Investigation of Spectral Clustering for Signed Graph Matrix Representations

Signed graphs [11], which allow for both favorable and adverse relationships, are becoming a common model choice for various data analysis applications, e.g., correlation clustering [1] and spectral clustering [9]. Unlike for unsigned graphs, there is no collective agreement of a matrix representation that relates to the unsigned graph Laplacian. There currently exists three proposed matrix representations: [9] proposes a zero row-sum preserving Laplacian (signed Laplacian), [8] proposes a physics preserving Laplacian (Physics Laplacian), and [4] proposes an expansion of the signed Laplacian into an unsigned Laplacian with twice the number of degrees-of-freedom (Gremban's expansion.) We investigate these three proposed matrix representations with respect to the quality of traditional (unsigned graph) spectral clustering concepts. We provide three numerical tests that use a stochastic block model with negative edge weights. We observe that the best matrix representation for spectral clustering depends on the underlying structure of a signed graph, which may be unavailable. However, since the Gremban's expansion matrix provides higher quality clusters for more combinations of inner and outer-connection probabilities for positive and negative valued edges, we conclude that the Gremban's expansion is the most robust representation to use for spectral clustering.