DiPhon: Diffusion on Graphons for Scalable Graph Generation

Existing graph diffusion models often struggle with scalability, as they are typically tied to a fixed number of nodes and require retraining when moving to larger graph sizes. This paper addresses the challenge of creating a size-agnostic generative framework that can learn structural statistics from small graphs and transfer that knowledge to generate significantly larger graphs.

The authors propose DiPhon, a framework that leverages graphons—the mathematical limit objects of dense graph sequences—to define a continuous diffusion process. By formulating a Jacobi stochastic differential equation (SDE) on the graphon space, the authors ensure that the diffusion process remains bounded within the interval [0, 1], which is essential for representing edge probabilities. They then discretize this continuous process into a graph-level diffusion model. A key technical contribution is the moment-alignment strategy, which ensures that the discrete graph process matches the first moment of the continuous graphon dynamics exactly, while keeping second-moment discrepancies under control.

DiPhon allows for a principled approach to scalable graph generation. Because the model operates on the graphon limit, it inherits size-agnostic statistical properties. Empirically, the authors demonstrate that models trained on small graphs can be used to generate progressively larger graphs at inference time without any retraining, while successfully preserving the core topological properties of the original graph distribution. The model also provides a tractable way to estimate the marginal score through graph denoising, allowing for efficient reverse-time sampling.

This work provides a theoretical and practical bridge between small-scale training data and large-scale graph generation. By moving away from fixed-size node representations and toward a graphon-based diffusion, DiPhon offers a more flexible and scalable paradigm for applications like molecular design, where the ability to generalize across different scales is critical.