  In the edge-2star model with hard constraints we prove the existence of an
open set of constraint parameters, bisected by a line segment on which there
are nonunique entropy-optimal graphons related by a symmetry. At each point in
the open set but off the line segment there is a unique entropy-optimizer,
bipodal and varying analytically with the constraints. We also show that
throughout another open set, containing a different portion of the same line of
symmetry, there is instead a unique optimal graphon, varying analytically with
the parameters. We explore the extent of these open sets, determining the point
at which a symmetric graphon ceases to be a local maximizer of the entropy.
Finally, we prove some foundational theorems in a general setting, relating
optimal graphons to the Boltzmann entropy and the generic structure of large
constrained random graphs.
