing gas stations in cities. Coincidence? Maybe, but probably not. There are theoretical grounds to expect a power close to 3/4. Geoffrey West of the Santa Fe Institute and his colleagues Jim Brown and Brian Enquist have argued that a 3/4-power law is exactly what you’d expect if natural selection has evolved a transport system for conveying energy and nutrients as efficiently and rapidly as possible to all points of a three-dimensional body, using a fractal network built from a series of branching tubes — precisely the architecture seen in the circulatory system and the airways of the lung, and not too different from the roads and cables and pipes that keep a city alive.

These numerical coincidences seem to be telling us something profound. It appears that Aristotle’s metaphor of a city as a living thing is more than merely poetic. There may be deep laws of collective organization at work here, the same laws for aggregates of people and cells. ...

[For more on city size, see: Why Has Globalization Led to Bigger Cities?, by Edward Glaeser.]