Signed Laplacian for spectral clustering revisited

Classical spectral clustering is based on a spectral decomposition of a graph Laplacian, obtained from a graph adjacency matrix representing positive graph edge weights describing similarities of graph vertices. In signed graphs, the graph edge weights can be negative to describe disparities of graph vertices, for example, negative correlations in the data. Negative weights lead to possible negative spectrum of the standard graph Laplacian, which is cured by defining a signed Laplacian. We revisit comparing the standard and signed Laplacians and argue that the former is more natural than the latter, also showing that the negative spectrum is actually beneficial, for spectral clustering of signed graphs.