TwinS-GCN: Spectral conjugate for Spectral Graph Convolutional Networks

2026-09-26

AI Summary

This study addresses the challenges of over-smoothing and long-range dependency capture in deep graph convolutional networks (GCNs), where conventional symmetric operators fail to distinguish information flow directions. To overcome these limitations, this work proposes TwinS-GCN, an architecture that decomposes the shift operator into dissipative and non-dissipative components via spectral conjugation under non-holomorphic kernels, constructing sum and ratio filters to achieve lossless directional signal transmission. Combined with Chebyshev reparameterization, it yields stable vertex-domain layers with real coefficients. Theoretically, we prove that ratio filters exhibit global impulse responses and superior long-range coverage on directed cycles compared to polynomial filters. Empirically, the proposed model efficiently accomplishes graph transfer tasks at shallow depths while achieving node classification performance competitive with state-of-the-art methods.

Abstract

Graph convolutional networks propagate information by repeated local aggregation through a graph shift operator; i.e., a $K$-layer network reaches $K$ hops neighborhood. On the one hand, such spreading can lead to oversmoothing. On the other hand, long-range dependencies demand the depth. Transporting information on long distances and without attenuation requires the shift to distinguish a direction of flow, which a symmetric operator cannot perform but a directed one can fulfill. A natural way to extract pure-directionality is to take the skew-symmetric part of the shift operator through the Cartesian split, which, however, generally does not commute with the shift itself, meaning that the filters built on it are not shift-invariant. We instead use the spectral conjugate; i.e., the image of the operator under $\tau:z\mapsto \bar{z}$, which commutes with the shift and splits it into a dissipative and a non-dissipative part. Two filter families follow: a sum filter, whose non-dissipative component transports signal without energy loss, and a ratio filter, ratio in the pair of components rather than polynomial in the shift. Both arise from non-holomorphic kernels, placing them outside the holomorphic class underlying classical spectral convolution. On the directed cycle, the ratio filter becomes an IIR filter with global impulse response, for which we prove a long-range reach gap against every degree-$K$ polynomial filter. Chebyshev reparameterization gives stable vertex-domain layers with real coefficients, yielding TwinS-GCN, which solves graph transfer tasks at reduced depth and is competitive with state-of-the-art graph convolutional networks on node classification benchmarks.

Problem

• Graph Convolutional Networks
• Oversmoothing
• Long-range Dependencies
• Shift-invariance
• Spectral Conjugate

Innovation

• Spectral Graph Convolution
• Spectral Conjugate
• Non-holomorphic Kernels
• IIR Filter
• Chebyshev Reparameterization