  The unconstrained exponential family of random graphs assumes no prior
knowledge of the graph before sampling, but it is natural to consider
situations where partial information about the graph is known, for example the
total number of edges. What does a typical random graph look like, if drawn
from an exponential model subject to such constraints? Will there be a similar
phase transition phenomenon (as one varies the parameters) as that which occurs
in the unconstrained exponential model? We present some general results for
this constrained model and then apply them to get concrete answers in the
edge-triangle model with fixed density of edges.
