HOG-Diff: Higher-Order Guided Diffusion for Graph Generation

TL;DR

Graph generation must capture complex, non-Euclidean topology, including higher-order interactions. HOG-Diff introduces a coarse-to-fine diffusion framework guided by higher-order skeletons extracted from cell complexes via cell complex filtering, implemented with a generalized Ornstein-Uhlenbeck bridge. The approach yields faster score-matching convergence and tighter reconstruction-error bounds, with strong empirical results on molecular and generic graphs. This work demonstrates the value of explicit higher-order topology in diffusion-based graph generation and offers interpretable topological guidance for generation.

Abstract

Graph generation is a critical yet challenging task as empirical analyses require a deep understanding of complex, non-Euclidean structures. Diffusion models have recently made significant achievements in graph generation, but these models are typically adapted from image generation frameworks and overlook inherent higher-order topology, leaving them ill-suited for capturing the topological properties of graphs. In this work, we propose Higher-order Guided Diffusion (HOG-Diff), a principled framework that progressively generates plausible graphs with inherent topological structures. HOG-Diff follows a coarse-to-fine generation curriculum guided by higher-order topology and implemented via diffusion bridges. We further prove that our model exhibits a stronger theoretical guarantee than classical diffusion frameworks. Extensive experiments on both molecular and generic graph generation tasks demonstrate that our method consistently outperforms or remains competitive with state-of-the-art baselines. Our code is available at https://github.com/Yiminghh/HOG-Diff.

Figures (16)

• Figure 1: Overview of HOG-Diff. The dashed line above illustrates the classical generation process, where graphs quickly degrade into random structures with uniformly distributed entries. In contrast, as shown in the coloured region below, HOG-Diff adopts a coarse-to-fine generation curriculum based on the diffusion bridge, explicitly learning higher-order structures during intermediate steps with theoretical guarantees.
• Figure 2: Cell Complex transformations. ( a) An example graph. ( b) Lifting: closed 2D disks are glued to the boundary of the rings to form the 2-cell complex. ( c) The resulting cell complex and the corresponding homeomorphisms to the closed balls for three representative cells of different dimensions in the complex. ( d) Black elements represent higher-order structures extracted through 2-cell filtering, while grey elements denote corresponding peripheral structures pruned by the filtering operation.
• Figure 3: Visualization of molecular graphs at different stages of the reverse generative process. Model trained on the ZINC250k dataset.
• Figure 4: Training curves of the score-matching process. The entire process of HOG-Diff is divided into two stages, i.e., $K=2$, referred to as coarse and fine, respectively. The combined loss of these two stages is labelled as Coarse+Fine.
• Figure 5: Visual illustration of cell complexes. (a) Triangle. (b) Tetrahedron. (c) Sphere. (d) Torus.

Theorems & Definitions (9)

• Definition 1: Regular cell complex
• Proposition 1: Cell complex filtering
• Theorem 1: Informal
• Theorem 2
• proof
• Definition 2: $\beta$-smooth
• proof
• proof
• Definition 3: Simplicial complexes