Statistical models of network structure are widely used in network science to reason about the properties of complex systems-where the nodes and edges represent entities and their relationships. Recently, a number of generative network models (GNM) have been developed that accurately capture characteristics of real world networks, but since they are typically defined in a procedural manner, it is difficult to identify commonalities in their structure. Moreover, procedural definitions make it difficult to develop statistical sampling algorithms that are both efficient and correct. In this paper, we identify a family of GNMs that share a common latent structure and create a Bayesian network (BN) representation that captures their common form. We show how to reduce two existing GNMs to this representation. Then, using the BN representation we develop a generalized, efficient, and provably correct, sampling method that exploits parametric symmetries and deterministic context-specific dependence. Finally, we use the new representation to design a novel GNM and evaluate it empirically.