Scaling and Kinetic Exchange Like Behavior of Hirsch Index and Total Citation Distributions: Scopus-CiteScore Data Analysis

Abstract

We analyze the data distributions $f(h)$, $f(N_c$) and $f(N_p)$ of the Hirsch index $(h)$, total citations ($N_c$) and total number of papers ($N_p$) of the top scoring 120,000 authors (scientists) from the Stanford cite-score (or c-score) 2022 list and their corresponding $h$ ($3 \le h \le 284$), $N_c (1009 \le N_c \le 428620$) and $N_p$ ($3\le N_p \le 3791$) statistics from the Scopus data. For reasons explained in the text, we divided the data of these top scorers (c-scores in the range 5.6125 to 3.3461) into six successive equal-sized Groups of 20,000 authors or scientists. We tried to fit, in each Group, $f(h)$, $f(Nc)$ and $f(Np)$ with Gamma distributions, viewing them as the ``wealth distributions'' in the fixed saving-propensity kinetic exchange models and found $f(h) \sim h^{γ_h} \mathrm{exp} (-h/T_h)$ with fitting noise level or temperature level ($T_h$) and average value of $h$, and the power $γ_h$ determined by the ``citation saving propensity'' in each Group. We further showed that using some earlier proposed power law scaling like $h = D_c N_c^{α_c}$ (or $h = D_p N_p^{α_p}$) with $α_c = 1/2 = α_p$, we can derive the observed $f(h)$ from the observed $f(N_c)$ or $f(N_p)$, with $D_c = 0.5$, but $D_p$ depending on the Group considered. This observation suggests that the average citations per paper ($N_c/N_p$) in each group ($= (D_p/D_c)^2 =4D_p^2$) vary (from 58 to 29) with the c-score range of the six Groups considered here, implying different effective Dunbar-like coordination numbers of the scientists belonging to different groups or networks.