A Universal Rank-Size Law

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.

[1]  A. Hansen.  Grand challenges in interdisciplinary physics , 2014, Front. Phys..

[2]  Gerardo G. Naumis,et al.  The tails of rank-size distributions due to multiplicative processes: from power laws to stretched exponentials and beta-like functions , 2007, New Journal of Physics.

[3]  P. Verhulst.  Recherches mathématiques sur la loi d’accroissement de la population , 2022, Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Bruxelles.

[4]  M. Ausloos,et al.  Evidence of economic regularities and disparities of Italian regions from aggregated tax income size data , 2014, 1411.7880.

[5]  M. Ausloos,et al.  Primacy analysis in the system of Bulgarian cities , 2013, 1309.0079.

[6]  Yuen Ren Chao,et al.  Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology , 1950 .

[7]  Constantino Tsallis,et al.  Numerical indications of a q-generalised central limit theorem , 2005, cond-mat/0509229.

[8]  Guohua Peng.  Zipf’s law for Chinese cities: Rolling sample regressions , 2010 .

[9]  Thorsten Gerber,et al.  Handbook Of Mathematical Functions , 2016 .

[10]  Kerstin Vogler,et al.  Table Of Integrals Series And Products , 2016 .

[11]  Iryna A. Voloshynovska,et al.  Characteristic Features of Rank-Probability Word Distribution in Scientific and Belletristic Literature , 2011, J. Quant. Linguistics.

[12]  J. Kwapień,et al.  Physical approach to complex systems , 2012 .

[13]  Claude E. Shannon,et al.  A Mathematical Theory of Communications , 1948 .

[14]  Lucien Benguigui,et al.  Beyond the power law - a new approach to analyze city size distributions , 2007, Comput. Environ. Urban Syst..

[15]  Kenneth Levenberg.  A METHOD FOR THE SOLUTION OF CERTAIN NON – LINEAR PROBLEMS IN LEAST SQUARES , 1944 .

[16]  M. Ausloos,et al.  generalized Lavalette function , 2014 .

[17]  Colin Rose,et al.  Mathematical Statistics with Mathematica , 2002 .

[18]  J. Henderson,et al.  Handbook of Regional and Urban Economics , 2015 .

[19]  Nikolay K. Vitanov,et al.  On nonlinear dynamics of interacting populations: Coupled kink waves in a system of two populations , 2009 .

[20]  Wentian Li,et al.  Beyond Zipf’s Law: The Lavalette Rank Function and Its Properties , 2016, PloS one.

[21]  Randy A. Becker,et al.  The Formation of Economic Agglomerations: Old Problems and New Perspectives , 1999 .

[22]  Bertrand M. Roehner,et al.  Driving Forces in Physical, Biological and Socio-economic Phenomena: A Network Science Investigation of Social Bonds and Interactions , 2007 .

[23]  Bruce J. West,et al.  Fractal physiology for physicists: Lévy statistics , 1994 .

[24]  Germinal Cocho,et al.  On the behavior of journal impact factor rank-order distribution , 2006, J. Informetrics.

[25]  X. Gabaix.  Zipf's Law for Cities: An Explanation , 1999 .

[26]  Marcel Ausloos.  Punctuation effects in English and Esperanto texts , 2010, ArXiv.

[27]  Marcel Ausloos,et al.  A scientometrics law about co-authors and their ranking: the co-author core , 2012, Scientometrics.

[28]  D. Stauffer.  Introduction to statistical physics outside physics , 2003, cond-mat/0310037.

[29]  M. Ausloos,et al.  Cross ranking of cities and regions: population versus income , 2015 .

[30]  Steven Brakman,et al.  The Return of Zipf: Towards a Further Understanding of the Rank‐Size Distribution , 1999 .

[31]  A. Gadomski.  Kinetic Approach to the Nucleation-and-Growth Phase Transition in Complex Systems , 2001 .

[32]  Manolis I. A. Lourakis.  A Brief Description of the Levenberg-Marquardt Algorithm Implemented by levmar , 2005 .

[33]  Mark Jefferson,et al.  The Law of the Primate City , 1939 .

[34]  Ioan-Iovitz Popescu,et al.  On a Zipf's Law extension to impact factors , 2003, Glottometrics.

[35]  D. Helbing,et al.  Growth, innovation, scaling, and the pace of life in cities , 2007, Proceedings of the National Academy of Sciences.

[36]  Adam Gadomski,et al.  Ranking structures and Rank-Rank Correlations of Countries. The FIFA and UEFA cases , 2014, ArXiv.

[37]  X. Gabaix.  Zipf's Law and the Growth of Cities , 1999 .

[38]  G. Cocho,et al.  Universality of Rank-Ordering Distributions in the Arts and Sciences , 2009, PloS one.

[39]  Cross Ranking of Cities and Regions: Population vs. Income , 2015, 1506.02414.

[40]  Robert A. Fairthorne,et al.  Empirical hyperbolic distributions (Bradford-Zipf-Mandelbrot) for bibliometric description and prediction , 1969, J. Documentation.

[41]  M. Ausloos.  Two-exponent Lavalette function: a generalization for the case of adherents to a religious movement. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[42]  Steve Pressé,et al.  Nonuniversal power law scaling in the probability distribution of scientific citations , 2010, Proceedings of the National Academy of Sciences.

[43]  S. Low,et al.  The "robust yet fragile" nature of the Internet. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[44]  H. E. McKean,et al.  Tables of the Incomplete Beta Function , 1968 .

[45]  K. Pearson.  Tables of the incomplete beta-function , 1951 .

[46]  J. Poot,et al.  A CENTURY OF THE EVOLUTION OF THE URBAN SYSTEM IN BRAZIL , 2013 .

[47]  H. Simon,et al.  ON A CLASS OF SKEW DISTRIBUTION FUNCTIONS , 1955 .

[48]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

[49]  Marcel Ausloos,et al.  Toward fits to scaling-like data, but with inflection points & generalized Lavalette function , 2014, 1404.3605.

[50]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[51]  Bruce M. Hill,et al.  The Rank-Frequency Form of Zipf's Law , 1974 .