Hyperbolic Diffusion Models for Hierarchical Network Generation with Controllable Assortativity

Abstract

The generation of complex networks that accurately reflect the topological features of real world systems remains a fundamental challenge in network science and machine learning. A critical limitation of existing graph generation methodologies is their inability to simultaneously preserve hierarchical structural properties and provide fine grained control over degree degree correlations, commonly quantified as assortativity. This paper proposes a novel generative framework termed the hyperbolic diffusion model, which leverages the continuous representations of hyperbolic geometry to intrinsically capture the scale free and hierarchical nature of complex networks. By defining a continuous time diffusion process on a hyperbolic manifold, the proposed model learns the underlying distribution of graph structures through a stochastic differential equation framework. Furthermore, we introduce a conditioning mechanism during the reverse denoising process that allows for explicit control over the assortative mixing of the generated networks, enabling the synthesis of both highly assortative social like networks and disassortative biological or technological networks. Through extensive theoretical analysis and comprehensive synthetic evaluations, we demonstrate that this approach surpasses traditional random graph models and contemporary deep generative models in matching target topological metrics while maintaining structural fidelity. The findings suggest that integrating non Euclidean geometry with diffusion based generative modeling provides a highly flexible and robust paradigm for simulating complex network topologies.