A set of master variables for the two-star random graph

The two-star random graph is the simplest exponential random graph model with nontrivial interactions between the graph edges. We propose a set of auxiliary variables that control the thermodynamic limit where the number of vertices N tends to infinity. Such `master variables' are usually highly desirable in treatments of `large N' statistical field theory problems. For the dense regime when a finite fraction of all possible edges are filled, this construction recovers the mean-field solution of Park and Newman, but with an explicit control over the 1/N corrections. We use this advantage to compute the first subleading correction to the Park-Newman result, which encodes the finite, nonextensive contribution to the free energy. For the sparse regime with a finite mean degree, we obtain a very compact derivation of the Annibale-Courtney solution, originally developed with the use of functional integrals, which is comfortably bypassed in our treatment.