A couple of weeks ago I tripped over a column entitled “Math and the City” published in the New York Times. The title interested me, so I had a look. It was a guest column written by Steven Strogatz, who was standing in for regular Times columnist Olivia Judson, who is referred to in a way that suggests she is very well known, although I suspect she is well known to Times readers like CBC on air personalities are well known to Canadians of the type who watch the CBC. I know even less about Strogatz but his article addressed a mathematical relationship in city size that I vaguely recall from school called Zipf’s Law.
According to Strogatz, Zipf’s Law describes:
… [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.
My immediate reaction to any such powerful and readily verifiable statement as this is to check it out, which I did within a few minutes of finishing the column. On the Web I found that those who like Zipf’s Law, like Strogatz, like it a lot. Many responded within a few days of Strogatz’s column with examples of city size distributions from varied countries that demonstrate or test Zipf’s construct. A good one can be found at the Infrastructurist blog site. Another site I found provides a compilation of cities by size for most of the major countries in the world. The data are a bit old (the Canadian populations are from 1991, the American numbers from 1994) but amply justify the authors’ faith in Zipf.
The prevalence of opinion and hard data didn’t stop me though. I immediately downloaded the list of Canadian Census Metropolitan Areas (CMAs) from the Statistics Canada Web site and the list of Stantard Metropolitan Statistical Areas from the United States Census Bureau site and tested Zipf north of the Rio Grande. The results are a bit different between Canada and the United States but neither undermines Zipf much for anyone but the dedicatedly literal.
The figure following illustrates the relative size of the ten largest CMAs in Canada and the ten largest SMSAs in the United States with the curve predicted by Zipf. The fit is pretty good in both cases with the Canadian metropoli arranged a bit erratically around Zipf’s curve but invariably close to it, while their American equivalents describe a curve that fits closely to Zipf’s slope but which lies significantly above the Zipf line. While the average deviation of the top ten Canadian urban centres from Zipf’s prediction (16.5 per cent) is considerably less than their American counterparts (73.4 per cent), the US distribution would be remarkably close to the Zipfian ideal if we could only get New York up to 24 million souls.
Ten Largest Canadian CMAs, 2006, and Ten Largest US SMSAs, 2003, Relative to Zipf’s Law
When we plot all 33 of Canada’s CMAs, the resulting image really is remarkable. Although the average deviation goes up slightly to 19.3 per cent, the visual fit is outstanding. The same can be said for the distribution of Ame rican SMSAs, all 364 of which are larger than predicted by Zipf but whose overall distribution closely follows his expected curve. The only noticeable difference is the tendency of US SMSAs to be above the Zipf line, particularly for mid-sized urban areas.
All Canadian CMAs Relative to Zipf’s Law, 2006
All US SMSAs Relative to Zipf’s Law, 2003
The relationship that Zipf’s Law illuminates is not hard to observe. Most countries and regions are dominated by a single large city or urban region. The reasons aren’t that hard to fathom either. Large scale is a well-recognized economic advantage. Once a community has the edge in size the benefits accumulate. All roads do tend to lead to Rome and once they do young people are inclined to follow them. Theorists have suggested that Zipf Law holds because the associations of many people in large places draw many more people to them. They also suggest that the rich get richer as the thriving economies of populous communities attract the downtrodden. Most certainly the bigger the place the more familiar it is and the more obvious it likely becomes to those who can’t find a good time, an education, a marriage partner, or a job where they grew up.
