doi:10.1016/j.physa.2007.08.002 
Copyright © 2007 Elsevier B.V. All rights reserved.
Tail universalities in rank distributions as an algebraic problem: The beta-like function
G.G. Naumisa, 
, 
and G. Cochob
aDepartamento de Fisica-Quimica, Instituto de Fisica, Universidad Nacional Autónoma de México, Apdo.Postal 20-364, 01000 México, D.F., Mexico
bDepartamento de Sistemas Complejos, Instituto de Fisica, Universidad Nacional Autónoma de México, Apdo.Postal 20-364, 01000 México, D.F., Mexico
Received 4 May 2007;
revised 4 July 2007.
Available online 8 August 2007.
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Abstract
Although power laws of the Zipf type have been used by many workers to fit rank distributions in different fields like in economy, geophysics, genetics, soft-matter, networks, etc. these fits usually fail at the tail. Some distributions have been proposed to solve the problem, but unfortunately they do not fit at the same time the body and the tail of the distribution. We show that many different data in rank laws, like in granular materials, codons, author impact in scientific journal, etc. can be very well fitted by the integrand of a beta function (that we call beta-like function). Then we propose that such universality can be due to the fact that systems made from many subsystems or choices, present stretched exponential frequency-rank functions which qualitatively and quantitatively are well fitted with the beta-like function distribution in the limit of many random variables. We give a plausibility argument for this observation by transforming the problem into an algebraic one: finding the rank of successive products of numbers, which is basically a multinomial process. From a physical point of view, the observed behavior at the tail seems to be related with the onset of different mechanisms that are dominant at different scales, providing crossovers and finite size effects.
Keywords: Ranking distributions; Power law distribution; Zipf law; Multiplicative processes
PACS classification codes: 89.75.Fb; 87.10.+e; 89.75.Da; 89.65.Gh; 89.65. −s; 87.23.Cc
Fig. 1. Population ranking of four representative municipalities from Mexico and Spain. The solid lines are the fits obtained from Eq. (1). The inset presents the corresponding values of a and b used in the fits. The correlation coefficients 
are bigger than 0.986 in all cases.
Fig. 2. Impact factor as a function of the rank for physics, computer science and agroscience. Fits using the beta-like function are shown as solid lines. Inset: values of a and b. The correlation coefficients are bigger than 
.
Fig. 3. Frequency of codons (normalized to 1000) as a function of the rank for the genome of four different species, with their corresponding fits shown as solid lines. Inset: values of a and b used for the fits in the beta-like distribution. The correlation coefficients are bigger than 
.
Fig. 4. Rank-ordered distribution of stick–slip events in a slowly sheared granular media. Circles are data taken from Ref. [4], and the solid line is a fit using Eq. (1), with a=1.08 and b=0.40. The correlation coefficient is 
.
Fig. 5. Successive products of three numbers p1=0.5202, p2=0.3125, p3=0.1673 as a function of the rank (bold solid line) for N=77, and a fitting using Eq. (1), with a=9.36, b=14.53.
Fig. 6. Path of decreasing rank in the n1,n2 and n3 space, for N=15 and three random numbers p1=0.5202, p2=0.3125, p3=0.1673.
Fig. 7. Values of n1 (thin solid line), n2 (gray line) and n3 (solid bold line) as a function of the rank, for N=20 and p1=0.5202, p2=0.3125, p3=0.1673.
Fig. 8. Path of decreasing ranks in the n2 and n3 plane for p1
p2
p3, where the n1 coordinate was eliminated using that n1+n2+n3=N. The dotted line corresponds to all the n2MAX(r), which defines the envelope of the ranking sequence.
Fig. 9. Numerical results for the ranking of the successive product of three numbers such that p1p2p3. The dotted line is the prediction using Eq. (16).
Fig. 10. Ranking of the successive product of three numbers such that p1p2p3, for p1=0.99999, p2=6.2×10-6, p3=3.8×10-6. The dashed line is the prediction made from Eq. (18), compared with the numerical result for N=100 iterations (solid line).
Fig. 11. A plot of Eq. (23) using C1=2, D=1, E=1, β=1 and α=1.
Table 1.
Numerical values of a and b and the correlation coefficients 
for all of the examples presented in this work
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