  We consider a family of directed exponential random graph models parametrized
by edges and outward stars. Much of the important statistical content of such
models is given by the normalization constant of the models, and in particular,
an appropriately scaled limit of the normalization, which is called the free
energy. We derive precise asymptotics for the normalization constant for finite
graphs. We use this to derive a formula for the free energy. The limit is
analytic everywhere except along a curve corresponding to a first order phase
transition. We examine unusual behavior of the model along the phase transition
curve.
