Principia Bibliometrica: Modeling Citation and Download Data in Legal Scholarship

Abstract

A stretched exponential function taking the form f(x) = e^(〖-x〗^β ); x∈ [0,∞), β∈(0,1] characterizes decay, diffusion, and relaxation phenomena known as Kohlrausch-Williams-Watts processes. Recent work 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 (β ≈ 0.805823) models impact factors from 2007 through 2019. The shape parameter for impact factors has changed dramatically relative to citation data for 2006 to 2013, when β was approximately 1.027133. A compressed exponential function (β ≈ 1.219476) 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. This result is consistent with the observation that power laws rarely model the entirety of a distribution, but only its tail.

Impact factors and SSRN downloads evidently measure different aspects of academic influence. 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 dissemination of preprints among legal scholars.

Mathematical relationships connecting the β-generalized exponential function, the Weibull distribution, and the Gini coefficient enable academic inequality in citation and download rates to be expressed as a very simple function of β. The Gini coefficient can be defined as G=1-2^(-1/β). From 2006–2013 to 2007–2019, the Gini coefficient of inequality in law review impact factors rose dramatically, from G ≈ 0.490760 to G ≈ 0.576911. Inequality in SSRN downloads is lower: G ≈ 0.433568. These figures straddle the Gini coefficient for inequality in American household income in 2021, which was 0.494.

Keywords: Bibliometrics, Power Law, Exponential Function, Stretched Exponential Function, Kohlrausch-Williams-Watts, Impact Factor, Ssrn