Taylor’s law quantifies the scaling properties of the fluctuations of the number of innovations occurring in open systems. Urn-based modeling schemes have already proven to be effective in modeling this complex behaviour. Here, we present analytical estimations of Taylor’s law exponents in such models, by leveraging on their representation in terms of triangular urn models. We also highlight the correspondence of these models with Poisson–Dirichlet processes and demonstrate how a non-trivial Taylor’s law exponent is a kind of universal feature in systems related to human activities. We base this result on the analysis of four collections of data generated by human activity: (i) written language (from a Gutenberg corpus); (ii) an online music website (Last.fm); (iii) Twitter hashtags; (iv) an online collaborative tagging system (Del.icio.us). While Taylor’s law observed in the last two datasets agrees with the plain model predictions, we need to introduce a generalization to fully characterize the behaviour of the first two datasets, where temporal correlations are possibly more relevant. We suggest that Taylor’s law is a fundamental complement to Zipf’s and Heaps’ laws in unveiling the complex dynamical processes underlying the evolution of systems featuring innovation.

中文翻译：

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创新过程中的泰勒定律


泰勒定律量化了开放系统中发生的创新数量波动的缩放特性。基于瓮的建模方案已被证明可以有效地模拟这种复杂的行为。在这里，我们通过利用三角瓮模型的表示，对此类模型中的泰勒定律指数进行分析估计。我们还强调了这些模型与泊松-狄利克雷过程的对应关系，并证明了非平凡的泰勒定律指数如何成为与人类活动相关的系统中的一种普遍特征。我们基于对人类活动生成的四个数据集的分析得出这一结果：（i）书面语言（来自古腾堡语料库）； (ii) 在线音乐网站（Last.fm）； (iii) Twitter 主题标签； (iv) 在线协作标记系统 (Del.icio.us)。虽然在最后两个数据集中观察到的泰勒定律与普通模型预测一致，但我们需要引入概括来充分表征前两个数据集的行为，其中时间相关性可能更相关。我们认为泰勒定律是齐普夫定律和希普斯定律的基本补充，可以揭示创新系统演化背后的复杂动态过程。