Rank-size distribution
Encyclopedia
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 law
s 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.
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
s 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
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
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
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
) 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
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
, although its largest city, New York City
New York City
New York is the most populous city in the United States and the center of the New York Metropolitan Area, one of the most populous metropolitan areas in the world. New York exerts a significant impact upon global commerce, finance, media, art, fashion, research, technology, education, and...
, has more than twice the population of second-place Los Angeles
Los Ángeles
Los Ángeles is the capital of the province of Biobío, in the commune of the same name, in Region VIII , in the center-south of Chile. It is located between the Laja and Biobío rivers. The population is 123,445 inhabitants...
, 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
Democratic Republic of the Congo
The Democratic Republic of the Congo is a state located in Central Africa. It is the second largest country in Africa by area and the eleventh largest in the world...
, the capital of Kinshasa
Kinshasa
Kinshasa is the capital and largest city of the Democratic Republic of the Congo. The city is located on the Congo River....
is more than eight times larger than the second-largest city, Lubumbashi
Lubumbashi
Lubumbashi is the second largest city in the Democratic Republic of the Congo, second only to the nation's capital Kinshasa, and the hub of the southeastern part of the country. The copper-mining city serves as the capital of the relatively prosperous Katanga Province, lying near the Zambian border...
. 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
Gibrat's law
Gibrat's law, sometimes called Gibrat's rule of proportionate growth is a rule defined by Robert Gibrat stating that the size of a firm and its growth rate are independent. The law proportionate growth gives rise to a distribution that is log-normal...
of proportionate growth.
Further reading
- Douglas R. WhiteDouglas R. WhiteDouglas R. White is an American complexity researcher , social anthropologist, sociologist, and social network researcher at the University of California, Irvine.-Biography:...
, Laurent Tambayong, and Nataša KejžarNataša KejžarNataša Kejžar is an olympic swimmer, born October 14, 1976 in Jesenice, Slovenia, Yugoslavia. She studied at Faculty of Computer and Information Science, Ljubljana and achieved her PhD degree in Statistics in 2007. She is...
. 2008. Oscillatory dynamics of city-size distributions in world historical systems. Globalization as an Evolutionary Process: Modeling Global Change. Ed. by George ModelskiGeorge ModelskiGeorge Modelski is Professor of Political Science Emeritus in the University of Washington. Modelski has done work on long-term processes in global politics and economics, as well as the world urban macrodynamics and world system evolution.-Books:*Globalization as Evolutionary Process: Modeling...
, Tessaleno Devezas, and William R. Thompson. London: Routledge. ISBN 9780415773614
See also
- Pareto distribution
- Pareto principlePareto principleThe Pareto principle states that, for many events, roughly 80% of the effects come from 20% of the causes.Business-management consultant Joseph M...
- The Long TailThe Long TailThe Long Tail or long tail refers to the statistical property that a larger share of population rests within the tail of a probability distribution than observed under a 'normal' or Gaussian distribution...
- The Use of Agent-Based Models in Regional Science--an agent-based simulation study that explains Rank-size distribution