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:Laplacian egenvictors 在图形中捕捉自然社区结构, 并广泛用于光谱群集和多重学习。 但是, Laplacian egenvictors 用作多比例图形比较的嵌入器的使用有限。 我们在这里建议, 嵌入式的 Laplacian 差异性( ELD) 是一种简单而快速的方法, 用来比较基于图形群落结构相似性的图表( 可能不同大小 ) 。 ELD 在普通的低维空间中, 将图形作为点云来代表点云, 可以有效地计算出基于自然的瓦塞斯坦的直径距离。 通过任何基于图形的嵌入器比较图形, 使用任何基于点的点云彩色方法来代表点云。 我们显示, ELDD会成为通过任何基于任何比例的精确度数据来比较图表的主要挑战, 使用简单的直径, 使用简单的直径直径的直径, 以直径的直径比直方的平方的平面的直径, 显示, 。