Laplacian eigenvectors capture natural community structures on graphs and are widely used in spectral clustering and manifold learning. The use of Laplacian eigenvectors as embeddings for the purpose of multiscale graph comparison has however been limited. Here we propose the Embedded Laplacian Discrepancy (ELD) as a simple and fast approach to compare graphs (of potentially different sizes) based on the similarity of the graphs' community structures. The ELD operates by representing graphs as point clouds in a common, low-dimensional space, on which a natural Wasserstein-based distance can be efficiently computed. A main challenge in comparing graphs through any eigenvector-based approaches is the potential ambiguity that could arise due to sign-flips and basis symmetries. The ELD leverages a simple symmetrization trick to bypass any sign ambiguities. For comparing graphs that do not have any ambiguities due to basis symmetries (i.e. the spectrums are simple), we show that the ELD becomes a natural pseudo-metric that enjoys nice properties such as invariance under graph isomorphism. For comparing graphs with non-simple spectrums, we propose a procedure to approximate the ELD via a simple perturbation technique to resolve any ambiguity from basis symmetries. We show that such perturbations are stable using matrix perturbation theory under mild assumptions that are straightforward to verify in practice. We demonstrate the excellent applicability of the ELD approach on both simulated and real datasets.

翻译：拉普拉斯特征向量能够捕捉图上的自然社区结构，广泛应用于谱聚类和流形学习。然而，将拉普拉斯特征向量作为嵌入用于多尺度图比较的研究仍然有限。本文提出嵌入拉普拉斯差异（ELD），作为一种基于图社区结构相似性来比较图（可能具有不同大小）的简单且快速的方法。ELD通过将图表示为公共低维空间中的点云，在该空间上可以高效计算基于Wasserstein的自然距离。通过任何基于特征向量的方法比较图的主要挑战在于符号翻转和基对称性可能带来的歧义。ELD利用简单的对称化技巧来规避符号歧义。对于比较不存在基对称性歧义（即谱简单）的图，我们证明ELD成为一种自然的伪度量，具有图同构不变性等优良性质。对于比较具有非简单谱的图，我们提出通过简单的扰动技术来近似ELD，以解决基对称性带来的任何歧义。利用矩阵扰动理论，我们在实践中易于验证的温和假设下证明了此类扰动的稳定性。我们在模拟数据集和真实数据集上展示了ELD方法的卓越适用性。