Cliques (or quasi-cliques) are frequently used to model communities: a set of nodes where each pair is (equally) likely to be connected. However, when observing real-world communities, we see that most communities have more structure than that. In particular, the nodes can be ordered in such a way that (almost) all edges in the community lie below a hyperbola. In this paper we present three new models for communities that capture this phenomenon. Our models explain the structure of the communities differently, but we also prove that they are identical in their expressive power. Our models fit to real-world data much better than traditional block models, and allow for more in-depth understanding of the structure of the data
