Urban Systems Diversity Measures

Digest the complex temporal dynamics of hierarchical communities is hard to attain in a way that captures the changes in rank and size of its members. Particularly, in urban systems, scaling laws and rank clocks approaches have proved to capture much of this dynamic at macro and micro scales respectively, correlating the variation of urban attributes with city size.
Nevertheless, the former highly dependents on a coherent city definition and the latter lose the actual population in the analysis. Fully aware that these problems are perhaps intractable, here we argue that adding simple diversity measures to the analysis could give some insights about the self-organization process that these urban hierarchical structures experience over time.
Taking some ideas from linguistics [1] and biology we looked at the behaviour of the rank itself (measured as the number of different cities occupying one rank over time) and relate it with the mean clock rank shifts and the cities total turnover from one year to another, to compose a rational picture of the complex temporal evolution of the urban system in terms of its population size.
We selected 10 urban systems (UK, France, Italy, Spain, Mexico, USA, Colombia, Canada, Japan, Ex-Soviet Union) as a case of study and apply these diversity measures over a 100 years period (1900 to 2010) divided into 12 points roughly corresponding with national official Census. Our findings emphasize the differences between European systems and their Asian and American counterparts, reinforcing the notion that there is no ultimate rank-size universality to be found in cities. For example, the corpus of cities that are present in lower ranks at all years sampled, it is much larger for European systems that for the American ones, detecting the fundamental differences in terms of foundations dates between the two continents. American systems tend to have greater variety at middle ranges reflecting a stronger interaction between their cities through the last 110 years.