Scaling behavior of the Hirsch index for failure avalanches, percolation clusters, and paper citations - Ghosh, Asim; Chakrabarti, Bikas K.; Ram, Dachepalli R. S.; Mitra, Manipushpak; Maiti, Raju; Biswas, Soumyajyoti; Banerjee, Suchismita (Frontiers in Physics, 2022)
A popular measure for citation inequalities of individual scientists has been the Hirsch index (h). If the number n_c of citations are plotted against the number n_p of papers (each having those many citations), then h corresponds to the fixed point (where n_c = h = n_p) of the above-mentioned (non-linear) citation function of the scientist. The same index can be estimated (from h=s=n_s) for the avalanche or cluster of size (s) distributions (n_s) in 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), lattice occupation probability (p) or Kolkara index (k) near the bundle failure threshold (σ_c) or percolation threshold (p_c) or critical value of Kolkata index k_c, good fit to Widom-Stauffer like scaling h/[N/log N] = f(N[σ_c -σ]^α), h/[N/log N]=f(N|p_c -p|^α) or h/[N_c/log N_c]=f(N_c|k_c -k|^α) respectively, with asymptotically defined scaling function f, for systems of size N (total number of fibers or lattice sites) or N_c (total number citations), and α denoting the appropriate scaling exponent. We also show that if the number (N_m) of members of parliaments or national assemblies of different countries (with population N) is identified as their respective h-index, then the data fits the scaling relation N_m N /log N, resolving a major recent controversy.