Rank-size distribution

Rank-size distribution or the rank-size rule (or law) describes the remarkable regularity in many phenomena including the distribution of city sizes around the world, sizes of businesses, particle sizes (such as sand), lengths of rivers, frequencies of word usage, wealth among individuals, etc. All are real-world observations that follow power laws such as those called Zipf's law, the Yule distribution, or the Pareto distribution. If one ranks the population size of cities in a given country or in the entire world and calculates the natural logarithm of the rank and of the city population, the resulting graph will show a remarkable log-linear pattern. This is the rank-size distribution.

In the case of city populations, the resulting distribution in a country, region or the world will be characterized by a largest city, with other cities decreasing in size respective to it, initially at a rapid rate and then more slowly. This results in a few large cities, and a much larger number of cities orders of magnitude smaller. For example, a rank 3 city would have ⅓ the population of a country's largest city, a rank four city would have ¼ the population of the largest city, and so on.

When any log-linear factor is ranked, the ranks follow the Lucas numbers, which consist of the sequentially additive numbers 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, etc. Like the more famous Fibonacci sequence, each number is approximately 1.618 (the Golden ratio) times the preceding number. For example, the third term in the sequence above, 4, is approximately 1.6183 or 4.236 (which is approximately 4); the fourth term in the sequence, 7, is approximately 1.6184 or 6.854 (which is approximately 7); the eight term in the series, 47, is approximately 1.6188 or 46.979 (which is approximately 47). With higher and higher values, the figures converge. An equiangular spiral is sometimes used to visualize such sequences.

One study claims that the rank size rule "works" because it is a "shadow" or coincidental measure of the true phenomenon. The true value of rank size is thus not as an accurate mathematical measure (since other power-law formulas are more accurate, especially at ranks lower than 10) but rather as a handy measure or “rule of thumb” to spot power laws. When presented with a ranking of data, is the third-ranked variable approximately ⅓ the value of the highest-ranked one? Or, conversely, is the highest-ranked variable approximately ten times the value of the tenth-ranked one? If so, the rank size rule has possibly helped spot another power law relationship. While Zipf's law works well in many cases it tends to not fit the largest cities in many countries. A 2002 study found that, Zipf’s Law was rejected for 53 of 73 countries, which is far more than would be expected based on random chance. The study also found that variations of the Pareto exponent are better explained by political variables than by economic geography variables like proxies for economies of scale or transportation costs. A 2004 study showed that Zipf's law did not work well for the five largest cities in six countries. In the richer countries, the distribution was flatter than predicted. For instance, in the United States, although its largest city, New York City, has more than twice the population of second-place Los Angeles, the two cities' metropolitan areas, also the two largest in the country, are much closer in population. In metropolitan-area population, New York City is only 1.3 times larger than Los Angeles. In other countries, the largest city would dominate much more than expected. For instance, in DR Congo, the capital of Kinshasa is more than eight times larger than the second-largest city, Lubumbashi. When considering the entire distribution of cities, including the smallest ones, the rank-size rule does not hold. Instead, the distribution is lognormal. This follows from Gibrat's law of proportionate growth.