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Variational Bayesian Flow Network for Graph Generation

Yida Xiong, Jiameng Chen, Xiuwen Gong, Jia Wu, Shirui Pan, Wenbin Hu

TL;DR

This work addresses graph generation by integrating node–edge coupling into the generative process. It introduces Variational Bayesian Flow Networks (VBFN), which lift classical factorized BFN beliefs to a joint Gaussian with a structured precision operator, enabling coupled Bayesian updates that inherently respect graph geometry. The method derives a closed-form posterior update as an SPD linear system and enforces coupling through a representation-derived dependency graph and Laplacian priors, while avoiding label leakage. Empirically, VBFN improves fidelity, diversity, and structural coherence on synthetic graphs and molecular benchmarks, outperforming strong baselines and demonstrating efficient, scalable inference via iterative solvers. This coupled Bayesian framework advances discrete graph generation by embedding geometry directly into the belief dynamics, with practical impact on molecular design and related domains needing reliable, structure-consistent graph samples.

Abstract

Graph generation aims to sample discrete node and edge attributes while satisfying coupled structural constraints. Diffusion models for graphs often adopt largely factorized forward-noising, and many flow-matching methods start from factorized reference noise and coordinate-wise interpolation, so node-edge coupling is not encoded by the generative geometry and must be recovered implicitly by the core network, which can be brittle after discrete decoding. Bayesian Flow Networks (BFNs) evolve distribution parameters and naturally support discrete generation. But classical BFNs typically rely on factorized beliefs and independent channels, which limit geometric evidence fusion. We propose Variational Bayesian Flow Network (VBFN), which performs a variational lifting to a tractable joint Gaussian variational belief family governed by structured precisions. Each Bayesian update reduces to solving a symmetric positive definite linear system, enabling coupled node and edge updates within a single fusion step. We construct sample-agnostic sparse precisions from a representation-induced dependency graph, thereby avoiding label leakage while enforcing node-edge consistency. On synthetic and molecular graph datasets, VBFN improves fidelity and diversity, and surpasses baseline methods.

Variational Bayesian Flow Network for Graph Generation

TL;DR

This work addresses graph generation by integrating node–edge coupling into the generative process. It introduces Variational Bayesian Flow Networks (VBFN), which lift classical factorized BFN beliefs to a joint Gaussian with a structured precision operator, enabling coupled Bayesian updates that inherently respect graph geometry. The method derives a closed-form posterior update as an SPD linear system and enforces coupling through a representation-derived dependency graph and Laplacian priors, while avoiding label leakage. Empirically, VBFN improves fidelity, diversity, and structural coherence on synthetic graphs and molecular benchmarks, outperforming strong baselines and demonstrating efficient, scalable inference via iterative solvers. This coupled Bayesian framework advances discrete graph generation by embedding geometry directly into the belief dynamics, with practical impact on molecular design and related domains needing reliable, structure-consistent graph samples.

Abstract

Graph generation aims to sample discrete node and edge attributes while satisfying coupled structural constraints. Diffusion models for graphs often adopt largely factorized forward-noising, and many flow-matching methods start from factorized reference noise and coordinate-wise interpolation, so node-edge coupling is not encoded by the generative geometry and must be recovered implicitly by the core network, which can be brittle after discrete decoding. Bayesian Flow Networks (BFNs) evolve distribution parameters and naturally support discrete generation. But classical BFNs typically rely on factorized beliefs and independent channels, which limit geometric evidence fusion. We propose Variational Bayesian Flow Network (VBFN), which performs a variational lifting to a tractable joint Gaussian variational belief family governed by structured precisions. Each Bayesian update reduces to solving a symmetric positive definite linear system, enabling coupled node and edge updates within a single fusion step. We construct sample-agnostic sparse precisions from a representation-induced dependency graph, thereby avoiding label leakage while enforcing node-edge consistency. On synthetic and molecular graph datasets, VBFN improves fidelity and diversity, and surpasses baseline methods.
Paper Structure (73 sections, 5 theorems, 64 equations, 4 figures, 6 tables, 2 algorithms)

This paper contains 73 sections, 5 theorems, 64 equations, 4 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Graph generative models.
  4. Diffusion and flow matching.
  5. Bayesian Flow Networks.
  6. Background
  7. Problem Definition
  8. Bayesian Flow Networks
  9. Belief parameter.
  10. Sender and receiver distributions.
  11. Bayesian update and flow distribution.
  12. Training objective.
  13. Limitation for graphs.
  14. Methodology
  15. Variational Lifting of Factorized Geometry
  16. ...and 58 more sections

Key Result

Theorem 4.1

Assume the structured prior Eq. eq:method_prior with $\bm\Omega_{\mathrm{prior}}\succ\bm 0$ and the structured sender Eq. eq:method_sender with $\bm\Omega_{\mathrm{obs}}\succ\bm 0$. Define the fused posterior precision at time $t$ Then $\bm P_t\succ\bm 0$ and where the posterior belief parameter $\bm\theta_t$ is the unique solution of the SPD linear system

Figures (4)

  • Figure 1: Illustration of VBFN framework. (a): From $G=(X, A)$, construct a no-leakage dependency graph $\mathcal{H}$, assign edge weights $\{w_{uv}\}$, and form the masked weighted Laplacian $\bm{\mathcal{L}}$. Then instantiate $\bm\Omega_{\mathrm{prior}}$ and obtain $\bm\Omega_{\mathrm{obs}}$. (b): With fixed message precision $\bm\Omega_{\mathrm{obs}}$, sender–receiver message–matching defines the training loss, while inference uses a coupled Bayesian update solved by iterative solvers.
  • Figure 2: Metrics versus sampling steps on Planar, QM9, and ZINC250k. Higher is better for V.U.N./Valid; Lower is better for Ratio/FCD.
  • Figure 3: Visualization of generated graphs on synthetic graph datasets.
  • Figure 4: Visualization of generated graphs on molecular graph datasets.

Theorems & Definitions (10)

  • Theorem 4.1: Joint Bayesian update with structured precisions
  • Proposition 4.2: Classical BFNs as a special case
  • Proposition 4.3: Posterior belief parameter as a coupled energy minimizer
  • Proposition 4.4: Structured KL under tied channel covariance
  • Proposition 4.5: Positive definiteness of the update operator
  • proof
  • proof
  • proof
  • proof
  • proof