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.

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Abstract

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