Relations Among Different Inequality Measures in Complex Systems: From Kinetic Exchange to Earthquake Models

TL;DR

This work unifies inequality analysis across socio-economic and geophysical models by computing Lorenz-curve–based indices $g$, $p$, and $k$ for four model classes, including two wealth-exchange systems, a Pareto distribution, and two earthquake models. The central finding is that the composite ratio $p/(2k-1)$ remains near unity across models, with small model-dependent deviations and a near-critical point around $g \approx k \approx 0.86$, a feature linked to SOC-like precursor behavior. The results suggest that inequality indices on Lorenz curves provide a compact, cross-domain diagnostic framework for capturing and comparing heterogeneity in disparate complex systems. The study thus highlights potential universal patterns in the statistics of extreme events, whether in markets or fault dynamics, and offers a quantitative tool for cross-disciplinary analysis.

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, the values of p/(2k-1) (across the wealth exchange models and the two-fractal overlap model) remain a little above unity (theoretically predicted value) and deviating a little higher by a maximum of 4% near g = k nearly equal to 0.86, which was identified earlier to be the precursor point of criticality in several self-organized critical models (k = 0.80 corresponds to Pareto's 80-20 law). The Burridge-Knopoff model shows larger deviations from the k and g relation, but does not access the high inequality regime where an intersection occurs. This and some other quantitatively similar behaviors of the inequality indices across socio-economic and geophysical models may provide a coherent and comparative framework for identifying the subtle features in the statistics of such disparate dynamical systems.

Figures (7)

• Figure 1: Schematic diagram of Lorenz curve and inequality measures where x-axis represents cumulative fraction of population from poor to rich and y-axis depicts the cumulative fraction of their wealth. Gini index $g$ is given by the ratio of the area of yellow region and yellow$+$pink region. Pietra index $p$ is the maximum distance between the equality line and Lorenz curve and Kolkata index is the ordinate point $k$ where Lorenz curve cuts the off diagonal line (perpendicular to the equality line) at the point $(k,L(k))$.
• Figure 2: B-Model with Exchange Range: Results of numerical analysis of the variations of inequality indices Gini ($g$), Pietra ($p$) and Kolkata ($k$) indices with range of exchanges ($R$) for B-Model (with total population, $N=100$ and ensemble averages over $10^3$ realizations). (a) Variation of $g$, $p$ and $k$ with range $R$; (b) $g$ vs $k$-index curve showing initial straight line fitting with slope $3/8$ and the $g=k\approx 0.85$ point, also $g$ vs $p$-index curve; (c) Change of $p/(2k-1)$ with $g$ and $k$; (d) Change of $p/g$ with $g$ and $k$.
• Figure 3: B-Model with Savings: Results of numerical analysis of the variations of inequality indices Gini ($g$), Pietra ($p$) and Kolkata ($k$) indices with saving propensity ($\lambda$) for B-Model (with total population, $N=100$ and ensemble averages over $10^3$ realizations). (a) Variation of $g$, $p$ and $k$ with savings $\lambda$; (b) $g$ vs $k$-index curve showing initial straight line fitting with slope $3/8$ and the $g=k\approx 0.87$ point, also $g$ vs $p$-index curve; (c) Change of $p/(2k-1)$ with $g$ and $k$; (d) Change of $p/g$ with $g$ and $k$.
• Figure 4: C-Model with Savings: Results of numerical analysis of the variations of inequality indices Gini ($g$), Pietra ($p$) and Kolkata ($k$) indices with saving propensity ($\lambda$) for C-Model (with total population, $N=100$ and ensemble averages over $10^3$ realizations). (a) Variation of $g$, $p$ and $k$ with saving propensity $\lambda$; (b) $g$ vs $k$-index curve showing initial straight line fitting with slope $3/8$ and the $g=k\approx 0.86$ point, also $g$ vs $p$-index curve; (c) Change of $p/(2k-1)$ with $g$ and $k$; (d) Change of $p/g$ with $g$ and $k$.
• Figure 5: P-Model: Results of numerical analysis of the variations of inequality indices Gini ($g$), Pietra ($p$) and Kolkata ($k$) indices with ($\alpha$) for P-Model for minimum income=$1$. (a) Variation of $g$, $p$ and $k$ with exponent $\alpha$; (b) $g$ vs $k$-index curve showing initial straight line fitting with slope $3/8$ and the $g=k\approx 0.88$ point, also $g$ vs $p$-index curve; (c) Change of $p/(2k-1)$ with $g$ and $k$; (d) Change of $p/g$ with $g$ and $k$.