Abstract
A mere hyperbolic law, like the Zipf’s law power function, is often inadequate to describe rank-size relationships. An alternative theoretical distribution is proposed based on theoretical physics arguments starting from the Yule-Simon distribution. A modeling is proposed leading to a universal form. A theoretical suggestion for the “best (or optimal) distribution”, is provided through an entropy argument. The ranking of areas through the number of cities in various countries and some sport competition ranking serves for the present illustrations.
Highlights
Approaches of hierarchical type lie behind the extensive use of models in theoretical physics [1], the more so when extending them into new “applications” of statistical physics ideas [2, 3], e.g. in complex systems [4] and phenomena, like in fluid mechanics [5, 6] mimicking agent diffusion
Researchers have detected the validity of power laws, for a number of characteristic quantities of complex systems [7,8,9,10,11]
To go deeper is a fact of paramount relevance, along with the exploration of more grounding concepts
Summary
Approaches of hierarchical type lie behind the extensive use of models in theoretical physics [1], the more so when extending them into new “applications” of statistical physics ideas [2, 3], e.g. in complex systems [4] and phenomena, like in fluid mechanics [5, 6] mimicking agent diffusion. The reason on why Zipf ’s law is found to be a valid tool for describing rank-sizes rule is still a puzzle In this respect, it seems that no theoretical ground is associated to such a statistical property of some sets of data [18, 19]. Zipf ’s law cannot be viewed as a universal law, and several circumstances rely on data whose rank and size relationship is not of hyperbolic nature. Such a statement is true even in the urban.
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