Random Hyperbolic Graphs with Arbitrary Mesoscale Structures

2025-06-03

AI Summary

Real-world networks exhibit both geometric properties—such as sparsity, small-worldness, power-law degree distribution, and high clustering—and non-geometric mesoscale structures—e.g., arbitrary community mixing patterns. Classical Random Hyperbolic Graphs (RHGs) model only node similarity and popularity, but their strict triangle inequality constraint prevents accurate representation of non-geometric inter-community connections. To address this, we propose the Random Hyperbolic Block Model (RHBM), the first model to explicitly integrate block structure into hyperbolic geometry under a maximum-entropy framework. RHBM decouples intra- and inter-block similarity, enabling violation of the triangle inequality to capture realistic inter-community links. Experiments demonstrate that RHBM preserves RHG’s macroscopic properties while precisely and controllably reproducing target community structures—outperforming RHG significantly on synthetic benchmarks.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

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.

Problem

• Extends RHGs to model non-geometric community structures
• Addresses limitations in controlling mesoscale mixing patterns
• Introduces block structures within a maximum-entropy framework

Innovation

• Extends RHGs with block structures
• Uses maximum-entropy framework
• Preserves non-geometric community structures