Zipf's Law

I tripped over this law on one of the NYTimes blogs and found it fascinating.

As stated in the blog,

Quote:

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The mathematics of cities was launched in 1949 when George Zipf, a linguist working at Harvard, reported a striking regularity in the size distribution of cities. He noticed that if you tabulate the biggest cities in a given country and rank them according to their populations, the largest city is always about twice as big as the second largest, and three times as big as the third largest, and so on. In other words, the population of a city is, to a good approximation, inversely proportional to its rank. Why this should be true, no one knows.

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Zipf originally applied the law to the frequency of words in a text, but it applies to a wide variety of data sets with no good explanation.

So, infrastructure evolves just as creatures do--through natural selection. It also follows logic that we'd converge methods, as well (noting the lung=plumbing example).

Relevant to my interests.

That's pretty neat. I took the liberty of graphing this in Excel for we visual folks:
http://img194.imageshack.us/img194/8...slawcities.jpg
As you can see, the top 25 US cities adhere pretty closely to Zipf's Law, although the actual populations overshoot the predicted populations pretty consistently (albeit not by much). Apparently our largest city needs to be a little bit bigger in order to put it all back into proportion.

Nice work on the graph! Zipf's law seems to hold consistently for the top 4 or 5, after which it seems like there are tiers or clusters, with 2 or 3 cities at the same level. Of course, deciding where to draw the line on a metro area is fuzzy these days, too, rendering the population numbers iffy.

I wonder if this has some connection to a sort of critical mass effect. Once a city is known as 'the largest', all of a sudden twice as many people want to live there sort of thing.

Hah, interesting. Good article about it on Wikipedia:

http://en.wikipedia.org/wiki/Zipf%27s_law

Check out the log log graph for word frequencies.

Apparently it also applies to random strings of words so it could just be an artefact... cool though.

This only seems to hold true for America. Looking at the other 3 of the top 4 most populous nations:

China:

1. Shanghai - 18,580,000
2. Beijing - 12,133,000
3. Guangzhou - 10,045,800
4. Shenzhen - 8,464,300
5. Xianggang (Hong Kong) - 7,000,000

India:

1. Mumbai - 13,922,125
2. Delhi - 12,259,230
3. Bangalore - 5,310,318
4. Kolkata (Calcutta) - 5,080,519
5. Chennai - 4,590,267

Indonesia:

1. Jakarta - 10,810,400
2. Bandung - 2,884,300
3. Surabaya - 2,729,300
4. Medan - 2,059,300
5. Bekasi - 1,994,850