Networks are commonly used to represent key processes in biology; examples include transcriptional regulatory networks, protein-protein interaction (PPI) networks, metabolic networks, etc. Databases store many such networks, as graphs, observed or inferred. Generative models for these networks have been proposed. For PPI networks, current models are based on duplication and divergence (D&D): a node (gene) is duplicated and inherits some subset of the connections of the original node. An early finding about biological networks is modularity: a higher-level structure is prevalent consisting of well connected subgraphs with less substantial connectivity to other such subgraphs. While D&D models spontaneously generate modular structures, neither have these structures been compared with those in the databases nor are D&D models known to maintain and evolve them. Given that the preferred generative models being based on D&D, the network inference models are also based on the same principle. We describe NEMo (Network Evolution with Modularity), a new model that embodies modularity. It consists of two layers: the lower layer is a derivation of the D&D process thus node-and-edge based, while the upper layer is module-aware. NEMo allows modules to appear and disappear, to fission and to merge, all driven by the underlying edge-level events using a duplication-based process. We also introduce measures to compare biological networks in terms of their modular structure. We present an extensive study of six model organisms across six public databases aimed at uncovering commonalities in network structure. We then use these commonalities as reference against which to compare the networks generated by D&D models and by our module-aware model NEMo. We find that, by restricting our data to high-confidence interactions, a number of shared structural features can be identified among the six species and six databases. When comparing these characteristics with those extracted from the networks produced by D&D models and our NEMo model, we further find that the networks generated by NEMo exhibit structural characteristics much closer to those of the PPI networks of the model organisms. We conclude that modularity in PPI networks takes a particular form, one that is better approximated by the module-aware NEMo model than by other current models. Finally, we draft the ideas for a module-aware network inference model that uses an altered form of our module-aware NEMo as the core component, from a parsimony perspective.