SigMaNet: One Laplacian to Rule Them All

TL;DR

SigMaNet introduces a novel Laplacian matrix for spectral graph convolutional networks that effectively handles directed and weighted graphs with positive and negative weights, demonstrating superior performance across multiple benchmarks.

Contribution

The paper proposes the Sign-Magnetic Laplacian ($L^{\sigma}$), a new parameter-free Laplacian that extends spectral GCN theory to directed, signed, and weighted graphs, outperforming existing methods.

Findings

SigMaNet achieves top performance in 15 out of 21 benchmark cases.

It outperforms more complex architectures on various metrics.

The approach effectively encodes edge direction and weight without sensitivity to magnitude.

Abstract

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 ($L^{\sigma}$), a new Laplacian matrix that we introduce ex novo in this work. $L^{\sigma}$ 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. $L^{\sigma}$ 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. $L^{\sigma}$ 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.