Last week, over at the Gray Lady’s house, there was a story about something called Zipf’s Law, which supposedly predicts the relative populations of our cities. It’s named after a fellow named George Zipf who several decades ago “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.” So, if Country X’s largest city has 12 million people, the second largest would have 6 million, the third largest 4 million–anyway, you get the idea. The author noted that, even though Nobel laureates have looked into it, “no one knows” why Zipf’s law works. It just kind of does. (Bigger chart after the jump)
But since the story lacked any hard data whatsoever, we decided to put together a chart comparing the relative size of the 10 largest cities in the world’s 10 largest countries. Instead of relying on political boundaries, we used the population size of the regions surrounding cities. For instance, in the case of New York City, we counted it as having 19 million people, the size of its metropolitan area, even though the city itself only has around 8 million inhabitants. This is because metropolitan areas more closely represent actual population distribution than city boundaries, which are often the result of historical and political quirks and therefore are unscientific for comparison’s sake.
As the chart shows, the countries with the largest populations — China, India, and the United States — largely buck the trend as their second through tenth largest cities are relatively bigger than Zipf’s law (the black ribbon on the chart) would predict. Other nations like Bangladesh and Japan fall way below the Zipf curve, with the largest city overwhelmingly dominating the nation’s population distribution.
Even so, countries like Nigeria and Brazil have cities whose populations come close to following Zipf’s law.
Why should any of this be the case? Your guess is as good as Paul Krugman’s
Click on the chart to expand
Yonah Freemark is an independent researcher currently working in France on comparative urban development as part of a Gordon Grand Fellowship from Yale University, from which he graduated in May 2008 with a BA in architecture. He blogs about transportation and land use issues at The Transport Politic and is a regular contributor to The Infrastructurist.
May 27th, 2009 at 12:54 pm
Is this 3D ribbon chart really the best way to represent this data? I’m finding it a little ambiguous.
May 27th, 2009 at 6:27 pm
Complete bollocks for the UK
London 10m , Glasgow 1m
May 27th, 2009 at 9:22 pm
basically, the law doesn’t hold at all. It doesn’t work for large countries (China, India, US), works for one large country (Brazil), and fails again for other large countries (Indonesian, Russia, Bangladesh, Japan.
It fails for France, the UK, Australia, Greece, Argentina, and Texas, and Mexico. What countries does it actually even work for?
May 28th, 2009 at 2:18 am
No, it doesn’t seem to work. I think it might be interesting to take all the cities in the world and create a similar chart. If it isn’t true for the whole, it probably isn’t true for the parts. Worldwide, doesn’t population by city decline linearly or exponentially, as theorized by Zipf? Maybe, it would be possible to classify cities by their status and proceed from there. For example, it seems that the megalopoli of the world tend to fall in the 20-25 million range. They are often major world capitals and international business centers. Their existence precludes the existence of anything sumilar happening close by. Megacities are created by internal population. Either the city in question is a major destination or it isn’t. Within a given country, power tends to concentrate. Power draws power. If a city is a tier 2 city it has other characteristics. Los Angeles is a major center for the western Unites States and some of northern Mexico, but New York has the most power. Guadalajara will never be as powerful or populous as Mexico City. As much as LA grows, for example, it will never have the UN or the NYSE, or a variety of other critical institutions, each of which draws additional people and additional institutions. Perhaps there is a ratio that can describe the additional population that a tier 1 city can draw versus a tier 2 city, regardless of the country. Dividinf the results by country doesn’t make as much sense, because in large, populous countries, there is enough room and people for more tier 1 and 2 cities.
May 28th, 2009 at 5:22 am
Ronnie, those aren’t even Britain’s two largest cities. And your figures aren’t accurate. The population of the Greater London Urban Area is 8,278,251. Greater Glasgow is fifth with 1,168,270.
As it happens, Britain does show some horrible random variation in its top two: Greater London overperforms, whilst the West Midlands underperforms (no surprise there then). And in America, the random variation is largely that New York isn’t as big as you would predict it to be.
The trends actually look good as they go along. I dare say they’d fit better if scaled so, say, the tenth largest city was the point at which the curve was intersected, as small sample size, population fluctuation (e.g. Los Angeles overhauling Chicago), and public policy are always going to have the greatest effects at the top end.
There’s also some similarity to Benford’s Law going on. I wonder if Zipf’s Law can be re-expressed as something like y = k log[n](1+1/x), where y is the population of the city, k is a constant, n is the total number of cities, and x is the rank.
May 28th, 2009 at 10:44 pm
I expect iIt’s due to network effects — cities form something close to a scale-free network. Like links on the World Wide Web. Population is roughly proportional to number of visitors, because we’re an essentially trade-based economy, and number of visitors is proportional to number of links.
May 30th, 2009 at 10:23 am
[...] 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 [...]
June 5th, 2009 at 10:28 am
The flow of links on a network resembles Zipf’s Law, with few large (high flow) links and more smaller links (See e.g. Levinson, David and Bhanu Yerra (2006) Self Organization of Surface Transportation Networks Transportation Science 40(2) 179-188. Whether the largest link is twice the second largest link I think is secondary to the general power law structure. Zipf’s law is static, and the real world is dynamic, so any overtaking by one city of another city will break a strict interpretation.