Yesterday Professor Mike Batty introduced me to the rank-size rule, an idea popularised by George Kingsley Zipf as the relationship between the frequency of an observed phenomenon against the phenomenon’s rank in its group. This is best exemplified by the example of city sizes: essentially Zipf shows that for every really large city, there exist many smaller ones; however these smaller cities aren’t just a bit smaller than the large city, they are considerably smaller, in fact the difference in city size from the biggest cities to the smallest can be explained by a power law, this can be represented as:

Where Pn is the frequency of occurance of a phenomenon ranked nth, and the exponent alpha is usually roughly equal to 1.

The power law thus produces a plot where the 2nd item is 1/2 the size of the 1st, the 3rd item is a 1/3 the size of the 1st etc. This can be represented by a plot of surname frequency in Southwark by rank.

Plot of Surname Frequency against Rank in Southwark for all observed surname (using R)

It is clear from the graph that there are very few surnames which are popular and many which are relatively unique. Another interesting characteristic of a power law, such as the relationship between surname frequency and rank are self similar: if we examine any portion of the curve we should get the same curve, albeit at a different scale.

Plot of Surname frequency for Rank 300 - 6000

It is clear from the above graph that a subset of the full data gives a power law relationship. We can attempt to linearise this relationship by taking the log of the frequency and rank:

The fact that the line is not straight indicates that the relationship is not a true power law. The long tail is accentuated by the stepped line, frequencies are integers so when we get to increasingly unique surnames the ranks tend to cluster. In the rank-size distribution of cities, the characteristic fall in the long tail when linearised like this indicates that city size distributions are really log-normal, however this is not the case in terms of surnames. If we exclude some of the long tail, the relationship can look a bit more linear as this plot demonstrates:

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