A Universal Rank-Size Law

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.

Universal City-size distributions through rank ordering

“…Note that, since f (a,b) (r) is assumed to be normalized to 1 function, following [21], it is absolutely meaningful to use the Shannon entropy as defined in (4). The entropy measure in Eq.…”

“…* Electronic address: abhianik@gmail.com † Electronic address: sribbasu@gmail.com arXiv:1809.08786v3 [physics.soc-ph] 8 May 2019 that the domain is the natural numbers 1 to N instead of the interval (0, 1) of usual continuous beta distribution. Several RO distributions including the DGBD and its further generalizations have already been studied to provide good fits in the context of different count or rank-size data from arts and sciences [20][21][22][23][24][25][26]. In the present paper, our main contribution is to propose and illustrate that the DGBD can be successfully applied to provide a simple, yet excellent, universal fit to the city size data across different socio-economic countries.…”

Evidence for Gross Domestic Product growth time delay dependence over Foreign Direct Investment. A time-lag dependent correlation study

“…Therefore, a rank-size rule fit was attempted with a decreasing power law, with different exponents at low and high ranks, -in order to obtain an inflection point near the center of the data range, i.e. with the analytical form (Ausloos & Cerqueti, 2016) y(r) = −1 + m 1 N −m2 r −m2 (N + 1 − r) m3 (5.1) where r is the rank and N = 43. The best 3-fit parameters have been so obtained: m 1 ∈ [0.720; 0.959]; m 2 ∈ [0.077, 0.130]; m 3 ∈ [0.218, 0.273]: for a regression coefficient R 2 ∼ [0.977, 0.988], indicating a quite good agreement with Eq.…”

An Empirical Model for n-gram Frequency Distribution in Large Corpora

“…an analysis of low frequency words is often omitted. These difficulties reinforce the importance of empirical approaches, leading to many models ( [1,2,5,6,10], among others as surveyed in [7]). Still, due to its simplicity and universality, Zipf's law is widely used, as a first approximation or as a basis for improvements.…”