The Fractional Calculus and the Logarithmic Diffusion in Nation Wide Language Data with Time Stamps: How Stable Are the Usages of Already Popular Words?

Abstract

The usages of already popular words (ex. “cat”, “handle”) look like very stable. Then, how stable are they? In this paper, first, by analysing nation-wide text data such as 3 billion Japanese blog articles, we find that the uses of already popular words are changing logarithmically. In addition, this logarithmic behaviour is also observed in various languages and media such as newspaper articles, page views of Wikipedia (English, Chinese etc.). Second, we show that empirical properties are basically explained the random walk with power law forgetting, which is related to the fractional calculus. In the model, the parameter characterizes the speed of forgetting corresponds to the border between the stationary and the non-stationary.