The GHS and other correlation inequalities for the two-star model

TL;DR

This paper investigates correlation inequalities in the two-star exponential random graph model, deriving new inequalities and properties of edge density as a function of parameters using tools from statistical mechanics.

Contribution

It introduces new correlation inequalities, including the GHS inequality, for the two-star model, and analyzes the monotonicity and concavity of average edge density.

Findings

Derived first and second order correlation inequalities.

Proved the GHS inequality for the model.

Showed edge density is increasing and concave in parameter h.

Abstract

We consider the two-star model, a family of exponential random graphs indexed by two real parameters, $h$ and $\alpha$, 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, if $\alpha, h\ge 0$, we derive first and second order correlation inequalities and then prove the so-called GHS inequality. As a consequence, under the above conditions on the parameters, the average edge density turns out to be an increasing and concave function of the parameter $h$, at any fixed size of the graph. Some of our results can be extended to more general classes of exponential random graphs.