Eigenspaces of networks reveal the overlapping and hierarchical community structure more precisely

Identifying community structure is fundamental for revealing the structure-functionality relationship in complex networks, and spectral algorithms have been shown to be powerful for this purpose. In a traditional spectral algorithm, each vertex of a network is embedded into a spectral space by making use of the eigenvectors of the adjacency matrix or Laplacian matrix of the graph. In this paper, a novel spectral approach for revealing the overlapping and hierarchical community structure of complex networks is proposed by not only using the eigenvalues and eigenvectors but also the properties of eigenspaces of the networks involved. This gives us a better characterization of community. We first show that the communicability between a pair of vertices can be rewritten in term of eigenspaces of a network. An agglomerative clustering algorithm is then presented to discover the hierarchical communities using the communicability matrix. Finally, these overlapping vertices are discovered with the corresponding eigenspaces, based on the fact that the vertices more densely connected amongst one another are more likely to be linked through short cycles. Compared with the traditional spectral algorithms, our algorithm can identify both the overlapping and hierarchical community without increasing the time complexity O(n(3)), where n is the size of the network. Furthermore, our algorithm can also distinguish the overlapping vertices from bridges. The method is tested by applying it to some computer-generated and real-world networks. The experimental results indicate that our algorithm can reveal community structure more precisely than the traditional spectral approaches.