Complex-Weighted Convolutional Networks: Provable Expressiveness via Complex Diffusion

TL;DR

This paper introduces Complex-Weighted Convolutional Networks (CWCN), a new GNN framework using complex weights for diffusion, which enhances expressiveness and addresses oversmoothing and heterophily issues.

Contribution

The paper proposes a novel complex-weighted diffusion framework and CWCN model that are theoretically expressive and practically effective for node classification tasks.

Findings

CWCN achieves competitive results on benchmark datasets.

Complex diffusion enhances GNN expressiveness and addresses oversmoothing.

Theoretical proof shows any node classification can be solved with appropriate complex weights.

Abstract

Graph Neural Networks (GNNs) have achieved remarkable success across diverse applications, yet they remain limited by oversmoothing and poor performance on heterophilic graphs. To address these challenges, we introduce a novel framework that equips graphs with a complex-weighted structure, assigning each edge a complex number to drive a diffusion process that extends random walks into the complex domain. We prove that this diffusion is highly expressive: with appropriately chosen complex weights, any node-classification task can be solved in the steady state of a complex random walk. Building on this insight, we propose the Complex-Weighted Convolutional Network (CWCN), which learns suitable complex-weighted structures directly from data while enriching diffusion with learnable matrices and nonlinear activations. CWCN is simple to implement, requires no additional hyperparameters beyond those of standard GNNs, and achieves competitive performance on benchmark datasets. Our results demonstrate that complex-weighted diffusion provides a principled and general mechanism for enhancing GNN expressiveness, opening new avenues for models that are both theoretically grounded and practically effective.