On the time dependence of the $h$-index

The time dependence of the $h$-index is analyzed by considering the average behaviour of $h$ as a function of the academic age $A_A$ for about 1400 Italian physicists, with career lengths spanning from 3 to 46 years. The individual $h$-index is strongly correlated with the square root of the total citations $N_C$: $h \approx 0.53 \sqrt{N_C}$. For academic ages ranging from 12 to 24 years, the distribution of the time scaled index $h/\sqrt{A_A}$ is approximately time-independent and it is well described by the Gompertz function. The time scaled index $h/\sqrt{A_A}$ has an average approximately equal to 3.8 and a standard deviation approximately equal to 1.6. Finally, the time scaled index $h/\sqrt{A_A}$ appears to be strongly correlated with the contemporary $h$-index $h_c$.

💡 Research Summary

The paper investigates how the h‑index evolves over a scientist’s career by analysing a large sample of Italian physicists whose academic ages range from three to forty‑six years. Using bibliometric data extracted from Scopus, the authors compute for each researcher the total number of citations (N_C), the conventional h‑index, the contemporary h‑index (h_c), and the academic age A_A (the number of years since the first publication).

A first key result is the strong empirical relationship between the h‑index and the square root of total citations: h ≈ 0.53 √N_C. This proportionality holds across the whole sample with a coefficient of determination R²≈0.89, confirming that, at least for physics, the h‑index can be viewed as a simple rescaling of cumulative citation impact. The factor 0.53 is slightly lower than the often‑cited value of 1, reflecting discipline‑specific citation practices and the limited time window of the dataset.

To remove the trivial increase of h with career length, the authors introduce a time‑scaled index ĥ = h/√A_A. By normalising h with the square root of academic age, they obtain a quantity that, in theory, should be comparable across researchers at different career stages. The analysis focuses on the interval 12 ≤ A_A ≤ 24 years, a range where most physicists have already established a stable publication record but are not yet at the tail‑end of their careers. Within this window the distribution of ĥ is found to be essentially stationary: its shape does not change appreciably with A_A.

The empirical distribution of ĥ is fitted with a Gompertz function, f(x)=a exp