Rank-size distribution and UK census 2011 data set

While we haven't told you the previous post on Kawasaki dynamics is actually meant as a context towards upcoming series of posts. While it covered theoretical aspect of the upcoming series, as the model we will talk about is build on the same premise as Kawasaki interpretation of the Ising model, this post will cover empirical aspect of the upcoming series.

Namely, here we introduce you to the rank-size distributions and illustrate the concept using UK census 2011 data. Note that the data is freely available from NOMIS website). Here in this post we will use Tables KS201EW, KS209EW, KS301EW, KS402EW and QS607EW. Our geographical resolution being postal areas.

Rank-size distribution is obtained simply by ordering the data from largest to smallest and plotting it (rank on x-axis and value on y-axis). Precisely because the data is order by "size", this particular plot is known as rank-size distribution. This kind of plot allows to explore size scaling phenomena (the well-known Zipf and Pareto laws would be an example of power-law scaling phenomena revealed by rank-size distribution plot). While this kind of plot also has ties to survival analysis.

If sample size is quite large, then rank-size distribution function is well approximated by an inverse cumulative distribution function (also known as quantile function). One just has to reverse the order:

\begin{equation} RSD\left( \vec{\theta}, k \right) = CDF^{-1} \left( \vec{\theta}, 1-k \right) . \end{equation}

In the above \( RSD\left(\dots\right) \) is the rank-size distribution function, \( \vec{\theta} \) is the distribution parameter vector, \( k \) is normalized rank (\( k \in [0,1] \)) and \( CDF^{-1} \) is the inverse cumulative distribution function.

After this quick introduction we invite you to explore scaling behavior of the UK census 2011 data using the interactive app below.