Empirical observations of ultraslow diffusion driven by the fractional dynamics in languages

Ultraslow diffusion (i.e., logarithmic diffusion) has been extensively studied theoretically but has hardly been observed empirically. In this paper, first, we find the ultraslow-like diffusion of the time series of word counts of already popular words by analyzing three different nationwide language databases: (i) newspaper articles (Japanese), (ii) blog articles (Japanese), and (iii) page views of Wikipedia (English, French, Chinese, and Japanese). Second, 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 $\beta \approx 0.5$, which is related to the fractional Langevin equation. The exponent $\beta$ characterizes the speed of forgetting and $\beta \approx 0.5$ 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 $\beta =1$ and the normal random walk for $\beta =0$. Third, 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.