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 ($\sigma$), lattice occupation probability ($p$) or Kolkara index ($k$) near the bundle failure threshold ($\sigma_c$) or percolation threshold ($p_c$) or critical value of Kolkata index $k_c$, good fit to Widom-Stauffer like scaling $h/[\sqrt{N}/log N]$ = $f(\sqrt{N}[\sigma_c -\sigma]^\alpha)$, $h/[\sqrt{N}/log N]=f(\sqrt{N}|p_c -p|^\alpha)$ or $h/[\sqrt{N_c}/log N_c]=f(\sqrt{N_c}|k_c -k|^\alpha)$ 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 $\alpha$ 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 $N_m \sim \sqrt N /log N$, resolving a major recent controversy.