TwinS-GCN: Spectral conjugate for Spectral Graph Convolutional Networks

Chun Hei Michael Chan; Flavia Petruso; Dimitri Van De Ville

cs.LGSep 26, 2026

TwinS-GCN: Spectral conjugate for Spectral Graph Convolutional Networks

Authors: Chun Hei Michael Chan, Flavia Petruso, Dimitri Van De Ville

Organizations: Neuro-X Institute and Department of Electrical Engineering ´Ecole polytechnique f´ed´erale de Lausanne (EPFL) Geneva, 1202, Switzerland

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 $τ:z\mapsto \bar{z}$ , 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.

Figures & tables

Figure 1: Isolating ${\mathbf{L}}^{\uparrow}$ allows $h_{\text{sum}}$ to translate the signal while the polynomial kernel $h$ inseparably moves and flattens it.

Figure 2: FIR models are obstructed by FIR wall whereas graph ratio filter overcomes it on $\mathcal{C}_{N}$ .

Panel of Figure 3 — Figure 3: a) Accuracy of different models on the graph transfer task. In the box are models with overlapping accuracy of $1$ . Models with performance lower than chance level ( $20\%$ ) at all depths are displayed in gray. b) Dirichlet energy computed after each convolution layer before activation, averaged within batch. Only models ( $10$ layers) with accuracy beyond chance level are shown. Input and target signal’s energy are also shown for each graph. In both case, error bars are over trials.

	Model	Chameleon-F	Squirrel-F	Texas	Wisconsin	Cornell	Mean Rank
	MLP	$36.73\pm 4.71$	$36.55\pm 1.82$	$82.16\pm 4.34$	$85.16\pm 4.95$	$\underline{75.86}\pm\underline{3.94}$	$6.70$
Undir.	GCN	$40.89\pm 4.12$	$39.47\pm 1.47$	$52.16\pm 5.16$	$48.92\pm 3.06$	$52.70\pm 5.30$	$10.20$
Undir.	GAT	$39.21\pm 3.08$	$35.62\pm 2.06$	$58.38\pm 6.63$	$49.41\pm 4.09$	$54.32\pm 5.05$	$10.20$
Dir.	MagNet	$42.36\pm 3.53$	$44.19\pm 1.39$	$\underline{85.31}\pm\underline{4.56}$	$83.40\pm 3.98$	$73.33\pm 4.17$	$\underline{4.20}$
	Dir-GNN	$44.13\pm 2.92$	$\underline{44.28}\pm\underline{2.17}$	$70.54\pm 6.52$	$67.39\pm 6.00$	$59.82\pm 6.90$	$6.40$
	FaberNet	$42.99\pm 3.33$	$\textbf{45.13}\pm\textbf{2.06}$	$83.15\pm 5.30$	$79.54\pm 5.18$	$72.61\pm 4.72$	$4.60$

Table 1: Node classification accuracy (in percentage) across benchmark datasets. Best result per dataset in bold , second best underlined . Additional information in Appendix D.2 .

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Graph and signals
Notation	Description
$G=(\mathcal{V},\mathcal{E},{\mathbf{W}})$	Directed weighted graph with nodes $\mathcal{V}$ , edges $\mathcal{E}$ , weights ${\mathbf{W}}$
$N=\|\mathcal{V}\|$	Number of nodes
$C$	Number of feature channels
$\bar{\mathcal{E}}$	Symmetrized edge set, $\mathcal{E}\cup\{(m,n):(n,m)\in\mathcal{E}\}$
${\mathbf{W}}\in\mathbb{R}^{N\times N}_{\geq 0}$	Non-negative weight (adjacency) matrix

Appendix

Table 2: Summary of notations used in the main text.

Panel of Figure 4 — Figure 4: a) Sensitivity profiles computed for shift, heat ( $t=5$ ), heat ( $t=20$ ), and ratio ( $K=1$ ) filters. b) Plot of the optimal filter sensitivity across distances. c) Plot of the $\mathcal{G}(K)$ function. Throughout $N$ is set to $200$ .

	$\varepsilon=10^{-1}$	$10^{-2}$	$10^{-3}$	$10^{-4}$
$\max_{k}\|a_{k}\|$ , target $z^{2}+1$	$1.0$	$1.0$	$1.0$	$1.0$
$\max_{k}\|a_{k}\|$ , target $\bar{z}$	$4.1\times 10^{1}$	$4.0\times 10^{2}$	$4.0\times 10^{3}$	$4.0\times 10^{4}$

Appendix

Table 3: Coefficient magnitude under a cluster of three.

Figure 5: Graphs used for the task (same as in Gravina et al. (2025) ).

	Model	depth	#hidden	non-lin.	norm.	bias	lr	$\bm{\alpha}$	$\bm{\beta}$	$\bm{q}$	Accuracy
Directed Cycle
	GCN	$1$	$8$	ReLU	No	No	$0.005$	–	–	–	$0.325\pm 0.054$
	MagNet	$6$	$4$	–	–	Yes	$0.005$	–	–	$0.05$	$0.267\pm 0.031$
	FaberNet	$10$	$32$	–	–	Yes	$0.005$	Yes	–	–	$1.000\pm 0.000$
	Dir-GNN	$10$	$16$	None	Yes	No	$0.005$	Yes	–	–	$1.000\pm 0.000$
	ADR-GNN	$2$	$4$	–	–	–	$0.005$	–	–	–	$0.283\pm 0.024$

Appendix

Table 4: Best hyperparameter configuration per topology for the graph transfer task , selected over depth and all searched axes by validation MSE. A dash marks an axis the architecture does not take.

\alpha

\beta

report whether the branch-mixing weights are learned;

q

is MagNet’s charge.

Dataset	#Nodes	#Edges	#Feat.	#Cls.
Cornell	183	295	1,703	5
Texas	183	309	1,703	5
Wisconsin	251	499	1,703	5
Chameleon †	2,277	36,101	2,325	5
Squirrel †	5,201	217,073	2,089	5

Appendix

Table 5: Statistics of the node classification benchmarks.

Model

depth

#hidden

dropout

norm.

bias

w.d.

feat.

\bm{K}

learn

\bm{\alpha}

learn

\bm{\beta}

model-specific

Accuracy

Chameleon-F

MagNet

3

128

0.2

–

Yes

0.005

0.005

raw

–

1

–

q{=}0

0.424\pm 0.035

Dir-GNN

5

128

0.5

Yes

0.005

0.0005

row

cat

–

\alpha{=}0.5

0.441\pm 0.029

FaberNet

4

128

0.6

–

Yes

0.01

0.0005

row

max

1

–

\alpha{=}0.5

, zero-ord.: No

0.430\pm 0.033

AdvDIFFormer

4

128

0.4

–

0.05

0

std

–

heads

2

K_{\mathrm{ord}}{=}1

0.415\pm 0.036

ADR-GNN

4

64

0.6

–

0.01

0.005

std

–

step

0.05

0.461\pm 0.041

Appendix

Table 6: Hyperparameter search results for benchmarking datasets . A dash marks an axis the architecture does not take.

\alpha

\beta

report whether the branch-mixing weights are learned;

q

is MagNet’s charge. Cluster tol are

1e^{-9}

unless specified.

	Model	#Params	Train (ms/epoch)	Inference (ms)	Build (ms)	Accuracy
Directed Cycle
	GCN	$25$	$0.67\pm 0.34$	$0.09\pm 0.00$	–	$0.325\pm 0.054$
	MagNet	$269$	$9.12\pm 0.28$	$1.80\pm 0.01$	–	$0.267\pm 0.031$
	FaberNet	$39{,}110$	$26.94\pm 0.78$	$4.44\pm 0.00$	–	$1.000\pm 0.000$
	Dir-GNN	$4{,}417$	$10.58\pm 0.06$	$1.96\pm 0.00$	–	$1.000\pm 0.000$
	ADR-GNN	$345$	$11.67\pm 0.52$	$1.61\pm 0.01$	–	$0.283\pm 0.024$

Appendix

Table 7: Runtime and parameter count per topology for the graph transfer task as specified in Appendix D.1 , for the best configuration selected in Table 4 . Train time is per-epoch, averaged over

100

epochs (train repeats is

5

) and inference time (infer repeats is

50

) is per forward pass over the full graph (

40

graphs/batch). Build time is provided specifically to TwinS-GCN models which require a one-time precomputing, per graph, of the operators. Accuracy is reported for reference.


Model	#Params	Train (ms/epoch)	Inference (ms)
GCN	$2{,}469$	$8.93\pm 1.80$	$1.02\pm 0.24$
MagNet	$6{,}917$	$17.34\pm 1.30$	$3.01\pm 0.68$
FaberNet	$27{,}205$	$38.95\pm 0.34$	$6.77\pm 0.60$
Dir-GNN	$4{,}938$	$9.33\pm 0.21$	$1.40\pm 0.12$
ADR-GNN	$30{,}181$	$134.74\pm 3.63$	$22.85\pm 4.63$

Appendix

Table 8: Runtime and parameter count per topology for the graph transfer task as specified in Appendix D.1 . The count is under an equivalent configuration (hidden=32, depth=4, polynomial degree (K)=1, dropout=0, bias=True, non-linearity=ReLU). Remaining information is as in Table 7 .

Model	#Params	Train (ms/epoch)	Inference (ms)	Accuracy
MLP	$117{,}701$	$1.44\pm 0.26$	$0.19\pm 0.05$	$0.852\pm 0.050$
MagNet	$111{,}429$	$6.78\pm 0.25$	$3.76\pm 0.74$	$0.834\pm 0.040$
Dir-GNN	$251{,}072$	$4.15\pm 0.12$	$1.49\pm 0.20$	$0.674\pm 0.060$
FaberNet	$6{,}029{,}317$	$55.83\pm 2.13$	$23.56\pm 6.79$	$0.795\pm 0.052$
AdvDIFFormer	$1{,}095{,}941$	$13.45\pm 0.50$	$2.67\pm 0.36$	$0.765\pm 0.048$
ADR-GNN	$197{,}125$	$18.80\pm 0.58$	$8.50\pm 0.29$	$0.845\pm 0.045$

Appendix

Table 9: Runtime and parameter count, illustrated with Wisconsin dataset benchmarking of datasets , for the best configuration selected in Table 6 . Train time is per-epoch, averaged over

100

epochs (train repeats is

5

) and inference time (infer repeats is

50

) is per forward pass over the full graph. Accuracy is reported to distinguish relevance across models.

Model	Cornell	Texas	Wisconsin	Chameleon-F	Squirrel-F
TwinS (S)	$20$	$10$	$50.0$	$1,230$	$22,540$
TwinS (R)	$50$	$40$	$90.0$	$3,810$	$58,540$
TwinS (C)	$50$	$40$	$90.0$	$3,810$	$58,540$

Appendix

Table 10: Build time (ms) per TwinS-GCN model, per graph.

Model	#Params	Train (ms/epoch)	Inference (ms)
MLP	$56{,}805$	$1.27\pm 0.44$	$0.17\pm 0.04$
Dir-GNN	$113{,}610$	$1.72\pm 0.09$	$0.39\pm 0.00$
MagNet	$173{,}157$	$10.67\pm 0.28$	$5.85\pm 0.30$
FaberNet	$692{,}677$	$14.61\pm 0.27$	$6.29\pm 0.12$
ADR-GNN	$84{,}517$	$15.03\pm 0.14$	$6.03\pm 0.67$
AdvDIFFormer	$79{,}973$	$6.93\pm 0.36$	$1.54\pm 0.00$

Appendix

Table 11: Runtime and parameter count, illustrated with Wisconsin dataset benchmarking of datasets , for equivalent configuration (hidden=32, depth=4, dropout=0, K=2, bias=True). Train time is per-epoch, averaged over

50

epochs (train repeats is

3

) and inference time (infer repeats is

50

) is per forward pass over the full graph.

Figure 6: Long-range dependency experiment on cycle graph with increasing orders.

Panel of Figure 7 — Figure 7: a) Measured wind velocity field. b) Derived temperature graph. c) Signal centered on source and target nodes used for the temperature displacement task.

Panel of Figure 8 — Figure 8: a) Estimated target signal using kernels $h,h_{\text{sum}},h_{\text{rat}}$ with $K=10$ . b) Mean NMSE across the 10 (source,target) pairs for estimated target signal as a function of filter order.

Graph Neural Networks

Graph Neural Networks learn from graph-structured data by combining information across connected nodes, edges, and larger structures.

See all 722 papers

Convex--Concave Quadratic Spectral Filtering for Graph Neural Networks
Jun 23, 2026Ranhui Yan, Jia Cai, Mengzhu Chen +1Graph Neural Networks Graph Robustness
School of Computing and Artificial Intelligence, Guangzhou Xinhua University, Guangzhou, Guangdong, China · School of Statistics and Data Science, Guangdong University of Finance and Economics,2026 Guangzhou, Guangdong, China
L2G-Net: Local to Global Spectral Graph Neural Networks via Cauchy Factorizations
Feb 21, 2026Samuel Fernández-Menduiña, Eduardo Pavez, Antonio OrtegaSpectral Graph Convolution Graph Signal Processing
University of Southern California.
Position: Spectral GNNs Are Neither Spectral Nor Superior for Node Classification
Mar 19, 2026Qin Jiang, Chengjia Wang, Michael Lones +2Spectral Graph Convolution Node Classification
Department of Computer Science, University of Heriot-Watt, Edinburgh, UK

Jun 23, 2026J/K move · Enter open · S save

Convex--Concave Quadratic Spectral Filtering for Graph Neural Networks

Ranhui Yan, Jia Cai, Mengzhu Chen, Haodong Yang

School of Computing and Artificial Intelligence, Guangzhou Xinhua University, Guangzhou, Guangdong, China · School of Statistics and Data Science, Guangdong University of Finance and Economics,2026 Guangzhou, Guangdong, China

Graph Neural Networks Graph Robustness Spectral Methods Spectral Graph Convolution

Open paper AskPDF arXiv

Spectral graph neural networks (GNNs) interpret message passing as frequency-selective filtering. While low-order spectral filters are efficient, their limited selectivity often leads to weak attenuation outside the passband, whereas high-order alternatives introduce optimization challenges. We propose DCQ-GNN, a spectral GNN based on a compact bank of adaptive convex--concave quadratic filters. By restricting the filter order to two while explicitly exploiting complementary curvature, DCQ-GNN improves spectral selectivity as quantified by Dirichlet energy and entropy measures without resorting to high-order polynomial expansions. The model fuses filter outputs through a node-adaptive gating mechanism to enable node-wise structure-aware spectral selection. We provide a formal spectral analysis grounded in Dirichlet energy attenuation, von Neumann entropy, and curvature polarity, and derive explicit characterizations of filter behavior across varying levels of homophily and structural perturbations. Extensive benchmarks on 10 datasets show that DCQ-GNN ties for the top average rank (3.0) on heterophilic graphs and obtains the second-best rank (4.2) on homophilic graphs, remaining competitive with representative high-order polynomial spectral filters. Furthermore, under strong structural perturbations, DCQ-GNN exhibits substantially smaller performance degradation compared to both first-order and high-order baselines. These results demonstrate that curvature-aware quadratic banks provide a robust and efficient alternative to high-order spectral models while preserving optimization stability and computational efficiency.

Abstract

Graph Neural Networks

Explore similar work

Convex--Concave Quadratic Spectral Filtering for Graph Neural Networks

L2G-Net: Local to Global Spectral Graph Neural Networks via Cauchy Factorizations

Position: Spectral GNNs Are Neither Spectral Nor Superior for Node Classification