Discretized kinetic theory on a network as a tool for the study of economic interactions
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The network of interpersonal connections is one of the main factors which affect the income distribution emerging from micro-to-macro models. In particular, the assortative or disassortative nature of the network could play a role. The insertion of a network structure into the evolution equations of kinetic theory is not trivial and turns out to impose constraints on the network itself. Our model (as discussed in Bertotti and Modanese, 2012) is based on a system of differential equations of the kinetic discretized-Boltzmann kind. Society is described as an ensemble of individuals divided into a finite number of income classes; the individuals exchange money through binary interactions, leaving the total wealth unchanged. The interactions occur with a certain predefined frequency, and several other parameters can also be adjusted. For instance, we can fix the probability that in an encounter between two individuals the one who pays is the rich or the poor; we can make the exchanged amount depend on the income classes (variable saving propensity), etc. After a sufficiently long time the solution of the equations reaches an equilibrium state characterized by an income distribution, which depends on the total income and on the interaction parameters, but not on the initial distribution. Akinetic model can be equipped with a network structure in a probabilistic way, through the introduction of correlation coefficients P(β|α), which give the conditioned probability that an individual withαlinks is connected to one withβlinks. An approach of this kind has been proposed by (Boguna et al. 2003), who have described the diffusion of epidemics through differential equations giving the probability for each individual to be infected and infect in turn others. It is well known that in this case the network structure is crucial, and that the hubs have a fundamental role in the transmission of the disease. Also in our model a network can be introduced in a similar way. However, some important differences naturally arise in the structure of the equations and in the conditions satisfied by the correlation coefficients. The reason is that in our model describing economic exchanges, money is conserved, while in a contagion process, unfortunately, the disease can multiply for free. We analyze these issues in general form and give results concerning some particular numerical solutions for the asymptotic distributions xαi(t), where t→ ∞ and xαi(t) denotes the density of individuals belonging to the i-th in-come class and having α economic links with other individuals.