Hyperbolae Are No Hyperbole: Modelling Communities That Are Not Cliques

TL;DR

This paper introduces three novel hyperbolic models for community structures in graphs that better capture real-world community patterns than traditional clique-based models, with improved data fit and interpretability.

Contribution

It presents three new hyperbolic community models that are theoretically equivalent in expressive power and outperform existing models in fitting real-world data.

Findings

Models fit real-world data better than traditional block models.

Hyperbolic models provide intuitive parameter interpretation.

Models are theoretically equivalent in expressive power.

Abstract

Cliques are frequently used to model communities: a community is a set of nodes where each pair is equally likely to be connected. But studying real-world communities reveals that they 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 or previously-proposed hyperbolic models, both of which are a special case of our model. Our models also allow for intuitive interpretation of the parameters, enabling us to summarize the shapes of the communities in graphs effectively.