Discretized kinetic theory on scale-free networks

TL;DR

This paper extends a kinetic model of income distribution by incorporating scale-free network structures, analyzing how network connectivity influences economic outcomes through both analytical and numerical methods.

Contribution

It introduces a probabilistic network structure into a kinetic income distribution model and analyzes its mathematical properties and effects on equilibrium distributions.

Findings

Equilibrium distributions depend on network connectivity and initial conditions.

Analytical results confirm existence, normalization, and positivity of solutions.

Numerical simulations align with previous models and highlight the impact of network heterogeneity.

Abstract

The network of interpersonal connections is one of the possible heterogeneous factors which affect the income distribution emerging from micro-to-macro economic models. In this paper we equip our model discussed in [1,2] with a network structure. The model is based on a system of $n$ differential equations of the kinetic discretized-Boltzmann kind. The network structure is incorporated in a probabilistic way, through the introduction of a link density $P(\alpha)$ and of correlation coefficients $P(\beta|\alpha)$, which give the conditioned probability that an individual with $\alpha$ links is connected to one with $\beta$ links. We study the properties of the equations and give analytical results concerning the existence, normalization and positivity of the solutions. For a fixed network with $P(\alpha)=c/\alpha^q$, we investigate numerically the dependence of the detailed and marginal equilibrium distributions on the initial conditions and on the exponent $q$. Our results are compatible with those obtained from the Bouchaud-Mezard model and from agent-based simulations, and provide additional information about the dependence of the individual income on the level of connectivity.