must do to obtain equivalence of ensembles, one can divide the entropy’s variables by volume, and consider the entropy density as a function of particle and energy density).

The equivalence of ensembles in statistical mechanics is basically a strenghening of the equivalence in thermodynamics between entropy $s_{e,t}$ and free energy $\psi_{\beta_1,\beta_2}$, corresponding to Lagrange multipliers $\beta_1$ and $\beta_2$, which follows from the concavity of the entropy. With the modeling of the thermodynamic states this now implies a bijection $(e,t)\longleftrightarrow
(\beta_1,\beta_2)$ such that $s_{e,t}$ and $\psi_{\beta_1,\beta_2}$ have the same optimizing states, at least off some manageable sets of parameter pairs corresponding to ‘phase coexistence’ where the bijection can degenerate to a many-to-one map [@Ge].

In exponential random graph models, which are mean field rather than short range, the entropy need not be concave [@TET] and indeed this fails in an obvious way for the specific model we are analyzing, the Strauss model, since even the domain $R$ of the entropy is not convex (see Figure 1). Therefore in the infinite node limit of the model the free energy density $\psi_{\beta_1,\beta_2}$, need not be equivalent to the entropy density $s_{e,t}$; $\psi_{\beta_1,\beta_2}$ can be obtained from $s_{e,t}$ by Legendre transform, but it may 