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Orderdisorder transition in conflicting dynamics leading to rankfrequency generalized beta distributions

机译：冲突动力学中的无序过渡导致秩频率广义β分布

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The behavior of rank-ordered distributions of phenomena present in a variety of fields such as biology, sociology, linguistics, finance and geophysics has been a matter of intense research. Often power laws have been encountered; however, their validity tends to hold mainly for an intermediate range of rank values. In a recent publication (Martnez-Mekler et al., 2009 [7]), a generalization of the functional form of the beta distribution has been shown to give excellent fits for many systems of very diverse nature, valid for the whole range of rank values, regardless of whether or not a power law behavior has been previously suggested. Here we give some insight on the significance of the two free parameters which appear as exponents in the functional form, by looking into discrete probabilistic branching processes with conflicting dynamics. We analyze a variety of realizations of these so-called expansionmodification models first introduced by Wentian Li (1989) [10]. We focus our attention on an orderdisorder transition we encounter as we vary the modification probability p. We characterize this transition by means of the fitting parameters. Our numerical studies show that one of the fitting exponents is related to the presence of long-range correlations exhibited by power spectrum scale invariance, while the other registers the effect of disordering elements leading to a breakdown of these properties. In the absence of long-range correlations, this parameter is sensitive to the occurrence of unlikely events. We also introduce an approximate calculation scheme that relates this dynamics to multinomial multiplicative processes. A better understanding through these models of the meaning of the generalized beta-fitting exponents may contribute to their potential for identifying and characterizing universality classes.
机译：在诸如生物学，社会学，语言学，金融学和地球物理学等多个领域中存在的现象的按次序分布的行为一直是一个深入研究的问题。经常会遇到功率法。但是，它们的有效性往往主要适用于等级值的中间范围。在最近的出版物中（Martnez-Mekler等，2009 [7]），β分布的功能形式的泛化已显示出非常适合于性质非常多样化的许多系统，适用于整个等级范围值，无论以前是否建议过幂律行为。在这里，我们通过研究具有冲突动力学的离散概率分支过程，对以函数形式的指数形式出现的两个自由参数的重要性给出了一些见识。我们分析了李文田（1989）[10]首次引入的这些所谓的扩展修改模型的各种实现。我们将注意力集中在改变修饰概率p时遇到的有序无序过渡上。我们通过拟合参数来表征这种过渡。我们的数值研究表明，拟合指数之一与功率谱尺度不变性表现出的远距离相关性相关，而另一项与无序元素的作用有关，导致这些特性的破坏。在没有长期相关性的情况下，此参数对不太可能发生的事件很敏感。我们还介绍了一种将这种动力学与多项式乘法过程相关联的近似计算方案。通过这些模型对广义的β拟合指数含义的更好理解可能有助于它们识别和表征通用性类别的潜力。

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