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.

[1]  Zhi-Hua Zhou,et al.  Line Orthogonality in Adjacency Eigenspace with Application to Community Partition , 2011, IJCAI.

[2]  F. Chung.  Laplacians and the Cheeger Inequality for Directed Graphs , 2005 .

[3]  V. N. Bogaevski,et al.  Matrix Perturbation Theory , 1991 .

[4]  V. Carchiolo,et al.  Extending the definition of modularity to directed graphs with overlapping communities , 2008, 0801.1647.

[5]  Youngdo Kim,et al.  Finding communities in directed networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[7]  Christos Faloutsos,et al.  EigenSpokes: Surprising Patterns and Scalable Community Chipping in Large Graphs , 2009, 2009 IEEE International Conference on Data Mining Workshops.

[8]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[9]  Michalis Vazirgiannis,et al.  Clustering and Community Detection in Directed Networks: A Survey , 2013, ArXiv.

[10]  Marina Meila,et al.  Clustering by weighted cuts in directed graphs , 2007, SDM.

[11]  Thomas Hofmann,et al.  Semi-supervised Learning on Directed Graphs , 2004, NIPS.

[12]  Bernhard Schölkopf,et al.  Learning from labeled and unlabeled data on a directed graph , 2005, ICML.

[13]  Srinivasan Parthasarathy,et al.  Symmetrizations for clustering directed graphs , 2011, EDBT/ICDT '11.

[14]  E A Leicht,et al.  Community structure in directed networks. , 2007, Physical review letters.