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Hierarchical Discrete Flow Matching for Graph Generation

Yoann Boget, Pablo Strasser, Alexandros Kalousis

Abstract

Denoising-based models, including diffusion and flow matching, have led to substantial advances in graph generation. Despite this progress, such models remain constrained by two fundamental limitations: a computational cost that scales quadratically with the number of nodes and a large number of function evaluations required during generation. In this work, we introduce a novel hierarchical generative framework that reduces the number of node pairs that must be evaluated and adopts discrete flow matching to significantly decrease the number of denoising iterations. We empirically demonstrate that our approach more effectively captures graph distributions while substantially reducing generation time.

Hierarchical Discrete Flow Matching for Graph Generation

Abstract

Denoising-based models, including diffusion and flow matching, have led to substantial advances in graph generation. Despite this progress, such models remain constrained by two fundamental limitations: a computational cost that scales quadratically with the number of nodes and a large number of function evaluations required during generation. In this work, we introduce a novel hierarchical generative framework that reduces the number of node pairs that must be evaluated and adopts discrete flow matching to significantly decrease the number of denoising iterations. We empirically demonstrate that our approach more effectively captures graph distributions while substantially reducing generation time.

Paper Structure

This paper contains 68 sections, 3 theorems, 34 equations, 4 figures, 13 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Hierarchical Graph Generation
  4. Graph Coarsening
  5. Graph Expansion
  6. Spanning Supergraph Density Minimization
  7. Hierarchical Graph Representations
  8. Properties
  9. D-Min Clustering
  10. Empirical Evaluation
  11. Hierarchical Discrete Flow Matching
  12. Discrete Flow Matching
  13. Conditional Discrete Flow Matching for Graphs
  14. Sampling and Generation
  15. Experiments
  16. ...and 53 more sections

Key Result

Proposition 3.1

The expanded graph ${\mathcal{H}}^\ell$ is a spanning supergraph of ${\mathcal{G}}^\ell_\text{unattr.}$, that is: Proofs are provided in Appendix ap:proofs. □ $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Top: Deterministic Graph Expansion. Each parent node spawns a fixed number of child nodes. Children of the same parent form cliques, while children of connected parents form bicliques. Bottom: Graph Refinement. A generative model prunes excess edges and optionally generates node and edge attributes.
  • Figure 2: SBM20k: Comparison of generated graphs with graphs from the dataset.
  • Figure 3: ego: Comparison of generated graphs with graphs from the dataset.
  • Figure 4: reddit: Comparison of generated graphs with graphs from the dataset.

Theorems & Definitions (3)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 1.1