  Spectral analysis has been successfully applied at the detection of community
structure of networks, respectively being based on the adjacency matrix, the
standard Laplacian matrix, the normalized Laplacian matrix, the modularity
matrix, the correlation matrix and several other variants of these matrices.
However, the comparison between these spectral methods is less reported. More
importantly, it is still unclear which matrix is more appropriate for the
detection of community structure. This paper answers the question through
evaluating the effectiveness of these five matrices against the benchmark
networks with heterogeneous distributions of node degree and community size.
Test results demonstrate that the normalized Laplacian matrix and the
correlation matrix significantly outperform the other three matrices at
identifying the community structure of networks. This indicates that it is
crucial to take into account the heterogeneous distribution of node degree when
using spectral analysis for the detection of community structure. In addition,
to our surprise, the modularity matrix exhibits very similar performance to the
adjacency matrix, which indicates that the modularity matrix does not gain
desired benefits from using the configuration model as reference network with
the consideration of the node degree heterogeneity.
