A recent body of work, by myself and many others, aims to develop a statistical theory of network data for problems a single network is observed. Of the models studied in this area, graphon models are probably most widely known in statistics. I will explain the relationship between three aspects of this work: (1) Specific models, such as graphon models, graphex models, and edge-exchangeable graphs. (2) Sampling theory for networks, specifically in the case statisticians might refer to as an infinite-population limit. (3) Invariance properties, especially various forms of exchangeability. I will also present recent results that show how statistically relevant results (such as central limit theorems) can be derived from such invariance properties.