Oscillatory Dynamics of Urban Hierarchies 900–2000 Vulnerability and Resilience

Abstract

We show the fallacies of the Zipfian and power-law views in quantitative analyses of city-size distributions and of the notion that they are superior, simpler, and more universal than q-exponential or Pareto II distributions because of the single-parameter assumption. Both sets of models have two parameters, but the q-exponential captures a scaling coefficient that is fundamental to understanding now the distribution of smaller sizes that is left out by the “cutoff” parameter of Zipfian and power-law. The additional parameter has a fundamental importance in understanding the historical dynamics of partially independent changes in the parameters of urban hierarchies and their interactions with the trading, larger resource bases and conflict networks (internal and external) in which city networks are embedded. We find, for the historical periods from 900 to 2000, combining Chandler and U.N. data, in three different regions of Eurasia, that q, as a small-city distribution parameter, has a time-lagged effect on $\beta$, as a measure of only power-law inflection on the top 10 regional cities in China, Europe and Mid-Asia. Taking other time-lag measures into account, we interpret this as showing that trade, economic and conflict or socio-political instability in towns and smaller-cities of urban hierarchy have a greater effect on the health and productivity of larger cities as reflected in Zipfian size distributions, with growth proportional to size. As a means to help the reader understand our modeling efforts, we try to provide foundational intuitions about the basis in “nonextensive” physics used to improve our understanding how q-exponentials are derived specifically for non-equilibrium networks and distributions with long-range process that interconnect different parts, such as trade and transport systems, warfare and interpolity rivalries. The ordinary entropic measure e is “extensive” in that interactions of random effects are additive; the nonextensive generalization of $e_{q}$ where $e_{q=1} = e$ is ordinary Boltzmann-Gibbs entropy and the concept of multiplicative departures from randomness in the range $1\ {<}\ q\ {<}\ 2$ helps to relate the physical processes subsumed in urban systems to the underlying foundation of certain Zipfian distributions from contributions to the non-Zipfian properties of the smaller cities in urban hierarchies.

Appendix A: Example of the European City-Size Distribution

Figures 16 and 17 show the raw European cumulative city-size distributions truncated to the top 20 cities for each of Chandler’s historical periods. The x-axis gives city rank. The y-axis in Fig. 16 is the natural log e of city size, unlabeled. Figure 17 shows the same data in a log10-log10 plot, where city rank is also logged and the y axis should be multiplied by 100. If the latter plot showed straight lines, the distributions would be Pareto I power-laws. Cutoffs can be seen for each of the lines, an indication that the Pareto II q-exponential is more relevant for these distributions. Minimum top-ranked city-sizes recorded by Chandler [9] are as small as 11,000 persons, but not lower, so that cities of sizes down to 5,000 are not included. Although the q-exponential has two parameters, a power-law fit to these data would also require a second parameter for each distribution, the size at which the distributions of higher sizes is power-law.