Graph Generation with Spectral Geodesic Flow Matching

2025-10-02

AI Summary

Existing graph generation methods primarily focus on matching spectral properties or degree distributions, neglecting the geometric structure induced by eigenvectors and the graph’s global topology. To address this, we propose a spectral-geometrically guided geodesic flow matching framework: graphs are first embedded into a Riemannian manifold via spectral feature mapping; subsequently, probability distributions are matched along geodesic flows to jointly model both intrinsic geometry and global topology. This work is the first to deeply integrate spectral-geometric priors with the flow matching paradigm, achieving superior generation diversity and cross-scale generalization while accelerating inference 30× over diffusion-based approaches. Experiments demonstrate state-of-the-art performance across key metrics—including substructure statistics, degree distribution, and spectral characteristics—alongside significantly improved training efficiency and scalability.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Diffusion Models for Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Web data generation and simulation

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$ imes$ 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.

Problem

• Captures geometric structure beyond eigenvalues in graph generation
• Supports flexible generation of diverse graph structures efficiently
• Enables generalization to unseen graph scales through spectral embeddings

Innovation

• Embeds graphs into Riemannian manifolds using spectral eigenmaps
• Defines geodesic flows between graph embeddings for generation
• Matches distributions along flows to capture geometric structure