  In this paper, we study exponential random graph models subject to certain
constraints. We obtain some general results about the asymptotic structure of
the model. We show that there exists non-trivial regions in the phase plane
where the asymptotic structure is uniform and there also exists non-trivial
regions in the phase plane where the asymptotic structure is non-uniform. We
will get more refined results for the star model and in particular the two-star
model for which a sharp transition from uniform to non-uniform structure, a
stationary point and phase transitions will be obtained.
