Soft Communities in Similarity Space

Abstract

The S1 model has been central in the development of the field of network geometry. It places nodes in a similarity space and connects them with a likelihood depending on an effective distance which combines similarity and popularity dimensions, with popularity directly related to the degrees of the nodes. The S1 model has been mainly studied in its homogeneous regime, in which angular coordinates are independently and uniformly scattered on the circle. We now investigate if the model can generate networks with targeted topological features and soft communities, that is, inhomogeneous angular distributions. To that end, hidden degrees must depend on angular coordinates, and we propose a method to estimate them. We conclude that the model can be topologically invariant with respect to the soft-community structure. Our results expand the scope of the model beyond the independent hidden variables limit and can have an important impact in the embedding of real-world networks.