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, 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 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 while retaining all non-spectral information. We demonstrate that the Dyson Diffusion Model learns graph spectra accurately and outperforms existing graph diffusion models.

翻译：扩散模型是生成建模的核心方法，已通过扩散邻接矩阵表示的方式被应用于图数据。对于包含n个节点的图，其邻接矩阵表示最多可达n!种，这一挑战仅能通过使用置换等变学习架构得到部分缓解。尽管现有图扩散模型具有计算效率，但除非在图中添加特定特征，否则难以区分某些图族。这一缺陷源于在学习架构中强制引入归纳偏置。本研究利用随机矩阵理论解析提取扩散过程的谱特性，从而能够将归纳偏置从架构转移到动态过程中。在此基础上，我们提出了Dyson扩散模型，该模型采用Dyson布朗运动来捕捉邻接矩阵上Ornstein-Uhlenbeck过程的谱动态，同时保留所有非谱信息。我们证明Dyson扩散模型能够准确学习图谱，并优于现有的图扩散模型。