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.