Rank–size Distribution
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Rank–size distribution is the distribution of size by rank, in decreasing order of size. For example, if a data set consists of items of sizes 5, 100, 5, and 8, the rank-size distribution is 100, 8, 5, 5 (ranks 1 through 4). This is also known as the rank–frequency distribution, when the source data are from a frequency distribution. These are particularly of interest when the data vary significantly in scales, such as city size or word frequency. These distributions frequently follow a
power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
distribution, or less well-known ones such as a
stretched exponential function The stretched exponential function f_\beta (t) = e^ is obtained by inserting a fractional power law into the exponential function. In most applications, it is meaningful only for arguments between 0 and +∞. With , the usual exponential functio ...
or parabolic fractal distribution, at least approximately for certain ranges of ranks; see below. A rank-size distribution is not a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
or
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
. Rather, it is a discrete form of a
quantile function In probability and statistics, the quantile function, associated with a probability distribution of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equ ...
(inverse cumulative distribution) in reverse order, giving the size of the element at a given rank.


Simple rank–size distributions

In the case of city populations, the resulting distribution in a country, a region, or the world will be characterized by its 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 one-third the population of a country's largest city, a rank 4 city would have one-fourth 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; the fourth term, 7, is approximately 1.6184, or 6.854; the eighth term, 47, is approximately 1.6188, or 46.979. With higher values, the figures converge. An
equiangular spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More ...
is sometimes used to visualize such sequences.


Segmentation

A rank-size (or rank–frequency) distribution is often segmented into ranges. This is frequently done somewhat arbitrarily or due to external factors, particularly for market segmentation, but can also be due to distinct behavior as rank varies. Most simply and commonly, a distribution may be split in two pieces, termed the head and tail. If a distribution is broken into three pieces, the third (middle) piece has several terms, generically middle,Illustrating the Long Tail
Rand Fishkin, November 24th, 2009
also belly,Digg that Fat Belly!
Robert Young, Sep. 4, 2006
torso, and body.The Small Head, the Medium Body, and the Long Tail .. so, where's Microsoft?
, 12 Mar 2005, Lawrence Liu's Report from the Inside
These frequently have some adjectives added, most significantly ''
long tail In statistics and business, a long tail of some probability distribution, distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involv ...
'', also ''fat belly'', ''chunky middle'', etc. In more traditional terms, these may be called ''top-tier'', ''mid-tier'', and ''bottom-tier''. The relative sizes and weights of these segments (how many ranks in each segment, and what proportion of the total population is in a given segment) qualitatively characterize a distribution, analogously to the skewness or kurtosis of a probability distribution. Namely: is it dominated by a few top members (head-heavy, like profits in the recorded music industry), or is it dominated by many small members (tail-heavy, like internet search queries), or distributed in some other way? Practically, this determines strategy: where should attention be focused? These distinctions may be made for various reasons. For example, they may arise from differing properties of the population, as in the
90–9–1 principle In Internet culture, the 1% rule is a general rule of thumb pertaining to participation in an internet community, stating that only 1% of the users of a website actively create new content, while the other 99% of the participants only lurk. Var ...
, which posits that in an internet community, 90% of the participants of a community only view content, 9% of the participants edit content, and 1% of the participants actively create new content. As another example, in marketing, one may pragmatically consider the head as all members that receive personalized attention, such as personal phone calls; while the tail is everything else, which does not receive personalized attention, for example receiving form letters; and the line is simply set at a point that resources allow, or where it makes business sense to stop. Purely quantitatively, a conventional way of splitting a distribution into head and tail is to consider the head to be the first ''p'' portion of ranks, which account for 1 - p of the overall population, as in the 80:20
Pareto principle The Pareto principle states that for many outcomes, roughly 80% of consequences come from 20% of causes (the "vital few"). Other names for this principle are the 80/20 rule, the law of the vital few, or the principle of factor sparsity. Manage ...
, where the top 20% (head) comprises 80% of the overall population. The exact cutoff depends on the distribution – each distribution has a single such cutoff point—and for power, laws can be computed from the Pareto index. Segments may arise naturally due to actual changes in the behavior of the distribution as rank varies. Most common is the king effect, where the behavior of the top handful of items does not fit the pattern of the rest, as illustrated at the top for country populations, and above for most common words in English Wikipedia. For higher ranks, behavior may change at some point, and be well-modeled by different relations in different regions; on the whole by a
piecewise function In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
. For example, if two different power laws fit better in different regions, one can use a broken power law for the overall relation; the word frequency in English Wikipedia (above) also demonstrates this. The Yule–Simon distribution that results from preferential attachment (intuitively, "the rich get richer" and "success breeds success") simulates a broken power law and has been shown to "very well capture" word frequency versus rank distributions. It originated from trying to explain the population versus rank in different species. It has also been shown to fit city population versus rank better.


Rank–size rule

The rank-size rule (or law) describes the remarkable regularity in many phenomena, including the distribution of city sizes, the sizes of businesses, the sizes of particles (such as sand), the lengths of rivers, the frequencies of word usage, and wealth among individuals. All are real-world observations that follow
power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
s, such as Zipf's law, the Yule distribution, or the
Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actua ...
. If one ranks the population size of cities in a given country or in the entire world and calculates the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
of the rank and of the city population, the resulting graph will show a log-linear pattern. This is the rank-size distribution.


Theoretical rationale

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 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 one-third 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.


Known exceptions to simple rank–size distributions

While Zipf's law works well in many cases, it tends to not fit the largest cities in many countries; one type of deviation is known as the King effect. A 2002 study found that Zipf's law was rejected in 53 of 73 countries, 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.Cuberes, David, The Rise and Decline of Cities, University of Chicago, September 29, 2004, 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 the Democratic Republic of the Congo, the capital,
Kinshasa Kinshasa (; ; ln, Kinsásá), formerly Léopoldville ( nl, Leopoldstad), is the capital and largest city of the Democratic Republic of the Congo. Once a site of fishing and trading villages situated along the Congo River, Kinshasa is now one o ...
, 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
log-normal In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
. This follows from Gibrat's law of proportionate growth. Because exceptions are so easy to find, the function of the rule for analyzing cities today is to compare the city systems in different countries. The rank-size rule is a common standard by which urban primacy is established. A distribution such as that in the United States or China does not exhibit a pattern of primacy, but countries with a dominant "
primate city A primate city is a city that is the largest in its country, province, Federated state, state, or region, and disproportionately larger than any others in the urban hierarchy. A ''primate city distribution'' is a rank-size distribution that has on ...
" clearly vary from the rank-size rule in the opposite manner. Therefore, the rule helps to classify national (or regional) city systems according to the degree of dominance exhibited by the largest city. Countries with a primate city, for example, have typically had a colonial history that accounts for that city pattern. If a normal city distribution pattern is expected to follow the rank-size rule (i.e. if the rank-size principle correlates with central place theory), then it suggests that those countries or regions with distributions that do not follow the rule have experienced some conditions that have altered the normal distribution pattern. For example, the presence of multiple regions within large nations such as China and the United States tends to favor a pattern in which more large cities appear than would be predicted by the rule. By contrast, small countries that had been connected (e.g. colonially/economically) to much larger areas will exhibit a distribution in which the largest city is much larger than would fit the rule, compared with the other cities—the excessive size of the city theoretically stems from its connection with a larger system rather than the natural hierarchy that central place theory would predict within that one country or region alone.


See also

*
Pareto principle The Pareto principle states that for many outcomes, roughly 80% of consequences come from 20% of causes (the "vital few"). Other names for this principle are the 80/20 rule, the law of the vital few, or the principle of factor sparsity. Manage ...
*
Long tail In statistics and business, a long tail of some probability distribution, distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involv ...


References


Further reading

* * * * Douglas R. White, Laurent Tambayong, and
Nataša Kejžar Nataša Kejžar (born 14 October 1976) is a swimmer for Slovenia at the 2000 Summer Olympics and a statistician. Kejžar was born 14 October 1976 in Jesenice, Slovenia, Yugoslavia. Coached by Ciril Globočnik, she started swimming in 1984, fin ...
. 2008. Oscillatory dynamics of city-size distributions in world-historical systems. ''Globalization as an Evolutionary Process: Modeling Global Change''. Ed. by George Modelski, Tessaleno Devezas, and William R. Thompson. London: Routledge.
The Use of Agent-Based Models in Regional Science
��an agent-based simulation study that explains rank–size distribution.


External links

* {{DEFAULTSORT:Rank-size distribution Functions related to probability distributions Geography Statistical laws