Rank Clocks
Supplementary Information for the Paper

Many systems of events that show scaling in their size distributions consist of unique events in time particularly those which are of a physical nature such as earthquakes, craters, solar flares and so on. The systems that we deal with in the paper consist of non-unique objects - cities in our case - that are fixed through time but changing in size. The problem that pervades this kind of analysis involves the definition of the objects in such a way that they are comparable through time, particularly when the objects can change in their physical area.

The three systems of cities that we deal with - for the US from 1790 to 2000, for the UK from 1901 to 2001, and for the World from 430BCE to 2000 - each define cities differently. In the US and World examples, the area of each city varies through time and the composition of the list of cities at each time also varies. In the US example, the Census Bureau who compiled the data have ensured that the cities identified are comparable from one decade to another; in short, cities do change by embracing their suburbs which might be defined as individual cities in their own right in earlier eras, but the US data set acknowledges this, consistently merging cities through time when appropriate. The World example does not attempt this in an explicit and controlled way. In the UK example, the cities are fixed in area so that they are exactly comparable from one time to another and the total number of cities is also fixed.

These differences are minimised in the analysis in that in each case, a smaller but fixed number of cities - 50 in total - are identified for analysis so that all results become comparable. Of the 266 US cities which enter the top 100 cities used in the original data set, some 135 (or 37%) of these enter the top 50 from 1790 to 2000. Of the 458 cities that are defined in the UK data, 71 (or 70.4%) of these enter the top 50 cities from 1901 to 2001. The World data set contains 390 cities in total organised into 50 top cities for each time period of which 12.8% appear in this list from 430BCE to 2000. These imply substantially different kinds of city systems but they are comparable. In the panel on the right, we also plot the total city populations for the top 50 cities for each of our examples. The trajectories show very different patterns of growth and change, thus making our three examples comparable in the most interesting way.

In the main paper, we did not show the Zipf plots and the rank clocks for the US and UK systems based on the top 50 cities at each time. We now show these, but also repeat the World system results which are based on 50 cities. This makes the data patterns in these plots and clocks comparable. The Zipf plots are shown below followed by the rank clocks. Readers can also compare these plots and clocks for the 50 city ranks to the respective full city ranks of 100 for the US and 458 for the UK systems which are in the main paper. The plots and clocks for the 50 city systems appear to have similar patterns to those for the full systems with a little greater volatility in shifting ranks as the city sets are reduced.

In the main paper, we do not show the plots and clocks for the model system. The full system contains too many cities (1500) to produce meaningful rank and distance clocks as the visualisation is too rich (see below) although we deal with only the top 50 of these at each time. However we show the plots and clocks for the model results based on selecting the top 50 cities and these are shown as a composite below. With 1500 cities, 248 (or 21%) enter the top 50 over 2500 time steps. This is proportionately a lot less than the 12.8% of the much smaller 390 city set that enter the top 50 World cities, and it is consistent with the fact that there is less volatility in the model data that in the World data. In the panel on the right alongside the Zipf plot and clocks, we show the growth in the total city populations. This is much smoother than any of our three examples, illustrating classic exponential growth in contrast to the growth in the city populations of the World data set which is almost double exponential. Again this indicates that the model is not able to simulate the volatility of city growth as it has appeared in world history, and in the massive transition beginning from Renaissance in Europe which led to the Industrial Revolution, and the subsequent urbanisation of the developing world.

Readers can now make comparisons visually of all the patterns for the model, the US, the UK and the World, each based on the top 50 cities.

The clocks are so rich in detail that a full analytic appreciation of any particular system would require the reader to explore the plots and clocks using the relevant software. We provide an example of this for the US system in a much stripped down program that readers can download here if they click on the following link. The zip file contains the program executable, the US data file, and a short two page manual showing the User how to run the program and explore its results. Please note that if you download this program, keep the data file in the same folder as the program executable. The program should then run under the Windows Operating System but will not run under any other. As the software has been written under Windows XP, it may not run under different variants and the author cannot take any responsibility for this.