Scaling behavior of the Hirsch index for failure avalanches, percolation clusters, and paper citations

Abstract

A popular measure for citation inequalities of individual scientists has been the Hirsch index (h). If for any scientist the number nc of citations is plotted against the serial number np of the papers having those many citations (when the papers are ordered from the highest cited to the lowest), then h corresponds to the nearest lower integer value of np below the fixed point of the non-linear citation function (or given by nc = h = np if both np and nc are a dense set of integers near the h value). The same index can be estimated (from h = s = ns) for the avalanche or cluster of size (s) distributions (ns) in the elastic fiber bundle or percolation models. Another such inequality index called the Kolkata index (k) says that (1 − k) fraction of papers attract k fraction of citations (k = 0.80 corresponds to the 80–20 law of Pareto). We find, for stress (σ), the lattice occupation probability (p) or the Kolkata Index (k) near the bundle failure threshold (σc) or percolation threshold (pc) or the critical value of the Kolkata Index kc a good fit to Widom–Stauffer like scaling h/[N/log⁡N] = f(N[σc−σ]α), h/[N/log⁡N]=f(N|pc−p|α) or h/[Nc/logNc]=f(Nc|kc−k|α), respectively, with the asymptotically defined scaling function f, for systems of size N (total number of fibers or lattice sites) or Nc (total number of citations), and α denoting the appropriate scaling exponent. We also show that if the number (Nm) of members of parliaments or national assemblies of different countries (with population N) is identified as their respective h − indexes, then the data fit the scaling relation Nm∼N/log⁡N, resolving a major recent controversy.

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

“…There seem to be considerable mismatches. The same is true when one uses the relation (2) between h and N C with the best fit value (4.5) of the prefactor (as suggested in [2]). Again the distributions of h and those obtained using relation (2) do not match (see the insets of Figs.…”

“…1. Other scaling relations h ∼ N 0.42 c suggested by Radicchi and Castellano [5] in 2013, or h ∼ N 0.50 c /logN c suggested by Ghosh et al [2] in 2022 do not give comparable good fits.…”

“…This is because of the lack of matches (inset of the Figs. 4) with α c = 0.42 [5] or α c = 1/2) with a log correction in relation (1) [2]).…”

“…A popular measure of the success of individual scientist or author (called scientist here generally) has been the Hirsch Index [1] or h-index, which can be viewed as the fixed point [2] of the non-linear function relating the monotonically decreasing number of publications (n p ) with increasing number of citations (n c ): n p = h = n c of the scientist. Mapping the citation function to a combinatorial Fermi one, Yong proposed [3] the relationship…”

“…Several attempts to check the validity of such a relationship between h and N c have been made, see e.g., Redner [4] (supporting the relation (1) with the exponent α c value equal to 0.5, from the data analysis for 255 scientists) and Radicchi and Castellano [5] ( analysing a much larger set of data for 83,897 scientists) who found the best fit value of the exponent α c ≃ 0.42. Ghosh et al [2] studied the Widom-Stauffer like scaling behavior of the Hirsch index for the fiber bundle as well as percolation models away from the "critical" breaking point or stress and percolation point respectively and proposed…”