Rank-Size Rule
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.

Why should simple rank be able to predict so easily such complex distributions? In short, why does the rank size rule work? One study has shown why this is so.

The distributions mentioned above such as Zipf, Pareto, Yule, etc., also called power laws, are all also related to the distribution known as the Fibonacci sequence and to that of the equiangular spiral. In the Fibonacci sequence, each term is approximately 1.618 (the Golden ratio) times the preceding term. The same ratio is seen in the Lucas numbers consisting of these sequentially additive numbers 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199 ,

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.

Primate City
A primate city is the leading city in its country or region, disproportionately larger than any others in the urban hierarchy. A 'primate city distribution' has one very large city with many much smaller cities and towns, and no intermediate-sized urban centres, in contrast to the linear 'rank-size distribution'. The 'law of the primate city' was first proposed by the geographer Mark Jefferson in 1939. He defines a primate city as being "at least twice as large as the next largest city and more than twice as significant." A primate city is number one in its country in most aspects, like politics, economy, media, culture and universities.

Significance

Not all countries have primate cities, but in those that do, the rest of the country depends on it for cultural, economic, political, and major transportation needs. On the other hand the primate city depends on the rest of the country as paying consumers of the cultural, economic, political and other services produced in the city.

The presence of a primate city in a country may indicate an imbalance in development usually a progressive core, and a lagging periphery, on which the city depends for labor and other resources. However, the urban structure is not directly dependent on a country's level of economic development.

Examples
Among the best known examples of primate cities are alpha world cities London and Paris. Budapest and Vienna have also been described as primate cities.

Rank-size usually indicates a country wherein all of the people have access to services because there are many cities of the differing sizes needed to spread services equally. A primate city can indicate and LDC wherein the people away from that city do not have access to services. Note: This rule is not always true and you need to think about whether or not this applies.