Production and distribution in a network: the system structure predominates theindividual proficiency

A system of producing firms is considered here. The firms own each other through fixed quotas of stocks, so they periodically share the incomes and the losses (i.e., they close the books) A network model is introduced to describe the system. The values of the firms evolve dynamically according to the financial flows (which depend on the topology of the network) and to the distribution of the individual productivities, nevertheless, on the long run, only the system structure matters. In the limit case in which the shares matrix is irreducible (it represents a strongly connected graph), the values of the firms are determined ratios of the total production, which evolves like a Brownian motion. So the values tend to be perfectly correlated. These ratios are determined only by the shares matrix, while the individual proficiencies affect only the total production of the system. When the shares matrix is reducible (the graph is connected, but not strongly), some firms increase their value much more than the other ones.