Dynamic analysis of the time-dependent progress of h-index

The h-index is a single-number metric that combines the effect of a researcher’s productivity and impact to measure her research ‘excellence.’ Google Scholar offers a free service where authors can check their record of citations, h-index and i10-index values. This paper employs dynamic analysis and attempts to shed some light on the reasons why h-index grows slowly. We first present a deterministic model where a closed-form expression of the h-index is found as a function of the rate of citations received, and the time since the author’s first publication. Next, we present a stochastic model where the rate of citations is a discrete random variable. For this case, we determine the probability density function of the random h-index using probability generating functions. We show that when the citation arrival rate is an affine linear function of time, the h-index always grows linearly in time. Finally, we consider the more general case of citation arrivals varying over time. For this case, simulation analysis demonstrates that as the variability increases, we find a decline in the h-index. We also find that when variability is low, the simulated h-index values move in parallel with the theoretical results.