Oscillatory Dynamics of Urban Hierarchies 900–2000 Vulnerability and Resilience

  • Douglas R. White
  • Laurent Tambayong
Part of the Intelligent Systems Reference Library book series (ISRL, volume 53)


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.


City size distributions Zipfian fallacies Primate cities q-exponentials Chandler city data Trade networks Socio-political instabilities War Secular cycles Resource to population ratios Detrending Dynamical models 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institute for Mathematical Behavioral SciencesUniversity of California at IrvineIrvineUSA
  2. 2.California State University at FullertonFullertonUSA

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