Twitter as an innovation process with damping effect

In the existing literature about innovation processes, the proposed models often satisfy the Heaps' law, regarding the rate at which novelties appear, and the Zipf's law, that states a power law behavior for the frequency distribution of the elements. However, there are empirical cases far from showing a pure power law behavior and such a deviation is mostly present for elements with high frequencies. We explain this phenomenon by means of a suitable "damping" effect in the probability of a repetition of an old element. We introduce an extremely general model, whose key element is the update function, that can be suitably chosen in order to reproduce the behaviour exhibited by the empirical data. In particular, we explicit the update function for some Twitter data sets and show great performances with respect to Heaps' law and, above all, with respect to the fitting of the frequency-rank plots for low and high frequencies. Moreover, we also give other examples of update functions, that are able to reproduce the behaviors empirically observed in other contexts.