Different from Laplacian or normal matrix, the properties of the adjacency eigenspace received much less attention. Recent work showed that nodes projected into the adjacency eigenspace exhibit an orthogonal line pattern and nodes from the same community locate along the same line. In this paper, we conduct theoretical studies based on graph perturbation to demonstrate why this line orthogonality property holds in the adjacency eigenspace and why it generally disappears in the Laplacian and normal eigenspaces. Using the orthogonality property in the adjacency eigenspace, we present a graph partition algorithm, AdjCluster, which first projects node coordinates to the unit sphere and then applies the classic k-means to find clusters. Empirical evaluations on synthetic data and real-world social networks validate our theoretical findings and show the effectiveness of our graph partition algorithm.

Compressive spectral clustering combines the distance preserving measurements of compressed sensing with the power of spectral clustering. Our analysis provides rigorous bounds on how small errors in the affinity matrix can affect the spectral coordinates and clusterability. This work generalizes the current perturbation results of two-class spectral clustering to incorporate multiclass clustering using k eigenvectors.