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), and a synthetic Pareto distribution. 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). In the Burridge-Knopoff model for some instances of time, the value of p/(2k-1) drops a little below unity. 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.
Summary
This paper investigates the relationships between different inequality measures—Gini index (g), Pietra index (p), and Kolkata index (k)—across a variety of complex systems. These systems include two kinetic wealth-exchange models (Banerjee model and Chakraborti/Yard-Sale model), a synthetic Pareto distribution, and two earthquake-simulating models (Chakrabarti-Stinchcombe two-fractal overlap model and the nonlinear Burridge-Knopoff model). The authors numerically compute the Lorenz function for each model and then estimate the variations of the inequality indices as they systematically vary parameters such as saving propensity (wealth-exchange models) or generation/block numbers (earthquake models). The key finding is that the ratio p/(2k-1) remains close to unity across the wealth-exchange models and the two-fractal overlap model, deviating by a maximum of 4% near g = k ≈ 0.86, which has been previously identified as a precursor to criticality in self-organized critical models. The Burridge-Knopoff model shows instances where this ratio drops below unity. The authors suggest that these quantitatively similar behaviors of inequality indices in disparate systems (socio-economic and geophysical) may provide a coherent framework for identifying subtle statistical features across these diverse dynamical systems. This is significant because it provides a potentially unifying perspective on inequality across seemingly unrelated domains.
Key Insights
- •The ratio p/(2k-1) remains close to unity across several models, with a maximum deviation of 4% near g = k ≈ 0.86. This supports a potential universality in inequality measures across different complex systems.
- •The point g = k ≈ 0.86, where the Gini and Kolkata indices are approximately equal, is identified as a potential precursor to criticality, aligning with previous findings in self-organized critical models.
- •In the Burridge-Knopoff model, the ratio p/(2k-1) can drop below unity, suggesting that the relationship between inequality measures in dynamical earthquake models may differ from those in wealth-exchange or fractal overlap models.
- •For the Pareto distribution, the ratio p/(2k-1) shows a higher maximum deviation above unity (7%) compared to the wealth exchange models.
- •The relationship k = 1/2 + (3/8)g appears to hold for lower g values across all the models considered, suggesting a consistent connection between Gini and Kolkata indices in the regime of lower inequality.
- •The study focuses primarily on numerical simulations and lacks rigorous theoretical derivations or proofs.
- •The authors assume that the steady-state distributions reached in the simulations are representative of the underlying dynamics and that finite-size effects are minimal after ensemble averaging.
Practical Implications
- •The research provides a framework for comparing inequality across different domains (socio-economic and geophysical), which could be useful for identifying common underlying mechanisms or patterns.
- •Researchers studying complex systems in economics, physics, and geophysics can use these findings to compare and contrast inequality in their respective fields.
- •The identification of g = k ≈ 0.86 as a potential precursor to criticality could be relevant for developing early warning systems for critical transitions in various systems.
- •Further research could focus on developing theoretical explanations for the observed relationships between the inequality measures and exploring the implications of deviations from these relationships in different systems.
- •Future studies can explore the impact of different model parameters, such as the friction laws in the Burridge-Knopoff model, on the inequality indices and their relationships.