Hierarchy, cities size distribution and Zipf's law
Department of Town and Regional Planning, and Center for the Study of Complex Dynamics, University of Florence, via Micheli, 2, 50121 Firenze, Italy
Corresponding author: a firstname.lastname@example.org
Revised: 5 April 2008
Published online: 28 May 2008
We show that a hierarchical cities structure can be generated by a self-organized process which grows with a bottom-up mechanism, and that the resulting distribution is power law. First we analytically prove that the power law distribution satisfies the balance between the offer of the city and the demand of its basin of attraction, and that the exponent in the Zipf's law corresponds to the multiplier linking the population of the central city to the population of its basin of attraction. Moreover, the corresponding hierarchical structure shows a variable spanning factor, and the population of the cities linked to the same city up in the hierarchy is variable as well. Second a stochastic dynamic spatial model is proposed, whose numerical results confirm the analytical findings. In this model, inhabitants minimize the transportation cost, so that the greater the importance of this cost, the more stable is the system in its microscopic aspect. After a comparison with the existent methods for the generation of a power law distribution, conclusions are drawn on the connection of hierarchical structure, and power law distribution, with the functioning of the system of cities.
PACS: 89.75.Da – Systems obeying scaling laws / 89.75.Fb – Structures and organization in complex systems / 89.65.Lm – Urban planning and construction
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2008