Statistical graph models aim at representing graphs as random realization among a set of possible graphs. To evaluate the quality of a model M with respect to an observed network G, most statistical model selection methods rely on the probability that G was generated by M, which is computed based on the entropy of the associated microcanonical ensemble. In this paper, we introduce another possible definition of the quality of fit of a model based on the edit distance expected value (EDEV). We show that adding a geometric structure to the microcanonical ensemble induces an alternative perspective which may lead to select models which could potentially generate more different graphs, but whose structure is closer to the observed network. Finally we introduce a statistical hypothesis testing methodology based on this distance to evaluate the relevance of a candidate model with respect to an observed graph.