Random Hyperbolic Graphs with Arbitrary Mesoscale Structures

TL;DR

This paper introduces the Random Hyperbolic Block Model (RHBM), an extension of Random Hyperbolic Graphs that incorporates community structures within a maximum-entropy framework, enabling better modeling of mesoscale network features.

Contribution

The paper presents RHBM, a novel model that combines geometric and block structures to improve mesoscale community modeling in networks.

Findings

RHBM effectively preserves community structures in synthetic networks.

RHBM overcomes limitations of purely geometric models in mesoscale pattern control.

The model highlights the importance of latent geometry in network structure representation.

Abstract

Real-world networks exhibit universal structural properties such as sparsity, small-worldness, heterogeneous degree distributions, high clustering, and community structures. Geometric network models, particularly Random Hyperbolic Graphs (RHGs), effectively capture many of these features by embedding nodes in a latent similarity space. However, networks are often characterized by specific connectivity patterns between groups of nodes -- i.e. communities -- that are not geometric, in the sense that the dissimilarity between groups do not obey the triangle inequality. Structuring connections only based on the interplay of similarity and popularity thus poses fundamental limitations on the mesoscale structure of the networks that RHGs can generate. To address this limitation, we introduce the Random Hyperbolic Block Model (RHBM), which extends RHGs by incorporating block structures within a maximum-entropy framework. We demonstrate the advantages of the RHBM through synthetic network analyses, highlighting its ability to preserve community structures where purely geometric models fail. Our findings emphasize the importance of latent geometry in network modeling while addressing its limitations in controlling mesoscale mixing patterns.