Complex-Weighted Convolutional Networks: Provable Expressiveness via Complex Diffusion

TL;DR

This work introduces complex-weighted diffusion on graphs as a principled way to overcome oversmoothing and heterophily in GNNs. By assigning Hermitian complex weights to edges, the authors prove that the steady-state of a complex random walk can achieve linear separability for any node-classification task, independent of the number of classes. They instantiate this idea as CWCN, a practical GNN that learns the complex-weighted structure via an edge- conditioning MLP and uses learnable complex matrices and nonlinearities per layer, with a training pipeline that treats the initial features as a complex matrix. Empirically, CWCN delivers competitive performance on standard benchmarks and exhibits clear advantages in heterophilic regimes, while offering strong theoretical guarantees and a simpler hyperparameter profile than several diffusion-based alternatives.

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.

Figures (6)

• Figure 1: Node features in polar coordinates at selected diffusion times (top) and evolution of node feature phases over time during diffusion (bottom). Complex random walk diffusion progressively separates the classes of a complex-weighted graph: nodes in the same class (same color) converge to complex numbers with the same phase. The figure corresponds to the dataset Texas geomgcn. From diffusion time 5 onward, the purple class is omitted for better visualization.
• Figure 2: Training accuracy (top) and examples of the processed graphs (bottom) for learnable complex versus real random walks, varying the numbers of classes in two heterophilic settings: a cycle graph with same-class nodes in opposite positions (left) and a ring of clusters with same-class clusters in opposite positions (right). The complex random walk consistently achieves higher mean accuracy (dots) than its real-valued counterpart (shade: standard deviation over 10 random seeds).
• Figure 3: Example of a balanced complex-weighted graph, where the node set is partitioned into three subsets satisfying the conditions of Proposition \ref{['th:charact-balanced']}.
• Figure 4: Illustration of how the triangle $T$ in the proof of Proposition \ref{['th:existence-balanced']} can be obtained as $T=T_1'\triangle T_2' \triangle T_3'$.
• Figure 5: Illustration of the proof of Proposition \ref{['th:existence-balanced']}: every cycle satisfies property (iii) of Proposition \ref{['th:charact-balanced']}.

Theorems & Definitions (21)

• Definition 1
• Remark 1
• Definition 2
• Definition 3
• Theorem 1
• proof : Proof sketch
• Proposition 1
• proof : Proof sketch
• Definition 4
• Definition 5