Random hyperbolic graphs with arbitrary mesoscale structures.

Area of Science:

• Network Science
• Complex Systems
• Graph Theory

Background:

• Real-world networks exhibit universal properties like sparsity, small-worldness, and community structures.
• Geometric network models, such as random hyperbolic graphs (RHGs), capture many network features by embedding nodes in a latent similarity space.
• However, purely geometric models struggle to represent non-geometric community structures where inter-group dissimilarity violates the triangle inequality.

Purpose of the Study:

• To address the limitations of existing geometric network models in capturing mesoscale community structures.
• To introduce a novel network model that incorporates block structures while retaining latent geometry.
• To enhance the ability of network models to represent diverse real-world network topologies.

Main Methods:

• Introduction of the random hyperbolic block model (RHBM), extending RHGs with block structures.
• Utilizing a maximum-entropy framework to incorporate community structures into the model.
• Analysis of synthetic networks to demonstrate the capabilities of RHBM.

Main Results:

• RHBM effectively preserves community structures, outperforming purely geometric models in this aspect.
• The model demonstrates flexibility in generating networks with specific mesoscale mixing patterns.
• Synthetic network analyses validate the advantages of RHBM in capturing non-geometric community features.

Conclusions:

• The random hyperbolic block model (RHBM) offers a significant advancement in modeling complex networks with community structures.
• RHBM highlights the importance of latent geometry while addressing its limitations in controlling mesoscale organization.
• This model provides a more accurate representation of real-world networks characterized by both geometric and block properties.