语言中分数动态驱动的超低扩散的经验观察：已经流行的单词的单词计数的动态统计特性
Empirical observations of ultraslow diffusion driven by the fractional dynamics in languages: Dynamical statistical properties of word counts of already popular words
摘要：
超低扩散（即对数扩散）已被广泛研究的理论，但几乎没有观察到经验。在本文中，首先，我们通过分析三个不同的全国性语言数据库：（i）报纸文章（日语），（ii）博客文章（日语），和（iii）维基百科的页面浏览量（英语，法语，中文和日语），发现已经流行的单词的字数的时间序列的超慢扩散。其次，我们用理论分析表明，这种扩散基本上是由指数为$β\approx 0.5$的幂律遗忘的随机行走模型解释的，它与分数朗之万方程有关。指数$β$表征遗忘的速度，并且$β\approx 0.5$对应于（i）平稳和非平稳之间的边界（或阈值）以及（ii）$β=1$的IID噪声和$β=0$的正常随机游走之间的中间动态。第三，流行词词数时间序列的生成模型是一种Poisson过程，其参数由上述随机游走模型采样得到，该模型不仅能近似地再现经验均方位移，而且能近似地再现功率谱密度和概率密度函数。
英文摘要：
Ultraslow diffusion (i.e. logarithmic diffusion) has been extensively studied theoretically, but has hardly been observed empirically. In this paper, firstly, we find the ultraslow-like diffusion of the time-series of word counts of already popular words by analysing three different nationwide language databases: (i) newspaper articles (Japanese), (ii) blog articles (Japanese), and (iii) page views of Wikipedia (English, French, Chinese, and Japanese). Secondly, we use theoretical analysis to show that this diffusion is basically explained by the random walk model with the power-law forgetting with the exponent $β\approx 0.5$, which is related to the fractional Langevin equation. The exponent $β$ characterises the speed of forgetting and $β\approx 0.5$ corresponds to (i) the border (or thresholds) between the stationary and the nonstationary and (ii) the right-in-the-middle dynamics between the IID noise for $β=1$ and the normal random walk for $β=0$. Thirdly, the generative model of the time-series of word counts of already popular words, which is a kind of Poisson process with the Poisson parameter sampled by the above-mentioned random walk model, can almost reproduce not only the empirical mean-squared displacement but also the power spectrum density and the probability density function.
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