Abstract
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.
Highlights
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
Novelties can be viewed as first time occurrences of some event and the mathematical object used to model an innovation process is an urn model with infinitely many colors, known as species sampling sequence[13,14,15]
The effect that we model refers to a damping factor in the probability of a repetition of an old element: using the metaphor of the urn, the number of balls of a given color in the urn increases with the number of times that color has been extracted according to a suitable update function that exhibits two different speeds, one before a certain threshold and a lower one after the threshold
Summary
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. We generalize the urn with triggering model by the introduction of a function F that drives the update mechanism of the number of balls of the same color of the extracted one when it is of an “old” color.
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