On Spectral Analysis of Directed Graphs with Graph Perturbation Theory

The eigenspace of the adjacency matrix of a graph possesses important information about the network structure. Recent works showed the existence of line orthogonality patterns when nodes are projected into the eigenspaces of graphs by either SVD or regular eigen decomposition. Further study proved why such phenomena hold true for simple graphs using matrix perturbation theory and several graph partition algorithms were introduced. However, the majority of the studies only focused on undirected graphs. In this paper, we will explore the adjacency eigenspace properties for directed graphs. With the aid of the graph perturbation theory, we emphasize on deriving rigorous mathematical results to explain several phenomena related to the eigenspace projection patterns for directed graphs. Furthermore, we will relax our original assumption and generalize the theories to the perturbed Perron-Frobenius simple invariant subspace so that the theories can adept to a much broader range of network structural types. Our theories have the potential for various applications such as graph partition for directed graphs.