# TwinS-GCN: Spectral conjugate for Spectral Graph Convolutional Networks

arXiv: https://arxiv.org/abs/2609.32940 · HTML: https://mlsanity.com/paper/2609.32940/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 38 of its 63 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 30 of 38)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [DeepGCNs: Can GCNs Go as Deep as CNNs?](https://mlsanity.com/paper/1904.03751/) (2019)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [A critical look at the evaluation of GNNs under heterophily: Are we really making progress?](https://mlsanity.com/paper/2302.11640/) (2023)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017)
- [On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topology](https://mlsanity.com/paper/2302.02941/) (2023)
- [AntisymmetricRNN: A Dynamical System View on Recurrent Neural Networks](https://mlsanity.com/paper/1902.09689/) (2019)
- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Edge Directionality Improves Learning on Heterophilic Graphs](https://mlsanity.com/paper/2305.10498/) (2023)
- [Discrete Signal Processing on Graphs: Frequency Analysis](https://mlsanity.com/paper/1307.0468/) (2013)
- [Anti-Symmetric DGN: a stable architecture for Deep Graph Networks](https://mlsanity.com/paper/2210.09789/) (2022)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)
- [Autoregressive Moving Average Graph Filtering](https://mlsanity.com/paper/1602.04436/) (2016)
- [Signal Processing on Directed Graphs](https://mlsanity.com/paper/2008.00586/) (2020)
- [On Oversquashing in Graph Neural Networks Through the Lens of Dynamical Systems](https://mlsanity.com/paper/2405.01009/) (2024)
- [HoloNets: Spectral Convolutions do extend to Directed Graphs](https://mlsanity.com/paper/2310.02232/) (2023)
- [Port-Hamiltonian Architectural Bias for Long-Range Propagation in Deep Graph Networks](https://mlsanity.com/paper/2405.17163/) (2024)

## Abstract (arXiv)

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 τ:zmapsto barz, 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.