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Spectral analysis has been successfully applied at the detection of communitystructure of networks, respectively being based on the adjacency matrix, thestandard Laplacian matrix, the normalized Laplacian matrix, the modularitymatrix, the correlation matrix and several other variants of these matrices.However, the comparison between these spectral methods is less reported. Moreimportantly, it is still unclear which matrix is more appropriate for thedetection of community structure. This paper answers the question throughevaluating the effectiveness of these five matrices against the benchmarknetworks with heterogeneous distributions of node degree and community size.Test results demonstrate that the normalized Laplacian matrix and thecorrelation matrix significantly outperform the other three matrices atidentifying the community structure of networks. This indicates that it iscrucial to take into account the heterogeneous distribution of node degree whenusing spectral analysis for the detection of community structure. In addition,to our surprise, the modularity matrix exhibits very similar performance to theadjacency matrix, which indicates that the modularity matrix does not gaindesired benefits from using the configuration model as reference network withthe consideration of the node degree heterogeneity.

Spectral methods for the detection of network community structure: a comparative analysis