  Zipf's law is one the most conspicuous empirical facts for cities, however,
there is no convincing explanation for the scaling relation between rank and
size and its scaling exponent. Based on the idea from general fractals and
scaling, this paper proposes a dual competition hypothesis of city develop to
explain the value intervals and the special value, 1, of the power exponent.
Zipf's law and Pareto's law can be mathematically transformed into one another.
Based on the Pareto distribution, a frequency correlation function can be
constructed. By scaling analysis and multifractals spectrum, the parameter
interval of Pareto exponent is derived as (0.5, 1]; Based on the Zipf
distribution, a size correlation function can be built, and it is opposite to
the first one. By the second correlation function and multifractals notion, the
Pareto exponent interval is derived as [1, 2). Thus the process of urban
evolution falls into two effects: one is Pareto effect indicating city number
increase (external complexity), and the other Zipf effect indicating city size
growth (internal complexity). Because of struggle of the two effects, the
scaling exponent varies from 0.5 to 2; but if the two effects reach equilibrium
with each other, the scaling exponent approaches 1. A series of mathematical
experiments on hierarchical correlation are employed to verify the models and a
conclusion can be drawn that if cities in a given region follow Zipf's law, the
frequency and size correlations will follow the scaling law. This theory can be
generalized to interpret the inverse power-law distributions in various fields
of physical and social sciences.
