SigMaNet: One Laplacian to Rule Them All
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This paper introduces SigMaNet, a generalized Graph Convolutional Network (GCN) capable of handling both undirected and directed graphs with weights not restricted in sign and magnitude. The cornerstone of SigMaNet is the introduction of a generalized Laplacian matrix: the Sign-Magnetic Laplacian (L^σ). The adoption of such a matrix allows us to bridge a gap in the current literature by extending the theory of spectral GCNs to directed graphs with both positive and negative weights. L^σ exhibits several desirable properties not enjoyed by the traditional Laplacian matrices on which several state-of-the-art architectures are based. In particular, L^σ is completely parameter-free, which is not the case of Laplacian operators such as the Magnetic Laplacian L^(q), where the calibration of the parameter q is an essential yet problematic component of the operator. L^σ simplifies the approach, while also allowing for a natural interpretation of the signs of the edges in terms of their directions. The versatility of the proposed approach is amply demonstrated experimentally; the proposed network SigMaNet turns out to be competitive in all the tasks we considered, regardless of the graph structure.
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