SigMaNet: One Laplacian to Rule Them All
This addresses a gap in graph neural network theory for researchers and practitioners working with complex, directed, and signed graphs, offering a versatile and high-performing solution.
The paper tackles the problem of extending spectral graph convolutional networks to directed graphs with both positive and negative weights, introducing a new Sign-Magnetic Laplacian that achieves best performance in 15 out of 21 cases and top-two performance in all cases.
This paper introduces SigMaNet, a generalized Graph Convolutional Network (GCN) capable of handling both undirected and directed graphs with weights not restricted in sign nor magnitude. The cornerstone of SigMaNet is the Sign-Magnetic Laplacian ($L^σ$), a new Laplacian matrix that we introduce ex novo in this work. $L^σ$ 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 other Laplacian matrices on which several state-of-the-art architectures are based, among which encoding the edge direction and weight in a clear and natural way that is not negatively affected by the weight magnitude. $L^σ$ is also completely parameter-free, which is not the case of other Laplacian operators such as, e.g., the Magnetic Laplacian. The versatility and the performance of our proposed approach is amply demonstrated via computational experiments. Indeed, our results show that, for at least a metric, SigMaNet achieves the best performance in 15 out of 21 cases and either the first- or second-best performance in 21 cases out of 21, even when compared to architectures that are either more complex or that, due to being designed for a narrower class of graphs, should -- but do not -- achieve a better performance.