Velarde Carlos, Robledo Alberto
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Mexico City, Mexico.
Instituto de Física y Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Mexico City, Mexico.
PLoS One. 2017 Oct 5;12(10):e0186015. doi: 10.1371/journal.pone.0186015. eCollection 2017.
We examine the relationship between two different types of ranked data, frequencies and magnitudes. We consider data that can be sorted out either way, through numbers of occurrences or size of the measures, as it is the case, say, of moon craters, earthquakes, billionaires, etc. We indicate that these two types of distributions are functional inverses of each other, and specify this link, first in terms of the assumed parent probability distribution that generates the data samples, and then in terms of an analog (deterministic) nonlinear iterated map that reproduces them. For the particular case of hyperbolic decay with rank the distributions are identical, that is, the classical Zipf plot, a pure power law. But their difference is largest when one displays logarithmic decay and its counterpart shows the inverse exponential decay, as it is the case of Benford law, or viceversa. For all intermediate decay rates generic differences appear not only between the power-law exponents for the midway rank decline but also for small and large rank. We extend the theoretical framework to include thermodynamic and statistical-mechanical concepts, such as entropies and configuration.
我们研究了两种不同类型的排序数据（频率和量级）之间的关系。我们考虑那些可以通过出现次数或测量大小这两种方式进行排序的数据，例如月球陨石坑、地震、亿万富翁等情况。我们指出这两种类型的分布是彼此的函数反函数，并首先根据生成数据样本的假设母体概率分布来具体说明这种联系，然后根据一个能再现它们的类似（确定性）非线性迭代映射来进行说明。对于随秩呈双曲线衰减的特殊情况，这些分布是相同的，即经典的齐普夫图，一种纯幂律。但当一个呈现对数衰减而另一个呈现反指数衰减时，它们的差异最大，就像本福特定律的情况那样，反之亦然。对于所有中间衰减率，不仅在中间秩下降的幂律指数之间，而且在小秩和大秩情况下都会出现一般差异。我们将理论框架扩展到包括热力学和统计力学概念，如熵和构型。
