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Generator-based Graph Generation via Heat Diffusion

Anthony Stephenson, Ian Gallagher, Christopher Nemeth

TL;DR

This work introduces G^3, a generator-based graph generation framework that defines a forward diffusion on graph matrices using the graph Laplacian heat equation and learns a neural surrogate generator to reverse this diffusion. By formulating the diffusion in terms of an explicit infinitesimal generator and employing generator matching, the method preserves permutation symmetry and graph locality while enabling scalable sampling via a reverse-time ODE. The approach yields competitive graph-generation performance across synthetic and real datasets, with significant speed advantages over existing diffusion-based baselines. Extensions to alternative Laplacians, directed graphs, and covariate-guided generation are discussed as promising avenues for future work, underscoring the framework’s flexibility and principled grounding in continuous-time stochastic processes.

Abstract

Graph generative modelling has become an essential task due to the wide range of applications in chemistry, biology, social networks, and knowledge representation. In this work, we propose a novel framework for generating graphs by adapting the Generator Matching (arXiv:2410.20587) paradigm to graph-structured data. We leverage the graph Laplacian and its associated heat kernel to define a continous-time diffusion on each graph. The Laplacian serves as the infinitesimal generator of this diffusion, and its heat kernel provides a family of conditional perturbations of the initial graph. A neural network is trained to match this generator by minimising a Bregman divergence between the true generator and a learnable surrogate. Once trained, the surrogate generator is used to simulate a time-reversed diffusion process to sample new graph structures. Our framework unifies and generalises existing diffusion-based graph generative models, injecting domain-specific inductive bias via the Laplacian, while retaining the flexibility of neural approximators. Experimental studies demonstrate that our approach captures structural properties of real and synthetic graphs effectively.

Generator-based Graph Generation via Heat Diffusion

TL;DR

This work introduces G^3, a generator-based graph generation framework that defines a forward diffusion on graph matrices using the graph Laplacian heat equation and learns a neural surrogate generator to reverse this diffusion. By formulating the diffusion in terms of an explicit infinitesimal generator and employing generator matching, the method preserves permutation symmetry and graph locality while enabling scalable sampling via a reverse-time ODE. The approach yields competitive graph-generation performance across synthetic and real datasets, with significant speed advantages over existing diffusion-based baselines. Extensions to alternative Laplacians, directed graphs, and covariate-guided generation are discussed as promising avenues for future work, underscoring the framework’s flexibility and principled grounding in continuous-time stochastic processes.

Abstract

Graph generative modelling has become an essential task due to the wide range of applications in chemistry, biology, social networks, and knowledge representation. In this work, we propose a novel framework for generating graphs by adapting the Generator Matching (arXiv:2410.20587) paradigm to graph-structured data. We leverage the graph Laplacian and its associated heat kernel to define a continous-time diffusion on each graph. The Laplacian serves as the infinitesimal generator of this diffusion, and its heat kernel provides a family of conditional perturbations of the initial graph. A neural network is trained to match this generator by minimising a Bregman divergence between the true generator and a learnable surrogate. Once trained, the surrogate generator is used to simulate a time-reversed diffusion process to sample new graph structures. Our framework unifies and generalises existing diffusion-based graph generative models, injecting domain-specific inductive bias via the Laplacian, while retaining the flexibility of neural approximators. Experimental studies demonstrate that our approach captures structural properties of real and synthetic graphs effectively.
Paper Structure (61 sections, 8 theorems, 58 equations, 14 figures, 7 tables, 2 algorithms)

This paper contains 61 sections, 8 theorems, 58 equations, 14 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Background
  4. Graph Matrix Representations
  5. Heat Equation on Graphs and the Heat Kernel
  6. Flow Matching
  7. Vector fields and probability paths.
  8. Surrogate vector field and objective.
  9. Generator Matching
  10. Semigroup and Kolmogorov equations.
  11. Surrogate generator and objective.
  12. Linear functional parameterisation.
  13. Graph Generation
  14. Graph Diffusions and Generators
  15. Semigroup viewpoint and long-time limit.
  16. ...and 46 more sections

Key Result

Proposition 1.1

Let $L\in\mathbb{R}^{n\times n}$ be the (combinatorial) Laplacian of an undirected graph, and define the heat kernel $H_s := e^{-sL}$ for $s\ge 0$. For any initial matrix $Y_0\in\mathbb{R}^{n\times n}$, the matrix-valued symmetric heat equation admits the unique solution In particular, the family of linear maps $(\mathcal{P}_s)_{s\ge 0}$ defined by $\mathcal{P}_s(Y):=H_s Y H_s$ forms the (matrix

Figures (14)

  • Figure 1: \ref{['fig:forward_process']} shows the effect of the symmetric heat diffusion process on a graph (bottom row) and its adjacency (top) as $t$ increases and the probability mass is spread over all possible edges. Colours indicate this value, with yellow and purple the respective extremes of the interval $[0,1]$ and edge transparency governed by the same quantity. \ref{['fig:example_enzyme_graphs']} shows a series of randomly drawn graphs from the (training) subset of the Enzyme dataset (top) and output graphs from $G^3$ (bottom). Node colouring reflects Forman-Ricci curvature forman2003bochner at that node.
  • Figure 1: $G^3$ Training
  • Figure 2: Spectral MMD as a function of maximum diffusion time $T$ and the number of planar graphs $N$ with $n=64$ nodes. Curves are Gaussian-smoothed functions of the observations shown, which themselves are averaged over 10 random seeds. Shaded regions indicate 66% confidence intervals under CLT assumptions. Learning rate set at $1\times 10^{-4}$ for training.
  • Figure 3: Spectral MMD as a function of the number of hidden neural network units per layer $w$ and the number of planar graphs $N$ with $n=64$ nodes. Curves are Gaussian-smoothed functions of the observations shown, which themselves are averaged over 10 random seeds. Shaded regions indicate 66% confidence intervals under CLT assumptions. Learning rate set at $1\times 10^{-4}$ for training.
  • Figure 4: SBM adjacency matrices generated by pre-specifying node labels to the conditional form of the $G^3$ model. Columns headers indicate the number of communities and the top row uses a given ratio of membership sizes with the bottom row using the sizes reversed in order.
  • ...and 9 more figures

Theorems & Definitions (17)

  • Definition 2.1: Heat equation on a graph
  • Proposition 1.1: Solution of the symmetric heat diffusion via the heat kernel
  • proof : Proof
  • Lemma 1.2: Dissipativity of the symmetric heat operator
  • proof
  • Corollary 1.3: Energy dissipation and Frobenius stability
  • proof
  • Lemma 1.4: Closed-form solution and semigroup contraction
  • proof
  • Lemma 1.5: Semigroup structure and infinitesimal generator
  • ...and 7 more