Discrete exponential families are now widely used to model social and other networks with heterogeneity and/or dependence among edge variables.  Models for graphs written in this way are called exponential family random graph models, or ERGMs.  Although the ERGM framework is extremely flexible, few techniques other than Markov chain Monte Carlo have historically been available for studying ERGM behavior.  By contrast, random graphs with independent edge variables (i.e., the Bernoulli graphs) are the subject of a large literature, and much is known regarding their properties.  In this talk, I describe a method for exploiting this knowledge by constructing families of Bernoulli graphs that bound the behavior of an arbitrary ERGM in a well-defined sense.  By determining the properties of these Bernoulli graphs (either analytically or via simulation), one can thus constrain the properties of the associated ERGM.  I show how this technique can be used to recapitulate the well-known ``density explosion'' of the edge-triangle model, and also demonstrate the application of this technique to the problem of checking the robustness of a spatial network model to an omitted source of edgewise dependence.