  In this work, we develop the Tsallis entropy approach for examining the
cross-shareholding network of companies traded on the Italian stock market. In
such a network, the nodes represent the companies, and the links represent the
ownership. Within this context, we introduce the out-degree of the nodes --
which represents the diversification -- and the in-degree of them -- capturing
the integration. Diversification and integration allow a clear description of
the industrial structure formed by the considered companies. The stochastic
dependence of diversification and integration is modelled through copulas. We
argue that copulas are well suited for modelling the joint distribution. The
analysis of the stochastic dependence between integration and diversification
by means of the Tsallis entropy gives a crucial information on the reaction of
the market structure to the external shocks, - on the basis of some relevant
cases of dependence between the considered variables. In this respect, the
considered entropy framework provides insights on the relationship between
in-degree and out-degree dependence structure and market polarisation or
fairness. Moreover, the interpretation of the results in the light of the
Tsallis entropy parameter gives relevant suggestions for policymakers who aim
at shaping the industrial context for having high polarisation or fair joint
distribution of diversification and integration. Furthermore, a discussion of
possible parametrisations of the in-degree and out-degree marginal
distribution, -- by means of power laws or exponential functions, -- is also
carried out. An empirical experiment on a large dataset of Italian companies
validates the theoretical framework.
