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A popular measure for citation inequalities of individual scientists has beenthe Hirsch index ($h$). If for any scientist the number $n_c$ of citations isplotted against the serial number $n_p$ of the paper having those manycitations (when the papers are ordered from highest cited to lowest) then $h$corresponds to the nearest lower integer value of $n_p$ below the fixed pointof the non-linear citation function (or given by $n_c = h = n_p$ if both $n_p$and $n_c$ are dense set of integers near the $h$ value). The same index can beestimated (from $h=s=n_{s}$) for the avalanche or cluster of size ($s$)distributions ($n_s$) in elastic fiber bundle or percolation models. Anothersuch inequality index, called the Kolkata index ($k$) says that $(1-k)$fraction of papers attract $k$ fraction of citations ($k=0.80$ corresponds tothe 80-20 law of Pareto). We find, for stress ($\sigma$), lattice occupationprobability ($p$) or Kolkata index ($k$) near the bundle failure threshold($\sigma_c$) or percolation threshold ($p_c$) or critical value of Kolkataindex $k_c$, 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 ofsize $N$ (total number of fibers or lattice sites) or $N_c$ (total number ofcitations), and $\alpha$ denoting the appropriate scaling exponent. We alsoshow that if the number ($N_m$) of members of parliaments or nationalassemblies of different countries (with population $N$) is identified as theirrespective $h-$index, then the data fit the scaling relation $N_m \sim \sqrt N/log N$, resolving a major recent controversy.

Scaling Behavior of the Hirsch Index for Failure Avalanches, Percolation Clusters and Paper Citations