Abstract
Power law distributions are omnipresent in physics, geophysics,
biology, lexicography, music as well as in social and financial
networks. Usually, the power law is restricted to a range of values
outside of which finite size corrections are often invoked. Here, we
introduce a functional form, which captures a generic behavior for
rank-frequency and rank-size relations for the full data range of a
variety of phenomena from the arts and sciences, both social and
natural. We have found that the two parameter beta-like distribution
$f(r)=A(N+1-r)^b/r^a$, where $r$ is the rank, $N$ the highest
rank value and $A$ is a normalization factor, fits the data with
correlation coefficient very close to unity. The parameter $a$ is
related to a usual scale invariance, while the crossover to a
different statistical regime is incorporated in the parameter $b$.
Though this crossover is frequently considered as a finite size effect this is may not necessarily be the case. The distribution parameters $a$ and $b$ may allow for a classification in universality classes of a variety of situations. Since we have observed that the competition between permanence and change appears to be a common feature in several of the phenomena being analyzed,
we comment on two formalisms that incorporate this competition in their dynamics: nonlinear chaotic maps with intermittent behavior and probabilistic expansion-modification equations. The ubiquity of our findings may be indicative of an underlying central limit theorem.
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