Spectral methods for the detection of network community structure: a comparativeanalysis

Abstract

Spectral analysis has been successfully applied to the detection of community structure ofnetworks, respectively being based on the adjacency matrix, the standard Laplacianmatrix, the normalized Laplacian matrix, the modularity matrix, the correlationmatrix and several other variants of these matrices. However, the comparisonbetween these spectral methods is less reported. More importantly, it is still unclearwhich matrix is more appropriate for the detection of community structure. Thispaper answers the question by evaluating the effectiveness of these five matricesagainst benchmark networks with heterogeneous distributions of node degree andcommunity size. Test results demonstrate that the normalized Laplacian matrix and thecorrelation matrix significantly outperform the other three matrices at identifyingthe community structure of networks. This indicates that it is crucial to takeinto account the heterogeneous distribution of node degree when using spectralanalysis for the detection of community structure. In addition, to our surprise, themodularity matrix exhibits very similar performance to the adjacency matrix,which indicates that the modularity matrix does not gain benefits from using theconfiguration model as a reference network with the consideration of the node degreeheterogeneity.