Zipf’s Law

A few weeks ago I wrote about Benford’s Law, the strange fact that the measurements of many things begin more often with lower valued digits than higher valued ones (1 is more common than 2, which is more common than 3, etc.).

Yesterday in the NY Times, Ed Glaeser wrote about a related phenomenon in city populations, known as Zipf’s Law. Apparently Zipf’s Law is a generalization of Benford’s (mathy reference, easier reading). Glaeser:

Zipf’s Law ” is one of the great curiosities of urban research.   The law claims that the number of people in a city is inversely proportional to the city’s rank among all cities.  In other words, the biggest city is about twice the size of the second biggest city, three times the size of the third biggest city, and so forth. …

My own view is that Zipf’s Law is really about the operation of agglomeration — the attraction of people to more people — and sprawl.  An initial population attracts more people who live nearby.  As long as each person attracts about the same number of new people, then … that gives us Zipf’s law.

I wonder if it is sensible to be surprised by such regularities. If you make the crude assumption that all people are essentially behaviorally identical then you should expect  some aggregate measures to be related to each other in some way. So we found one relationship. The fun part is trying to put meaning to it, to relate it to our common tendencies at an individual level. That’s what Glaeser is doing.

Of course, knowing nothing else one might be tempted to argue that chaos should rein, that we’re dissimilar enough that aggregates don’t relate. Of course in fine detail, that’s true. But broadly it appears it isn’t, or at least not in some respects.

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