We analyze the time series of hashtag numbers of social media data. We observe that the usage distribution of hashtags is characterized by a fat-tailed distribution with a size-dependent power law exponent and we find that there is a clear dependency between the growth rate distributions of hashtags and size of hashtags usage. We propose a generalized random multiplicative process model with a theory that explains the size dependency of the fat-tailed distribution. Numerical simulations show that our model reproduces these size-dependent properties nicely. We expect that our model is useful for understanding the mechanism of fat-tailed distributions in various fields of science and technology.
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