Scaling Separation and Reconstruction of Zipf's law

Details Scaling Separation and Reconstruction of Zipf's law Scaling Separation and Reconstruction of Zipf's law

2011-04-16

arxiv.org/abs/1104.3199v1

Physics

Physics and Society

27 pages, 5 figures

Scientific paper

Zipf's law is the well-known formulation describing the scaling relation between rank and size of elements in both physical and social systems. The Zipf distribution represents one of the ubiquitous general empirical observations across the individual sciences. However, there is no convincing explanation for the fundament of the empirical law, despite the frequency with which it has been observed, esp., in complex systems. In this paper, Zipf's law is reinterpreted by the process of scaling decomposition and reconstruction. A geometric sequence, p-sequence, abstracted from the general Zipf distribution, is converted into a self-similar hierarchy. Thus the discrete inverse power law is converted into a pair of exponential laws termed the generalized 2n principle. A new inverse power law is then derived from the exponential functions to describe the size-number scaling relation. By doing the underlying rationale can be revealed as entropy-maximizing, which harmonizes equity for individuals (parts) and efficient of the whole. Three simple mathematical experiments and two typical empirical cases are employed to support the theoretical results. The self-similar hierarchy proved to occur widely in physical and social systems and represent a source of considerable interest in many fields involving cities, rivers, earthquakes, fractals, route from bifurcation to chaos. A number of scaling phenomena such as fractal, 1/f{\beta} noise, allometric growth may be integrated by the ideas of hierarchical structure to explain the simple rules of complex systems.

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