Analysis of Spectral Space Properties of Directed Graphs Using Matrix Perturbation Theory with Application in Graph Partition

The eigenspace of the adjacency matrix of a graph possesses important information about the network structure. However, analyzing the spectral space properties for directed graphs is challenging due to complex valued decompositions. In this paper, we explore the adjacency eigenspaces of 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 that are unique for directed graphs. Furthermore, we relax the community structure assumption and generalize the theories to the perturbed Perron-Frobenius simple invariant subspace so that the theories can adapt to a much broader range of network structural types. We also develop a graph partitioning algorithm and conduct evaluations to demonstrate its potential.