On the Asymptotics of Constrained Exponential Random Graphs

Abstract. The unconstrained exponential family of random graphs assumes no prior knowl-edge of the graph before sampling, but in many situations partial information of the graph is already known beforehand. A natural question to ask is what would be a typical random graph drawn from an exponential model subject to certain constraints? In particular, will there be a similar phase transition phenomenon as that which occurs in the unconstrained exponential model? We present some general results for the constrained model and then apply them to get concrete answers in the edge-triangle model. 1.