Relations among different inequality measures in complex systems: From kinetic exchange to earthquake models

Abstract

We present a numerical study of several inequality measures across two kinetic wealth-exchange models with extreme inequality features (namely the Banerjee model, and the Chakraborti or yard-sale model) and two earthquake simulating models (namely the Chakrabarti-Stinchcombe two-fractal overlap model and the nonlinear dynamical Burridge-Knopoff model). For each model we compute numerically the Lorenz function for the respective models' wealth, overlap magnitude or avalanche distributions. We then estimate the variations of Gini (𝑔), Pietra (𝑝), and Kolkata (𝑘) indices in these models with systematic variations of saving propensity (for the two wealth-exchange models), with systematic variations of generation or block numbers (for the two earthquake simulating models). We find that for appropriate values of the respective model parameters, the inequality indices 𝑔 and 𝑘 in corresponding the distributions (of “wealth” or “avalanche”) show quantitatively similar behavior, namely 𝑔=𝑘≃0.86, which was identified earlier to correspond to the precursor point of criticality in self-organized critical models (𝑘=0.80 corresponds to that for Pareto's 80-20 law). The values of 𝑝/(2⁢𝑘−1) in all these (wealth exchange and earthquake) models remain a little above unity, as was predicted theoretically. These observations for the inequality indices 𝑔, 𝑘, and 𝑝 across the socioeconomic and geophysical models indicate the presence of unifying subtle features in the statistics of such disparate dynamical systems.