Eigenspaces of networks reveal the overlapping and hierarchical community structure moreprecisely

Abstract

Identifying community structure is fundamental for revealing the structure–functionalityrelationship in complex networks, and spectral algorithms have been shown to be powerfulfor this purpose. In a traditional spectral algorithm, each vertex of a network is embeddedinto a spectral space by making use of the eigenvectors of the adjacency matrix orLaplacian matrix of the graph. In this paper, a novel spectral approach for revealing theoverlapping and hierarchical community structure of complex networks is proposedby not only using the eigenvalues and eigenvectors but also the properties ofeigenspaces of the networks involved. This gives us a better characterization ofcommunity. We first show that the communicability between a pair of vertices canbe rewritten in term of eigenspaces of a network. An agglomerative clusteringalgorithm is then presented to discover the hierarchical communities using thecommunicability matrix. Finally, these overlapping vertices are discovered with thecorresponding eigenspaces, based on the fact that the vertices more densely connectedamongst one another are more likely to be linked through short cycles. Comparedwith the traditional spectral algorithms, our algorithm can identify both theoverlapping and hierarchical community without increasing the time complexityO(n3), wheren is the size of the network. Furthermore, our algorithm can also distinguish theoverlapping vertices from bridges. The method is tested by applying it to somecomputer-generated and real-world networks. The experimental results indicate that ouralgorithm can reveal community structure more precisely than the traditional spectralapproaches.