Soft communities in similarity space

Abstract

The $\mathbb{S}^1$ model has been a central geometric model in the development of the field of network geometry. It 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, heterogeneous angular distributions. Under these circumstances, 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 might have important implications, both in expanding the scope of the model beyond the independent hidden variables limit and in the embedding of real-world networks.