Diffusion models are central to generative modeling and have been adapted to graphs by diffusing adjacency matrix representations. The challenge of having up to $n!$ such representations for graphs with $n$ nodes is only partially mitigated by using permutation-equivariant learning architectures. Despite their computational efficiency, existing graph diffusion models struggle to distinguish certain graph families and their spectra, unless graph data are augmented with ad hoc features. This shortcoming stems from enforcing the inductive bias within the learning architecture. In this work, we leverage random matrix theory to analytically extract the spectral properties of the diffusion process, allowing us to push most of the inductive bias from the architecture into the dynamics. Building on this, we introduce the Dyson Diffusion Model, which employs Dyson's Brownian motion to capture the spectral dynamics of an Ornstein-Uhlenbeck process on the adjacency matrix. Furthermore, conditioned on the spectral dynamics, we formulate a Lie group diffusion, appropriately modeling the remaining degrees of freedom. Strikingly, the resulting learning problem becomes permutation invariant at the Lie algebra level. We demonstrate that the Dyson Diffusion Model learns graph spectra accurately and outperforms existing graph diffusion models.

翻译：扩散模型是生成式建模的核心方法，通过扩散邻接矩阵表示已适配至图结构。对于包含n个节点的图，存在多达n!种此类表示，当前通过使用置换等变学习架构仅能部分缓解这一挑战。尽管计算高效，现有图扩散模型仍难以区分某些图族及其谱特征，除非对图数据附加特殊特征。这一缺陷源于将归纳偏置强加于学习架构之中。本研究利用随机矩阵理论分析提取扩散过程的谱特性，从而将大部分归纳偏置从架构迁移至动力学过程。基于此，我们提出戴森扩散模型，采用戴森布朗运动捕获邻接矩阵上Ornstein-Uhlenbeck过程的谱动力学。进一步地，以谱动力学为条件，我们构建李群扩散模型以适当建模剩余自由度。引人注目的是，由此产生的学习问题在李代数层面实现置换不变性。实验证明，戴森扩散模型能够精确学习图谱，且性能优于现有图扩散模型。