  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.
