  What dynamics govern a time series representing the appearance of words in
social media data? In this paper, we investigate an elementary dynamics, from
which word-dependent special effects are segregated, such as breaking news,
increasing (or decreasing) concerns, or seasonality. To elucidate this problem,
we investigated approximately three billion Japanese blog articles over a
period of six years, and analysed some corresponding solvable mathematical
models. From the analysis, we found that a word appearance can be explained by
the random diffusion model based on the power-law forgetting process, which is
a type of long memory point process related to ARFIMA(0,0.5,0). In particular,
we confirmed that ultraslow diffusion (where the mean squared displacement
grows logarithmically), which the model predicts in an approximate manner,
reproduces the actual data. In addition, we also show that the model can
reproduce other statistical properties of a time series: (i) the fluctuation
scaling, (ii) spectrum density, and (iii) shapes of the probability density
functions.
