For Thine Is the Stretch, the Power, and the Glory: Bibliometrics as a Branch of Relaxation Kinetics

A stretched exponential function taking the form, f(x) = exp(-x^β), x ∈ [0, ∞), β ∈ (0, 1], characterizes decay, diffusion, and relaxation phenomena known as Kohlrausch-Williams-Watts (KWW) processes. Recent work, particularly on relaxation kinetics in metallic glasses, has described the conditions under the shape parameter β deviates from its usual value. Where β > 1, the corresponding exponential function is compressed rather than stretched.

The β-generalized exponential function provides good parametric fits for two measures of influence in legal academia: law review impact factors and Social Science Research Network (SSRN) downloads per author. A stretched exponential function models impact factors from 2007 through 2019. A compressed exponential function describes SSRN downloads per author by law school, except the single outlier atop the rankings. A power law distribution fits the SSRN data, but only for the top 100 schools.

The fact that the value of the exponent β straddles the value of 1, which characterizes an ordinary exponential function, is not a trivial artifact of fitting a model to observed bibliometric data. Treating the reciprocal of β as a rough measure of heterogeneity, h = 1/β, suggests that impact factors and SSRN downloads measure different aspects of academic influence in law. Law reviews, especially as the stigma of online publishing recedes, have become more heterogeneous. By contrast, the compressed rather than stretched exponential kinetics of SSRN data implies the presence of avalanche-like processes in the posting and sharing of preprints among legal scholars.