Soft communities in similarity space

TL;DR

This paper explores the $\ ext{S}^1$ network model's ability to generate networks with soft communities through heterogeneous angular distributions, extending its applicability to real-world network embedding.

Contribution

It introduces a method to incorporate heterogeneous angular distributions into the $\ ext{S}^1$ model, enabling the generation of networks with targeted topological features.

Findings

The model can produce networks with soft community structures.

Hidden degrees depend on angular coordinates in heterogeneous regimes.

The model remains topologically invariant with respect to soft communities.

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.