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

TL;DR

This paper investigates the scaling behavior of the Hirsch index and related inequality measures near critical thresholds in various systems, revealing universal scaling laws and resolving debates on parliamentary membership scaling.

Contribution

It introduces a universal scaling framework for the Hirsch index and inequality measures at critical points in failure, percolation, and social systems, unifying diverse phenomena.

Findings

Hirsch index scales with system size near critical points.

Kolkata index follows similar scaling laws at thresholds.

Parliament membership scales as sqrt(N)/log(N), resolving recent controversy.

Abstract

A popular measure for citation inequalities of individual scientists has been the Hirsch index ($h$). If for any scientist the number $n_c$ of citations is plotted against the serial number $n_p$ of the paper having those many citations (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 point of 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 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 Kolkata 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 of 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 fit the scaling relation $N_m \sim \sqrt N /log N$, resolving a major recent controversy.