Temporal dynamics of Hirsch index

We performed the analysis of the data from the Scopus database regarding temporal dynamics of $h$-index and $h_s(2015)$-index of a group of the continuously and consistently working scientists. We propose a model describing the temporal dynamics of $h$-index. Temporal dynamics of $h_s(2015)$-index demonstrates sigmoidal behaviour. The model takes into account: 1) changing the publication activity of the scientist (sigmoidal growth of number of publications at the early stages of scientific career is assumed); 2) the distribution of articles by the number of citations; 3) the dynamics of each specific article citation (typically, the number of citations at first increases and then gradually decreases). The dynamics of the $h$-index as a function of average productivity (number of articles published per year) is investigated. We used two types of citations distributions, i.e. Lotka distribution and geometric distribution. Both distributions lead to a qualitatively correct temporal dynamics of Hirsch index.