Zipf’s law and world city populations

Will Fitzgerald just wrote about an excellent article by Steven Strogatz on Zipf’s Law for the populations of cities. If you look at the biggest city, then the next biggest city, etc., there tends to be an exponential fall-off in size.

I was wondering what this looks like so here’s the classic zipfian plot (log-size vs. log-rank) for city population data from from populationdata.net:

If you fit a power law — that is, a line on the above logsize-logrank plot — you can use rank to predict the sizes of smaller cities very accurately, according to Will’s analysis. Larger cities are more problematic, lying off the line.

I was curious whether the power law holds within countries as well. The above plot was only for the countries that had more than 10 cities in the dataset — just eight countries. So here are those same cities again, but plotted against ranks within their respective countries.

The answer is — usually, yes, the power law looks like it holds within countries as well. (Country names are French in this data … Etats-Unis = USA, Allemagne = Germany, etc.) Russia seems to have the biggest difference between its head vs. tail cities. The tail cities have the linear logsize-logrank relationship, but the top 3 cities (Moscow, St. Petersburg, Nizhny Novgorod) seem to have their own different slope.

If you randomly subsample out of a Zipf distribution, the samples will be Zipfian as well, so this isn’t too surprising. If, on the other hand, you’re a fan of theories that power law population relationships might happen as a result of the structural dynamics of growth — for example, winners-win (i.e. rich-get-richer) growth patterns can sometimes result in zipf-distributed sizes — then there’s a case that these dynamics might be happening at both the world and country levels.

Also: this is the first time I’ve used Hadley Wickham‘s ggplot2 and it was great. All of the fun of lattice minus a lot of the pain, plus default display options that aren’t ugly as hell :)

Update: alternative view of those two above graphs.

This was brought to you via the following R code:

d=read.delim(‘cities.tsv‘,header=T)
bigs=names(table(d$country))[table(d$country) > 10]
x=d[d$country %in% bigs,]; x=x[order(-x$pop),]
plot(log10(pop) ~ log10(1:nrow(x)), data=x, main=’World city populations for 8 countries\nlogsize vs logrank’, col=’darkred’)
text(x=log10(1:nrow(x)), y=log10(x$pop), labels=x$city, pos=ifelse(1:nrow(x) %% 2 == 1, 4, 2), cex=.5, col=’gray30′)
# or better
library(ggplot2)
qplot(log10(1:nrow(x)), log10(pop), data=x) + geom_text(hjust=ifelse(1:nrow(x) %% 2 == 1, 0, 1),label=sprintf(” %s “,x$city),size=2,colour=’darkblue’)
library(plyr)
xr=ddply(x, .(country), function(x) { x=x[order(-x$pop),]; ranks=(1:nrow(x)); data.frame(x$city, logpop=log10(x$pop), logrank=log10(ranks)) })
qplot(logrank, logpop, country, data=xr, facets=~country, main=’world city populations by ranks, for 8 countries’)
# alternate views
xr=ddply(x, .(country), transform, within_country_rank=rank(-pop))
qplot(rank(-pop),pop, data=xr, log=’xy’,colour=country, main=’City population vs rank across countries’)
qplot(within_country_rank,pop, data=xr, log=’xy’,colour=country,main=’City population vs rank within country’)