According to Wikipedia and io9, Zipf's law can be applied to "big cities" and agglomerates of big cities (e.g. San Francisco and Oakland). More specifically, according to the Wikipedia article, the power law fits with a factor of 1.07:

When Zipf's law is checked for cities, a better fit has been found with b = 1.07; i.e. the n-th largest settlement is 1/(n^1.07) the size of the largest settlement

A similar fact is stated (without reference) in this Nature Scientific Report.

However, I have yet to find any example where this law works in European cities. Here are some examples:

  • Belgium: Largest city has 1,019,022 inhabitants. According to the law, the three subsequent cities should have 485,379 , 314,531 and 231,195 people. This respectively corresponds to a 6%, 23% and 15% difference between the expected value and actual value.

  • France: largest city is Paris with 2,229,621 inhabitants. The Zipf's power law is off by 24%, 37% and 10% with respects to reality.

  • Germany: Berlin has 3,426,354 inhabitants. Zipf's power law is off by 7%, 17% and 20% for the three following cities.

  • Italy: Rome's population 2,318,895. Zipf with 1.07 factor is off by 11%, 25% and 40%.

Out of curiosity I have also checked it out for Mexico and it fails drastically as well: Mexico city has about four million people and the following four cities in population size are all about 1.5 million people. However, it does seem to somewhat work for the largest US cities.

Is there any evidence this law indeed works for the population of European cities? Subsequently, how about for other "big cities"?

  • I haven't checked the other countries yet, but for the Belgium cities if you take a value in between the actual reported population and the population of the Larger Urban Zone it fits better. There may be some sort of area or population density factor taken into consideration. Apr 9 '16 at 18:18
  • Perhaps the problem with Europe is that national borders are fairly arbitrary. Try your computations using natural barriers instead.
    – jamesqf
    Apr 9 '16 at 20:13
  • @jamesqf, but aren't the borders of the biggest cities in the US also arbitrary?
    – ChrisR
    Apr 9 '16 at 22:06
  • 2
    @jamesqf. Europe is arguably the only continent where national borders are not arbitrary. They follow ethno-linguistic borders fairly precisely.
    – fdb
    Apr 10 '16 at 14:01
  • 1
    It also doenst fit (at first glance, without doing the math) when you look at european agglomerations: de.wikipedia.org/wiki/Agglomeration#Deutschland_2
    – mart
    Apr 13 '16 at 9:45

The observation that the population of the cities in any given country is inversely proportional to their rank was made, long before Zipf, by the physicist Felix Auerbach in his article “Das Gesetz der Bevölkerungskonzentration”, Petermanns Mitteilungen 59, 1913, pp. 74-76. Here Auerbach compiled lists of the largest cites in Germany, France, Italy, Britain, the US, and British India and observed that the product of the population of any city times its rank in the list is a constant, which is different in each country. However, he noted that this is not true of the largest half dozen or so cities, but only of the middle ranking and smaller cities.

I have not found this article on the internet, but I read it in our university library. Zipf does not credit Auerbach for this discovery in any of his publications, but this presumably has its own reason: Zipf was a notorious Nazi sympathiser; Auerbach was a Jew.

  • Do you have any source confirming that Zipf was a "notorious Nazi sympathizer?"
    – ChrisR
    Apr 10 '16 at 18:00
  • 2
    You can download his terrible book “National Unity and Disunity: the Nation as a bio-social organism”, Bloomington 1941 here: babel.hathitrust.org/cgi/…
    – fdb
    Apr 10 '16 at 18:11
  • Look for "Hitler" and "Jews" in the index.
    – fdb
    Apr 10 '16 at 18:13
  • Zipf probably didn't credit Auerbach because Zipf's law relates to frequency of words in books. And Auerbach did not research frequency of words in books. It is only later that people realize that Zipf's law is applicable to lots of other things like the frequency of digits in a list of numbers (any list, literally, from your grocery shopping receipt to list of GDP of all countries in the world) to the frequency of first names in a population
    – slebetman
    Apr 12 '16 at 9:14
  • @slebetman. In the book that I mentioned Zipf discusses the size of cities in great detail, without crediting Auerbach. Have a look at the book.
    – fdb
    Apr 12 '16 at 9:49

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