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Cliques are frequently used to model communities: a community is a set ofnodes where each pair is equally likely to be connected. But studyingreal-world communities reveals that they have more structure than that. Inparticular, the nodes can be ordered in such a way that (almost) all edges inthe community lie below a hyperbola. In this paper we present three new modelsfor communities that capture this phenomenon. Our models explain the structureof the communities differently, but we also prove that they are identical intheir expressive power. Our models fit to real-world data much better thantraditional block models or previously-proposed hyperbolic models, both ofwhich are a special case of our model. Our models also allow for intuitiveinterpretation of the parameters, enabling us to summarize the shapes of thecommunities in graphs effectively.

Hyperbolae Are No Hyperbole: Modelling Communities That Are Not Cliques