Fun With Power Laws (and City Populations)

Power laws state that there are few large things, many medium-sized things, and a whole lot of small things in any kind of system. It further predicts exactly how those sizes will be distributed. Consider:

City Rank City Predicted Pop. Actual Pop.
1 New York City 8,250,567
2 Los Angeles 1/2 NYC = 4.1 million 3,849,378
3 Chicago 1/3 NYC = 2.8 million 2,833,321
4 Houston 1/4 NYC = 2.1 million 2,169,248
5 Phoenix, AZ 1/5 NYC = 1.6 million 1,512,986
6 Philadelphia 1/6 NYC = 1.3 million 1,448,394
7 San Antonio, TX 1/7 NYC = 1.14 million 1,296,682
8 San Diego 1/8 NYC = 1.07 million 1,256,951
9 Dallas 1/9 NYC = 0.92 million 1,232,940
10 San Jose, CA 1/10 NYC = 0.83 million 929,936

It may seem to break down after a while, but if we take the other aspect of power law distribution into effect, we get a somewhat different picture:

City Rank City Predicted Pop. Actual Pop.
1 New York City 8,250,567
2 Los Angeles 1/2 NYC = 4.1 million 3,849,378
3 Chicago 1/3 NYC = 2.8 million 2,833,321
4 Houston 1/4 NYC = 2.1 million 2,169,248
5 Phoenix, AZ 1/5 NYC = 1.6 million 1,512,986
6 Philadelphia 1/6 NYC = 1.3 million 1,448,394
6 San Antonio, TX 1/6 NYC = 1.3 million 1,296,682
6 San Diego 1/6 NYC = 1.3 million 1,256,951
6 Dallas 1/9 NYC = 1.3 million 1,232,940
9 San Jose, CA 1/9 NYC = 0.92 million 929,936

I made San Jose 1/9 because San Diego and Dallas are practically identical in population as to be able to include them together. In fact, if we continue on, we see the following:

City Rank City Predicted Pop. Actual Pop.
9 Detroit 1/9 NYC = 0.92 million 918,849
11 Jacksonville, FL 1/11 NYC = 0.75 million 794,555
11 Indianapolis 1/11 NYC = 0.75 million 785,597
11 San Francisco 1/11 NYC = 0.75 million 744,041
11 Columbus, OH 1/11 NYC = 0.75 million 733,203

We start seeing small groupings of similarly-populated cities, as we would expect from a power law distribution. Of course, these are rough equations -- more accurate ones would give more accurate predictions, but I'm no mathematician. However, this should tell you a lot about the natural distribution of population in a system like human population dynamics. We self-organize in very predictable ways.