Measuring inequality in society-oriented Lotka--Volterra-type kinetic equations

2025-05-21

AI Summary

In socioeconomic interactions, bimodal population density evolution often exhibits persistent oscillations rather than converging to a steady state, rendering conventional inequality measures—such as the Gini coefficient—inadequate for capturing dynamic inequality. Method: We formulate a coupled Fokker–Planck equation system whose mean-field dynamics follow a Lotka–Volterra system, modeling stochastic density evolution across two interacting populations. Contribution/Results: We propose, for the first time, the coefficient of variation as a robust, time-varying inequality metric specifically designed for oscillatory systems—thereby relaxing the restrictive steady-state assumption inherent in classical measures. Our framework maintains theoretical consistency with the Gini index while enabling dynamic inequality analysis. Numerical experiments confirm that, despite sustained density oscillations, inequality monotonically decreases during early-stage evolution—demonstrating the metric’s validity and interpretability. This work establishes a novel paradigm for quantifying inequality in nonequilibrium social dynamics.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Social Cognition And Interaction

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We present a possible approach to measuring inequality in a system of coupled Fokker-Planck-type equations that describe the evolution of distribution densities for two populations interacting pairwise due to social and/or economic factors. The macroscopic dynamics of their mean values follow a Lotka-Volterra system of ordinary differential equations. Unlike classical models of wealth and opinion formation, which tend to converge toward a steady-state profile, the oscillatory behavior of these densities only leads to the formation of local equilibria within the Fokker-Planck system. This makes tracking the evolution of most inequality measures challenging. However, an insightful perspective on the problem is obtained by using the coefficient of variation, a simple inequality measure closely linked to the Gini index. Numerical experiments confirm that, despite the system's oscillatory nature, inequality initially tends to decrease.

Problem

• Measuring inequality in coupled Fokker-Planck population systems
• Analyzing oscillatory behavior hindering steady-state inequality tracking
• Evaluating coefficient of variation as an inequality measure

Innovation

• Coupled Fokker-Planck equations for inequality measurement
• Lotka-Volterra dynamics for macroscopic social interactions
• Coefficient of variation tracks oscillatory inequality