Randomly stopped extreme Zipf extensions

In this paper, we extend the Zipf distribution by means of the Randomly Stopped Extreme mechanism; we establish the conditions under which the maximum and minimum families of distributions intersect in the original family; and we demonstrate how to generate data from the extended family using any Zipf random number generator. We study in detail the particular cases of geometric and positive Poisson stopping distributions, showing that, in log-log scale, the extended models allow for top-concavity (top-convexity) while maintaining linearity in the tail. We prove the suitability of the models presented, by fitting the degree sequences in a collaboration and a protein-protein interaction networks. The proposed models not only give a good fit, but they also allow for extracting interesting insights related to the data generation mechanism.