Graph Generation with Spectral Geodesic Flow Matching

TL;DR

SFMG introduces a geometry-aware graph generator by combining spectral decomposition with geodesic flow matching on manifolds. It models eigenvalues in Euclidean space and eigenvectors on the Stiefel manifold, using three conditional flow-matching models to generate eigenvalues, eigenvectors, and the final adjacency (plus node features) from spectral information. Empirically, SFMG achieves state-of-the-art or competitive results across graph benchmarks and molecule generation, delivering up to 30× faster generation than diffusion-based methods while maintaining high fidelity in spectral and structural metrics. The approach highlights the value of integrating spectral geometry with manifold-aware generative modeling for scalable, structure-preserving graph synthesis.

Abstract

Graph generation is a fundamental task with wide applications in modeling complex systems. Although existing methods align the spectrum or degree profile of the target graph, they often ignore the geometry induced by eigenvectors and the global structure of the graph. In this work, we propose Spectral Geodesic Flow Matching (SFMG), a novel framework that uses spectral eigenmaps to embed both input and target graphs into continuous Riemannian manifolds. We then define geodesic flows between embeddings and match distributions along these flows to generate output graphs. Our method yields several advantages: (i) captures geometric structure beyond eigenvalues, (ii) supports flexible generation of diverse graphs, and (iii) scales efficiently. Empirically, SFMG matches the performance of state-of-the-art approaches on graphlet, degree, and spectral metrics across diverse benchmarks. In particular, it achieves up to 30$\times$ speedup over diffusion-based models, offering a substantial advantage in scalability and training efficiency. We also demonstrate its ability to generalize to unseen graph scales. Overall, SFMG provides a new approach to graph synthesis by integrating spectral geometry with flow matching.

Figures (9)

• Figure 1: Illustration of SFMG. SFMG adopts flow matching in $R^{k}$ and geodesic flow matching in $\mathbb{V}_k(n)$ to the first $k$ eigenvalues and eigenvectors. The last flow matching is used to recover the adjacency matrix from the generated Laplacian matrix and generate the feature matrix.
• Figure 2: Compared to other approaches, SFMG produces graphs with clearer structures and topologies that most closely resemble the ground truth.
• Figure 3: Randomly selected Ego-Small graphs from the training and generated ones by SFMG.
• Figure 4: Randomly selected Community-Small graphs from the training and generated ones by SFMG.
• Figure 5: Randomly selected Planar graphs from the training and generated ones by SFMG.