CKGConv: General Graph Convolution with Continuous Kernels

TL;DR

CKGConv proposes a general graph convolution by learning continuous kernels as functions of graph pseudo-coordinates derived from graph positional encodings, addressing the lack of canonical coordinates and irregular structures. It unifies spatial and spectral GNNs and, with generalized distance $\mathrm{GD}$ as pseudo-coordinates, achieves expressive power equivalent to $\mathrm{GD}$-WL, comparable to state-of-the-art Graph Transformers. Empirically, CKGCN delivers top-tier performance across multiple benchmarks and demonstrates strong long-range dependency modeling, while ablations show that non-linear, flexible kernels offer advantages over purely positive filters. The work further reveals complementary behavior with Graph Transformers, motivating hybrid architectures and the broad potential of extending continuous-kernel convolutions to other non-Euclidean spaces using graph pseudo-coordinates.

Abstract

The existing definitions of graph convolution, either from spatial or spectral perspectives, are inflexible and not unified. Defining a general convolution operator in the graph domain is challenging due to the lack of canonical coordinates, the presence of irregular structures, and the properties of graph symmetries. In this work, we propose a novel and general graph convolution framework by parameterizing the kernels as continuous functions of pseudo-coordinates derived via graph positional encoding. We name this Continuous Kernel Graph Convolution (CKGConv). Theoretically, we demonstrate that CKGConv is flexible and expressive. CKGConv encompasses many existing graph convolutions, and exhibits a stronger expressiveness, as powerful as graph transformers in terms of distinguishing non-isomorphic graphs. Empirically, we show that CKGConv-based Networks outperform existing graph convolutional networks and perform comparably to the best graph transformers across a variety of graph datasets. The code and models are publicly available at https://github.com/networkslab/CKGConv.

Figures (5)

• Figure 1: Continuous Kernel Graph Convolution (CKGConv)
• Figure 2: Adjacency matrices and learned continuous kernels across multiple channels for two graphs from the ZINC dataset.
• Figure 3: (a) Detailed Architecture of CKGCN with $L$ CKGConv blocks and task-dependent output head, (b) the detailed design of each CKGConv-block.
• Figure 4: The toy example for Anti-oversmoothing.
• Figure 5: The toy example for Edge Detection.

Theorems & Definitions (11)

• Proposition 3.1
• Proposition 4.1
• Proposition 4.2
• Lemma E.1
• proof : Proof of Proposition \ref{['prop:ckgconv_gdwl']}
• proof : Proof of Proposition \ref{['prop:ckgconv_deepset']}
• Lemma E.2
• proof : Proof of Lemma \ref{['lemma:A^k']}
• proof : Proof of Proposition \ref{['prop:ckgconv_spectral']}
• Proposition E.3