Line orthogonality in adjacency eigenspace with application to community partition

Authors:
Leting Wu;Xiaowei Ying;Xintao Wu;Zhi-Hua Zhou
Affiliations:
Software and Information Systems Dept., University of North Carolina at Charlotte;Software and Information Systems Dept., University of North Carolina at Charlotte;Software and Information Systems Dept., University of North Carolina at Charlotte;National Key Laboratory for Novel Software Technology, Nanjing University, China
Venue:
IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume Three
Year:
2011

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Abstract

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.