I guess this is a famous result, but it was new to me: George Zipf observed in 1949 that the populations of the cities in just about every country follow a power-law distribution. The largest city is always about twice as large as the second largest, and three times as big as the third largest, and so on. You can see this in action in R by plotting the populations of Canadian cities in 1996 (available as dataset cities in the DAAG package) against their rank on a log-log scale:
library(DAAG)
plot(sort(cities$POP1996,decreasing=TRUE), log="xy",
ylab="Population (thousands)",xlab="rank",main="Canadian Cities")
The nearly straight line (especially if you downweight the rank 1 city, Toronto) confirms the power-law distribution. The same result applies to cities of other countries, and even to other statistics than population, but no-one knows exactly why it holds. Oddly, while the result holds within nations, it breaks down somewhat when considered globally. Tim Gulden of George Mason University used light output from cities (estimated from satellite images) to
evaluate Zipf's laws for the population and economic activity of cities globally:
Co-author Richard Florida interprets these results:
While the population of cities tends to follow the Zipf law that Strogatz describes within a nation, this scaling does not hold for the whole collection of world cities. The distribution ends up being somewhat flatter - particularly among the largest cities. This may result from barriers to migration between countries.
Update May 29: From reader
Andreas, here's the chart of Danish cities: