英文摘要 |
In graph theory, the adjacency matrix is the basic data structure to represent a network; in linear algebra, the eigenvalues and corresponding eigenvectors of a matrix can be found out via an eigen analysis. Therefore, the eigen analysis of adjacency matrix is an approach to investigating topological structures of networks. After an eigen analysis of adjacency matrix, by inspecting the eigenvectors from the macro view, we can explore cohesive subgroups of the whole network; and by inspecting the elements of eigenvectors from the micro view, we can evaluate centrality of the individual nodes. In the literature, there have been many eigen analytical results of undirected networks thanks to symmetric and diagonalizable adjacency matrices. However, adjacency matrices of direct networks may be asymmetric and not diagonalizable so that the eigen analytical results of direct networks are rare to date. Therefore, in this paper, we try to apply the eigen analysis of adjacency matrix to explore topological structures of directed networks, including the strongly connected graph, simple bowtie structure, and recursive bowtie structure, to further the understandings of eigen properties of directed networks. As a result, we propose an eigen analysis algorithm for directed networks in theory and apply the algorithm to explore topological structures of blogroll networks in practice. |