  --- the companies populating a Stock market, along with their connections,
can be effectively modeled through a directed network, where the nodes
represent the companies, and the links indicate the ownership. This paper deals
with this theme and discusses the concentration of a market. A
cross-shareholding matrix is considered, along with two key factors: the node
out-degree distribution which represents the diversification of investments in
terms of the number of involved companies, and the node in-degree distribution
which reports the integration of a company due to the sales of its own shares
to other companies. While diversification is widely explored in the literature,
integration is most present in literature on contagions. This paper captures
such quantities of interest in the two frameworks and studies the stochastic
dependence of diversification and integration through a copula approach. We
adopt entropies as measures for assessing the concentration in the market. The
main question is to assess the dependence structure leading to a better
description of the data or to market polarization (minimal entropy) or market
fairness (maximal entropy). In so doing, we derive information on the way in
which the in- and out-degrees should be connected in order to shape the market.
The question is of interest to regulators bodies, as witnessed by specific
alert threshold published on the US mergers guidelines for limiting the
possibility of acquisitions and the prevalence of a single company on the
market. Indeed, all countries and the EU have also rules or guidelines in order
to limit concentrations, in a country or across borders, respectively. The
calibration of copulas and model parameters on the basis of real data serves as
an illustrative application of the theoretical proposal.
