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Discretized kinetic theory on scale-free networks

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 $P(α)$ and of correlation coefficients $P(β|α)$, which give the conditioned probability that an individual with $α$ links is connected to one with $β$ 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 $P(α)=c/α^q$, 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.