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.
89.75.Da Systems obeying scaling laws 89.75.Fb Structures and organization in complex systems 89.65.Lm Urban planning and construction