Scaling behavior of the Hirsch index for failure avalanches, percolation clusters, and paper citations

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 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 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 a 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 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 (σ c ) or percolation threshold (p c ) or the critical value of the Kolkata Index k c a 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 the 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 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 − indexes, then the data fit the scaling relation N m ∼ N / log ⁡ N , resolving a major recent controversy.