Constraint-Preserving Flow Matching for Spectral Graph Generation

Spectral graph generation models graphs through the eigenvalues and eigenvectors of their Laplacians, providing a compact and continuous alternative to direct graph construction.

谱图生成通过拉普拉斯矩阵的特征值和特征向量对图进行建模，为直接图构建提供了一种紧凑且连续的替代方案。

However, these eigensystems must retain the structural properties of graph Laplacians. Existing approaches preserve eigenvector orthonormality but do not enforce the Laplacian nullspace constraint, potentially producing eigensystems inconsistent with Laplacian structure.

然而，这些特征系统必须保留图拉普拉斯矩阵的结构属性。现有的方法虽然保持了特征向量的正交归一性，但并未强制执行拉普拉斯零空间约束，这可能导致生成的特征系统与拉普拉斯结构不一致。

To address this, we introduce Laplacian-Constrained Stiefel Flow Matching (LaSFM), which preserves the nullspace condition alongside eigenvector orthonormality and positive eigenvalues during generation.

为了解决这个问题，我们引入了拉普拉斯约束施蒂费尔流匹配（LaSFM），它在生成过程中不仅保持了特征向量的正交归一性和正特征值，还保留了零空间条件。

We characterize the underlying constrained Stiefel manifold and formulate flow matching on it. We further extend LaSFM to property-guided generation, steering samples toward desired graph properties while maintaining these constraints.

我们刻画了底层的约束施蒂费尔流形，并在其上制定了流匹配方法。我们进一步将 LaSFM 扩展到属性引导生成，在保持这些约束的同时，引导样本向所需的图属性靠拢。

Experiments demonstrate improved performance on both spectral and graph-level metrics, along with better alignment with target properties.

实验表明，该方法在谱指标和图级指标上均表现出更好的性能，并与目标属性具有更好的一致性。