in the grand canonical ensemble, with a variety of possible many-body interactions appropriate to the model’s use. Phase transitions, which require an infinite node limit, have been proven for them [@CD; @RY] using the recently developed ‘graphon’ formalism [@Lov] in place of the infinite volume limit formalism [@R1; @R2] used in statistical mechanics. Exponential random graph models are mean field models and therefore the analogues of the various statistical mechanics ensembles (microcanonical, grand canonical, pressure, $\ldots$) which are equivalent in the infinite volume limit for particle systems with short range interactions [@R1], need not be equivalent in these mean field models; see for instance [@TET]. (Equivalence of ensembles is discussed further in the Conclusion.) In this work we use the microcanonical ensemble of one of the best known exponential random graph models, one originally formulated by Strauss [@St], and give evidence of phase transitions which are not as accessible in the grand canonical ensemble. The transitions previously analyzed for a wide class of exponential random graphs are similar to liquid/gas transitions in that they are transitions between graphs of similar character, of the same (fluid-like) phase [@RY], while the transitions we focus on in the microcanonical ensemble are analogous to 