The GHS and other inequalities for the two-star model

We consider the two-star model, a family of exponential random graphs indexed by two real parameters, h and α, that rule respectively the total number of edges and the mutual dependence between them. Borrowing tools from statistical mechanics, we study different classes of correlation inequalities for edges, that naturally emerge while taking the partial derivatives of the (finite size) free energy. In particular, under a mild hypothesis on the parameters, we derive first and second order correlation inequalities and then prove the so-called GHS inequality. As a consequence, the average edge density turns out to be an increasing and concave function of the parameter h, at any fixed size of the graph.