A Unified Uncertainty Representation for Graph Neural Networks via Doubly-Spectral Stochastic Expansion

Fred Xu; Thomas Markovich; Florence Regol; Yizhou Sun

cs.LGSep 28, 2026

A Unified Uncertainty Representation for Graph Neural Networks via Doubly-Spectral Stochastic Expansion

Authors: Fred Xu, Thomas Markovich, Florence Regol, Yizhou Sun

Organizations: Department of Computer Science University of California, Los Angeles · Block, Inc. · Mila, Quebec AI Institute

Abstract

Reliable deployment of graph neural networks requires calibration, out-of-distribution (OOD) detection, and robustness to distribution shift, yet existing methods address these needs with separate models and objectives. We model uncertain node embeddings as random graph signals: graph Fourier filters capture structural variation, and a scalar orthogonal-polynomial chaos coordinate captures latent stochastic variation. The resulting doubly-spectral stochastic (DSS) expansion supplies task-matched readouts from one representation: the mean coefficient encodes class evidence for the energy-based OOD score, the higher-order coefficients encode structured logit variation, and quadrature averaging over the chaos coordinate defines the single predictive distribution used for prediction and calibration. A capacity theorem shows that, under a full-rank feature assumption, a restricted subfamily matches the chaos coefficients of any Gaussian-latent random graph signal, with exponentially decaying truncation error under a growth condition; the task-level claims are established empirically. DSS-GNN has two deployment modes: standalone, or as a residual branch beside a deterministic encoder (DSS-Hybrid). Standalone DSS-GNN achieves the lowest Brier score among the compared uncertainty-aware baselines on all 14 node classification benchmarks without post-hoc correction; DSS-Hybrid achieves the best AUROC on most node-OOD settings, competitive cross-graph OOD detection, and the strongest shifted accuracy on all 7 GOOD concept-shift benchmarks under standard empirical risk minimization (ERM). Cross-evaluating both modes on all three tasks shows that each remains effective on the other's tasks, with documented exceptions, and yields explicit deployment guidance.

Figures & tables

Figure 1: (a) Second-order correction $c_{i}$ (Proposition 2 ) on Minesweeper. Each node is positioned by loss curvature ( $\operatorname{tr}(\mathcal{H}_{i})$ ) and chaos energy ( $\mathcal{E}_{i}$ ); color shows correction magnitude. (b) Nodes binned by correction quartile: the correction concentrates on hard nodes (low accuracy, high Brier).

	Accuracy (%)			Brier Score
Dataset	DSS-GNN	TFE-GNN	G- $\Delta$ UQ	DSS-GNN	TFE-GNN	G- $\Delta$ UQ
Cora	86.63 $\pm$ 1.26	71.25 $\pm$ 1.90	84.35 $\pm$ 1.97	0.207 $\pm$ 0.019	0.449 $\pm$ 0.024	0.226 $\pm$ 0.023
Citeseer	80.20 $\pm$ 1.28	68.87 $\pm$ 1.87	70.85 $\pm$ 2.22	0.308 $\pm$ 0.010	0.463 $\pm$ 0.020	0.433 $\pm$ 0.026
PubMed	89.72 $\pm$ 0.31	82.84 $\pm$ 0.76	88.18 $\pm$ 0.66	0.156 $\pm$ 0.005	0.265 $\pm$ 0.011	0.179 $\pm$ 0.008
Texas	91.31 $\pm$ 3.44	84.92 $\pm$ 4.26	9.02 $\pm$ 1.83	0.240 $\pm$ 0.111	0.255 $\pm$ 0.083	0.779 $\pm$ 0.029
Cornell	85.11 $\pm$ 5.12	81.28 $\pm$ 5.45	21.06 $\pm$ 3.49	0.231 $\pm$ 0.079	0.290 $\pm$ 0.069	0.793 $\pm$ 0.023

Table 1: Node classification: test accuracy (%,

\uparrow

) and Brier score (

\downarrow

). Mean

\pm

standard deviation over 10 runs. Best result per dataset in bold . DSS-GNN uses the per-dataset configuration of Table 12 and the mean-logit readout.

Model	Cora			Amazon-Photo			Coauthor-CS			Cross-graph
	Structure	Feature	Label	Structure	Feature	Label	Structure	Feature	Label	Twitch	Arxiv
GNNSafe	87.52	93.44	92.80	99.58	98.55	97.35	99.60	99.64	97.23	66.82	71.06
GNNSafe++	90.62	95.56	92.75	99.82	99.64	97.51	99.99	99.97	97.89	95.36	74.77
Graph-EBM	61.14	72.42	92.69	75.22	86.49	97.24	72.75	89.28	97.91	44.43	52.80
MC-dropout	87.50	93.12	93.03	98.67	98.50	96.80	99.51	99.53	97.25	68.58	66.34
Deep Ensemble	87.87	93.63	93.75	98.58	98.43	97.35	98.18	98.45	94.97	72.25	67.72

Table 2: OOD detection (AUROC, %) on the nine node-OOD settings and the two cross-graph settings (AUPR, FPR95, and ID accuracy for the latter are in Table 3 ). GNNSafe/GNNSafe++ and GPN [ 30 ] values as reported by Wu et al. [36] on this benchmark (their Tables 1–2; their appendix lists

81.77/93.24

for GPN on Coauthor-CS feature/label; GPN out-of-memory on Arxiv); Graph-EBM [ 9 ] , MC-dropout (

M{=}20

), and Deep Ensembles (

M{=}5

) run in our pipeline on the same GCN backbone (3 seeds; the latter two with the same score propagation). Bold: best.

	Twitch				Arxiv
Model	AUROC	AUPR	FPR95	ID Acc.	AUROC	AUPR	FPR95	ID Acc.
GNNSafe	66.82	70.97	76.24	70.40	71.06	80.44	87.01	53.39
GNNSafe++	95.36	97.12	33.57	70.18	74.77	83.21	77.43	53.50
Graph-EBM	44.43	57.91	94.84	63.04	52.80	65.88	97.40	53.45
MC-dropout	68.58	81.25	94.84	64.10	66.34	74.88	89.69	53.56
Deep Ensemble	72.25	83.19	94.29	66.94	67.72	75.98	87.32	47.58

Table 3: Cross-graph OOD (AUROC, AUPR:

\uparrow

; FPR95, false positive rate at 95% true positive rate:

\downarrow

; ID: in-distribution; bold: best). GNNSafe/GNNSafe++ and GPN values as reported by Wu et al. [36] (OOM: out of memory on a 24 GB GPU); Graph-EBM [ 9 ] , MC-dropout, and Deep Ensembles run in our pipeline (3 seeds).

Model	GOOD-CBAS	GOOD-WebKB	GOOD-Twitch	GOOD-Cora	GOOD-Cora	GOOD-Arxiv	GOOD-Arxiv
	color	university	language	word	degree	time	degree
ERM	82.43	27.16	51.59	64.03	60.30	65.64	54.81
IRM	82.00	26.06	49.78	63.93	60.26	65.54	56.72
VREx	82.86	26.61	55.75	64.03	60.53	65.92	56.68
Coral	81.57	28.07	51.80	64.04	60.30	65.79	55.14
DANN	83.57	29.36	51.67	63.96	60.23	65.67	55.34

Table 4: Test accuracy (%) on the concept-shift GOOD node benchmark for DSS-Hybrid (standard ERM training) and 13 baselines; ERM through TAR as reported by Zheng et al. [38] , G-

\Delta

UQ run in our pipeline. Best in bold, second-best underlined; OOM: out of memory.

Task	Standalone DSS-GNN	DSS-Hybrid
Calibration (14 datasets)	Best Brier on all 14 and best accuracy on 13 (Table 1 ); stronger than the hybrid on 13 of 14	Never loses beyond noise to its own GCN base; Roman-Empire $+27.7$ accuracy, $-0.33$ Brier
OOD detection (11 settings)	Effective on 10 of 11: Cora $79.8/88.2/94.0$ , Photo $96.5$ – $98.1$ , CS $94.3$ – $97.8$ , Arxiv $73.7$ (margin objective); the Twitch energy score does not separate	Best published AUROC on 7 of 11 (Table 2 ); stronger than the standalone on 10 of 11
GOOD concept shift (7 settings)	Above every published baseline on 5 of 7, at ERM level on the other 2	Strongest on all 7 (Table 4 )

Table 5: Both deployment modes on all three tasks. Each cell summarizes the full per-dataset comparison in Appendix G (Tables 24 , 25 , and 26 ); the standalone OOD and GOOD cells and the hybrid calibration cell are the cross-evaluations; the other cells restate Tables 1 , 2 – 3 , and 4 .

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Calibration	OOD Detection	Robust Classif.
MC-dropout [ 10 ]	(✓)	✗	✗
Deep Ensembles [ 17 ]	✓	(✓)	✗
G- $\Delta$ UQ [ 33 ]	✓	✓ ∗	✓
TFE-GNN [ 8 ]	(✓)	✗	✗
GNNSafe [ 36 ]	✗	✓	✗
GNNSafe++ [ 36 ]	✗	✓	✗

Appendix

Table 6: Tasks evaluated by each method. ✓= evaluated in the original paper; (✓) = evaluated as a secondary metric or in the appendix; ✓ ∗ = evaluated in the original paper but OOM on our benchmark; ✗= not evaluated in the original paper or not designed for that task.

Dataset(s)	Repository	License
Cora, Citeseer, PubMed	https://github.com/kimiyoung/planetoid	MIT
Texas, Cornell, Wisconsin,	https://github.com/bingzhewei/geom-gcn	not stated
Chameleon, Squirrel	https://github.com/bingzhewei/geom-gcn	not stated
Coauthor-CS, Amazon-Photo	https://github.com/shchur/gnn-benchmark	MIT
Roman-Empire, Amazon-Ratings,	https://github.com/yandex-research/heterophilous-graphs	MIT
Minesweeper, Tolokers, Questions	https://github.com/yandex-research/heterophilous-graphs	MIT

Appendix

Table 7: Dataset sources and licenses.

Dataset	Nodes	Edges	Classes	Features
Cora	2,708	10,556	7	1,433
Citeseer	3,327	9,228	6	3,703
PubMed	19,717	88,651	3	500
Texas	183	574	5	1,703
Cornell	183	557	5	1,703
Wisconsin	251	916	5	1,703

Appendix

Table 8: Node classification datasets for calibration experiments (Section 4.2 ). Results use 10 splits per dataset (Appendix D ); edges are counted in both directions. Cornell uses the corrected files of the geom-gcn repository, whose initial release duplicated the Texas features and labels. Datasets below the mid-rule are from the heterophilous suite of Platonov et al. [26] .

Dataset	OOD type	Nodes	Edges	Classes	Features
Cora	structure / feature / label	2,708	10,556	7	1,433
Amazon-Photo	structure / feature / label	7,650	238,162	8	745
Coauthor-CS	structure / feature / label	18,333	163,788	15	6,805
Twitch (DE $\to$ ES/FR/RU)	cross-graph	9,498	306,276	2	3,170
Arxiv ( $\leq$ 2015 $\to$ 2018–20)	cross-graph (temporal)	169,343	2,315,598	40	128

Appendix

Table 9: OOD detection datasets (Section 4.3 ). For node-OOD (Cora, Amazon-Photo, Coauthor-CS), OOD nodes are generated via structure (stochastic block model), feature (random interpolation of node features), or label (held-out classes) perturbations. For cross-graph OOD, the ID and OOD graphs are from different domains. Statistics are for the in-distribution graph.

Setting	Domain	Nodes	Edges	Classes	Features
GOOD-Cora / degree	node degree	19,793	126,842	70	8,710
GOOD-Cora / word	word diversity	19,793	126,842	70	8,710
GOOD-Arxiv / degree	node degree	169,343	2,315,598	40	128
GOOD-Arxiv / time	publication year	169,343	2,315,598	40	128
GOOD-CBAS / color	node color	700	3,962	4	4
GOOD-WebKB / university	university	617	1,138	5	1,703

Appendix

Table 10: GOOD benchmark settings (Section 4.4 ). All use concept shift. The domain column indicates the covariate that defines the distribution shift. Node counts are for the full graph before splitting; GOOD-Cora is built on the full Cora citation graph (CoraFull), not on the 7-class Planetoid subset of Table 8 .

Dataset	Architecture	Hidden	$L$	Optimizer	Overrides
Cora	Prop-first (rw)	64	2	Adam	dropout 0.8
Citeseer	Cheb	64	2	Adam	none
PubMed	Cheb	128	2	Adam	none
Texas	Prop-first	64	2	RMSprop	none
Cornell	Cheb	128	2	Adam	dropout 0.8
Wisconsin	Cheb	64	2	Adam	dropout 0.6

Appendix

Table 11: Per-dataset non-

P

DSS-GNN configuration for the calibration experiments (Table 1 ). “Prop-first” = propagate-once structure (spectral filtering followed by MLP); “TFE-adj” = TFE-style adjacency propagation; “Cheb” = Chebyshev filters on rescaled Laplacian. Unlisted hyperparameters use the shared defaults above;

P

\lambda_{\mathrm{reg}}

, and

S

per dataset are listed in Table 12 .

Dataset	$P$	$\lambda_{\mathrm{reg}}$	$S$	Accuracy	Brier	Val. Brier
Cora	1	0.01	4	86.63 $\pm$ 1.26	0.207 $\pm$ 0.019	0.203
Citeseer	2	0	4	80.20 $\pm$ 1.28	0.308 $\pm$ 0.010	0.315
PubMed	5	0.01	4	89.72 $\pm$ 0.31	0.156 $\pm$ 0.005	0.152
Texas	2	0.01	4	91.31 $\pm$ 3.44	0.240 $\pm$ 0.111	0.190
Cornell	1	0.01	4	85.11 $\pm$ 5.12	0.231 $\pm$ 0.079	0.166
Wisconsin	2	0	4	93.25 $\pm$ 3.39	0.104 $\pm$ 0.047	0.088

Appendix

Table 12: Selected configuration per dataset for the calibration experiments (Table 1 ): chaos order

P

, chaos energy regularization

\lambda_{\mathrm{reg}}

, quadrature size

S

, and that single configuration’s test accuracy (%) and Brier score (mean

\pm

SD over the same 10 splits). The last column reports the validation Brier of the selected configuration, recorded in a separate run on the same 10 splits (for Cornell, in the same run).

Dataset	$P=0$	$P=1$	$P=2$	$P=3$	$P=4$	$P=5$	$P=6$	$P=8$
Cora	85.12 $\pm$ 1.06	86.63 $\pm$ 1.26	86.33 $\pm$ 1.12	86.33 $\pm$ 0.82	86.63 $\pm$ 1.14	86.32 $\pm$ 1.13	86.61 $\pm$ 0.84	86.38 $\pm$ 0.96
Citeseer	65.39 $\pm$ 1.02	79.21 $\pm$ 0.79	80.20 $\pm$ 1.28	78.47 $\pm$ 0.80	71.50 $\pm$ 4.58	78.92 $\pm$ 1.13	76.77 $\pm$ 2.19	78.49 $\pm$ 1.08
PubMed	89.12 $\pm$ 0.57	89.33 $\pm$ 0.49	89.33 $\pm$ 0.66	89.38 $\pm$ 0.67	89.33 $\pm$ 0.54	89.72 $\pm$ 0.31	89.40 $\pm$ 0.57	89.39 $\pm$ 0.51
Texas	75.08 $\pm$ 23.70	88.36 $\pm$ 4.57	91.31 $\pm$ 3.44	89.18 $\pm$ 4.92	91.31 $\pm$ 3.30	87.54 $\pm$ 5.59	89.01 $\pm$ 2.82	87.37 $\pm$ 5.43
Cornell	83.62 $\pm$ 6.80	85.11 $\pm$ 5.12	85.74 $\pm$ 6.73	83.62 $\pm$ 5.39	84.26 $\pm$ 6.87	83.83 $\pm$ 7.38	83.62 $\pm$ 6.53	82.98 $\pm$ 7.61
Wisconsin	93.25 $\pm$ 3.46	92.87 $\pm$ 2.92	93.25 $\pm$ 3.39	88.50 $\pm$ 3.35	93.00 $\pm$ 3.46	90.75 $\pm$ 2.84	89.63 $\pm$ 4.82	90.50 $\pm$ 2.92

Appendix

Table 13: Accuracy (%) of DSS-GNN for varying chaos order

P

across all datasets.

Dataset	$P=0$	$P=1$	$P=2$	$P=3$	$P=4$	$P=5$	$P=6$	$P=8$
Cora	0.228 $\pm$ 0.013	0.207 $\pm$ 0.019	0.212 $\pm$ 0.016	0.209 $\pm$ 0.016	0.208 $\pm$ 0.019	0.215 $\pm$ 0.015	0.211 $\pm$ 0.011	0.211 $\pm$ 0.015
Citeseer	0.427 $\pm$ 0.008	0.346 $\pm$ 0.008	0.308 $\pm$ 0.010	0.326 $\pm$ 0.012	0.422 $\pm$ 0.027	0.327 $\pm$ 0.012	0.370 $\pm$ 0.023	0.331 $\pm$ 0.013
PubMed	0.162 $\pm$ 0.007	0.158 $\pm$ 0.007	0.160 $\pm$ 0.008	0.159 $\pm$ 0.008	0.161 $\pm$ 0.007	0.156 $\pm$ 0.005	0.159 $\pm$ 0.008	0.159 $\pm$ 0.007
Texas	0.269 $\pm$ 0.082	0.213 $\pm$ 0.111	0.240 $\pm$ 0.111	0.233 $\pm$ 0.096	0.253 $\pm$ 0.096	0.253 $\pm$ 0.086	0.249 $\pm$ 0.114	0.242 $\pm$ 0.082
Cornell	0.241 $\pm$ 0.087	0.231 $\pm$ 0.079	0.228 $\pm$ 0.088	0.250 $\pm$ 0.090	0.263 $\pm$ 0.129	0.247 $\pm$ 0.107	0.252 $\pm$ 0.079	0.273 $\pm$ 0.111
Wisconsin	0.103 $\pm$ 0.048	0.107 $\pm$ 0.041	0.104 $\pm$ 0.047	0.161 $\pm$ 0.060	0.103 $\pm$ 0.048	0.129 $\pm$ 0.037	0.153 $\pm$ 0.052	0.135 $\pm$ 0.040

Appendix

Table 14: Brier Score (

\downarrow

) of DSS-GNN for varying chaos order

P

across all datasets.

Setting	$P{=}0$	$P{=}1$	$P{=}2$
GOOD-Cora / word	63.24 $\pm$ 0.40	64.89 $\pm$ 0.21	64.01 $\pm$ 0.25
GOOD-Cora / degree	61.38 $\pm$ 0.81	62.60 $\pm$ 0.59	61.22 $\pm$ 0.07
GOOD-Arxiv / time	64.81 $\pm$ 0.27	65.43 $\pm$ 1.07	64.48 $\pm$ 0.27

Appendix

Table 15: Shifted test accuracy (%) of standalone DSS-GNN on GOOD concept shift as the chaos order varies at the validation-selected configuration (mean

\pm

SD over 3 seeds).

Dataset	Structure	Feature	Label
Cora	$+14.5$	$+5.4$	$-6.7$
Amazon-Photo	$-0.4$	$+1.7$	$+3.9$
Coauthor-CS	$+1.9$	$+1.1$	$+1.3$
Twitch (cross-graph)	$+18.2$
Arxiv (cross-graph)	$+7.3$

Appendix

Table 16: Effect of energy-score propagation on DSS-Hybrid: change in AUROC (points) from the raw to the propagated energy score at identical checkpoints (3 runs, fixed configuration).

Dataset	$\lambda_{\mathrm{reg}}{=}0$	$10^{-3}$	$10^{-2}$	$10^{-1}$
Cora	0.219	0.219	0.207	0.220
Citeseer	0.308	0.310	0.318	0.499
PubMed	0.159	0.156	0.156	0.157
Texas	0.366	0.246	0.213	0.251
Cornell	0.230	0.237	0.228	0.342
Wisconsin	0.103	0.107	0.108	0.174

Appendix

Table 17: Effect of chaos energy regularization

\lambda_{\mathrm{reg}}

on Brier score (

\downarrow

), with

P=2

fixed. Bold indicates the best

\lambda_{\mathrm{reg}}

per dataset.

Dataset	$S{=}2$	$S{=}3$	$S{=}4$	$S{=}6$	$S{=}8$
Cora	0.216	0.207	0.217	0.209	0.212
Citeseer	0.310	0.308	0.312	0.313	0.311
PubMed	0.157	0.156	0.156	0.158	0.156
Texas	0.216	0.221	0.213	0.217	0.218
Cornell	0.235	0.286	0.228	0.218	0.242
Wisconsin	0.142	0.109	0.115	0.103	0.108

Appendix

Table 18: Brier score (

\downarrow

) vs. number of quadrature nodes

S

, with

P=2

fixed. The theoretical minimum is

S=3

Dataset	ChebNet $K{=}4$	$P{=}0$	Best $P{>}0$	$\Delta$ Chaos
Cora	0.281	0.228	0.207	$-9.2\%$
Citeseer	0.470	0.427	0.308	$-27.9\%$
PubMed	0.171	0.162	0.156	$-3.7\%$
Texas	0.208	0.269	0.213	$-20.8\%$
Cornell	0.324	0.241	0.228	$-5.4\%$
Wisconsin	0.073	0.103	0.103	$\phantom{-}0\%$

Appendix

Table 19: Contribution decomposition: Brier score (

\downarrow

). ChebNet

K{=}4

is a vanilla single-filter diagnostic;

P{=}0

and Best

P{>}0

are the

P{=}0

column and the best

P{>}0

column of Table 14 (same per-dataset architecture). Boldface compares the controlled

P{=}0

versus Best

P{>}0

pair, and the rightmost column isolates the chaos contribution. Table 1 reports the selected order of Table 12 , which can differ from the best swept order.

Dataset	$P{=}0$	$P{=}1$	$P{=}2$	$P{=}3$	Ratio $P{=}2$ / $P{=}0$
Cora (2.7K nodes)	35	67	102	131	$2.9\times$
Amazon-Rat. (25K nodes)	464	526	576	646	$1.2\times$
ogbn-arxiv (169K nodes)	2002	2088	2174	2273	$1.1\times$
ogbn-arxiv peak memory (MiB)	1721	2813	3976	5141	$2.3\times$

Appendix

Table 20: Wall-clock time per epoch (ms) vs. chaos order

P

(Cora and Amazon-Ratings on an A100; ogbn-arxiv on one H100, 3 runs, with peak GPU memory in MiB).

Component	Comparison	Effect
Chaos expansion ( $P$ )	$P{=}0$ vs best $P{>}0$ , same architecture (Tables 13 , 14 , 19 )	Calibration: Citeseer Brier $-27.9\%$ and accuracy $+14.8$ points (the endpoint also beats plain GCN, $0.308$ vs $0.318$ Brier, Tables 1 / 23 ); Texas $-20.8\%$ ; Minesweeper $-11.1\%$
Chaos expansion ( $P$ )	OOD ablation varying only $P$ (Appendix E.2 )	Energy AUROC flat across $P$ (Cora-structure $91.0/91.2/90.9/90.8$ at one fixed configuration); detection reads the mean logit, so $P$ can be selected for calibration without changing detection
Chaos expansion ( $P$ )	GOOD ablation at the selected configurations (Table 15 )	Shifted accuracy: $P{=}1$ beats $P{=}0$ on GOOD-Cora word ( $+1.7$ ), degree ( $+1.2$ ), and GOOD-Arxiv time ( $+0.6$ )
Spectral backbone (chaos off)	$P{=}0$ vs GCN (Tables 13 , 23 )	Heterophilous accuracy comes from the spectral backbone: Roman-Empire $79.0$ at $P{=}0$ vs GCN $51.3$
Filtering design vs single filter	Per-dataset architecture at $P{=}0$ vs single-branch ChebNet (Table 19 )	Lower or equal Brier for the full design on 12 of 14 datasets (equal on Questions; Chameleon $0.362$ vs $0.776$ , Squirrel $0.416$ vs $0.798$ , Roman-Empire $0.298$ vs $0.368$ ); exceptions are Texas and Wisconsin
DSS residual branch (as a unit)	Hybrid vs identically trained GCN base, 14 datasets (Table 24 )	Accuracy: Roman-Empire $+27.7$ , Minesweeper $+6.6$ ; no dataset loses beyond noise (worst $-0.8$ )

Appendix

Table 21: Component-wise evidence. Each row names one component, the comparison that isolates it, and the measured effect.

Dataset	DSS-GNN	MC-dropout	Deep Ensemble	Plain GCN
Cora	0.207 (86.63)	0.2188 (87.36)	0.2134 (87.45)	0.2125 (87.47)
Citeseer	0.308 (80.20)	0.3230 (80.03)	0.3170 (79.77)	0.3176 (80.00)
PubMed	0.156 (89.72)	0.2176 (86.20)	0.2153 (86.16)	0.2155 (86.10)
Texas	0.240 (91.31)	0.5862 (64.43)	0.5686 (62.79)	0.5758 (64.43)
Cornell	0.231 (85.11)	0.6829 (52.77)	0.6826 (52.98)	0.6839 (52.34)
Wisconsin	0.104 (93.25)	0.6370 (50.38)	0.6373 (51.25)	0.6435 (50.25)

Appendix

Table 22: Brier score (

\downarrow

) with test accuracy (%) in parentheses for DSS-GNN (Table 1 ) and stochastic GCN baselines under the same 10-split protocol (means over 10 splits).

	Accuracy (%)			Brier Score
Dataset	DSS-GNN	GCN	GAT	DSS-GNN	GCN	GAT
Cora	86.63 $\pm$ 1.26	87.45 $\pm$ 1.61	87.75 $\pm$ 1.71	0.207 $\pm$ 0.019	0.213 $\pm$ 0.014	0.222 $\pm$ 0.015
Citeseer	80.20 $\pm$ 1.28	79.84 $\pm$ 0.96	80.57 $\pm$ 1.12	0.308 $\pm$ 0.010	0.318 $\pm$ 0.007	0.353 $\pm$ 0.007
PubMed	89.72 $\pm$ 0.31	86.21 $\pm$ 0.31	85.72 $\pm$ 0.28	0.156 $\pm$ 0.005	0.215 $\pm$ 0.005	0.232 $\pm$ 0.004
Texas	91.31 $\pm$ 3.44	64.43 $\pm$ 6.80	71.80 $\pm$ 7.06	0.240 $\pm$ 0.111	0.576 $\pm$ 0.031	0.555 $\pm$ 0.038
Cornell	85.11 $\pm$ 5.12	52.34 $\pm$ 9.99	49.57 $\pm$ 12.63	0.231 $\pm$ 0.079	0.684 $\pm$ 0.080	0.631 $\pm$ 0.073

Appendix

Table 23: Node classification: test accuracy (%,

\uparrow

) and Brier score (

\downarrow

) for DSS-GNN vs. standard GNN baselines. Mean

\pm

SD over 10 splits. Best per dataset in bold .

	Standalone DSS-GNN		DSS-Hybrid		GCN base
Dataset	Accuracy	Brier	Accuracy	Brier	Accuracy	Brier
Cora	86.63 $\pm$ 1.26	0.207 $\pm$ 0.019	84.33 $\pm$ 0.97	0.241 $\pm$ 0.015	84.20 $\pm$ 1.02	0.243 $\pm$ 0.016
Citeseer	80.20 $\pm$ 1.28	0.308 $\pm$ 0.010	74.71 $\pm$ 1.05	0.375 $\pm$ 0.009	74.32 $\pm$ 1.13	0.378 $\pm$ 0.009
PubMed	89.72 $\pm$ 0.31	0.156 $\pm$ 0.005	88.27 $\pm$ 0.56	0.177 $\pm$ 0.006	87.93 $\pm$ 0.56	0.181 $\pm$ 0.006
Texas	91.31 $\pm$ 3.44	0.240 $\pm$ 0.111	45.57 $\pm$ 7.68	0.737 $\pm$ 0.034	46.39 $\pm$ 7.55	0.737 $\pm$ 0.034
Cornell	85.11 $\pm$ 5.12	0.231 $\pm$ 0.079	55.11 $\pm$ 14.03	0.656 $\pm$ 0.142	47.66 $\pm$ 12.01	0.727 $\pm$ 0.068

Appendix

Table 24: Test accuracy (%) and Brier score, mean

\pm

SD over the same 10 splits. The standalone column is Table 1 ; the hybrid and its GCN base are a matched pair (identical training, budgets, and splits; the only difference is the DSS residual).

Setting	Standalone AUROC	Standalone ID acc.	Hybrid AUROC	Hybrid ID acc.	Objective (standalone)
Cora / structure	79.79 $\pm$ 1.46 ( $+$ BN)	69.50	94.32	n/r ∗	standard (no OOD exposure)
Cora / feature	88.15 $\pm$ 1.02 ( $+$ BN)	69.17	97.60	n/r ∗	standard (no OOD exposure)
Cora / label	94.03 $\pm$ 1.09 ( $+$ BN)	89.24	94.11	n/r ∗	standard (no OOD exposure)
Amazon-Photo / structure	96.53 $\pm$ 1.41 ( $+$ BN)	91.59	99.69	n/r ∗	standard (no OOD exposure)
Amazon-Photo / feature	98.10 $\pm$ 0.11 ( $+$ BN)	92.00	99.66	n/r ∗	standard (no OOD exposure)
Amazon-Photo / label	96.53 $\pm$ 0.10 ( $+$ BN)	94.81	97.52	n/r ∗	standard (no OOD exposure)

Appendix

Table 25: AUROC (%) and ID test accuracy (%) on the 11 OOD settings. Standalone: 3 runs, mean

\pm

SD; hybrid: Tables 2 – 3 (AUROC only for the nine node-OOD settings ∗ ). The last column gives the standalone training objective.

Setting	Standalone	Hybrid	Best baseline
GOOD-CBAS / color	82.38 $\pm$ 13.17 ( $+$ BN)	88.57	TAR 87.29
GOOD-WebKB / university	36.41 $\pm$ 1.60 ( $+$ BN)	40.77	TAR 30.83
GOOD-Twitch / language	60.28 $\pm$ 0.51 ( $+$ BN)	61.21	TAR 57.20
GOOD-Cora / word	64.89 $\pm$ 0.21 ( $+$ BN)	64.81	TAR 64.73
GOOD-Cora / degree	62.60 $\pm$ 0.59 ( $+$ BN)	62.73	TAR 61.73
GOOD-Arxiv / time	65.43 $\pm$ 1.07 ( $+$ BN)	66.76	TAR 66.08

Appendix

Table 26: Shifted (OOD) test accuracy (%) on the GOOD concept-shift settings. Standalone: mean

\pm

SD over 3 seeds, configuration selected on the GOOD OOD-validation split (Appendix G.1 ); hybrid and best baseline from Table 4 .

Dataset	$\|\Delta$ Brier $\|$	Disagree (%)	Dataset	$\|\Delta$ Brier $\|$	Disagree (%)
Cora	0.0002	0.03	Roman-Emp.	0.0000	0.00
Citeseer	0.0000	0.00	Amz-Rat.	0.0000	0.00
PubMed	0.0000	0.02	Minesweeper	0.0003	0.21
Texas	0.0021	0.00	Tolokers	0.0000	0.00
Cornell	0.0000	0.00	Questions	0.0000	0.00
Wisconsin	0.0000	0.00	CS	0.0001	0.00

Appendix

Table 27: Quadrature readout versus mean-logit readout at identical checkpoints under the full 10-split protocol of Table 1 : absolute Brier difference and the fraction of test nodes on which

\arg\max\bar{p}_{i}

and

\arg\max Z_{i,0}

disagree (mean over 10 splits). Accuracy differs by at most

0.04

points on every dataset. In hybrid mode the disagreement is

0.0000

on all 14 datasets.

Dataset	BatchNorm off (control)	Table 1	BatchNorm on
Cora	86.44 $\pm$ 1.56 / 0.216 $\pm$ 0.022	86.63 $\pm$ 1.26 / 0.207 $\pm$ 0.019	85.78 $\pm$ 1.76 / 0.220 $\pm$ 0.017
Citeseer	80.27 $\pm$ 1.18 / 0.307 $\pm$ 0.010	80.20 $\pm$ 1.28 / 0.308 $\pm$ 0.010	67.03 $\pm$ 1.85 / 0.480 $\pm$ 0.024
Wisconsin	93.62 $\pm$ 1.72 / 0.109 $\pm$ 0.029	93.25 $\pm$ 3.39 / 0.104 $\pm$ 0.047	91.75 $\pm$ 4.75 / 0.134 $\pm$ 0.061
Roman-Empire	78.82 $\pm$ 1.08 / 0.300 $\pm$ 0.012	78.84 $\pm$ 0.61 / 0.300 $\pm$ 0.007	78.60 $\pm$ 0.64 / 0.300 $\pm$ 0.009

Appendix

Table 28: Test accuracy (%) / Brier, mean

\pm

SD over 10 splits, under the protocol of Table 1 at one calibration configuration per dataset, without and with per-chaos-channel BatchNorm.

Setting (covariate split)	ERM GCN	DSS-Hybrid (ERM)
GOOD-CBAS / color	49.05	53.33
GOOD-WebKB / university	13.49	29.10
GOOD-Twitch / language	42.78	52.64
GOOD-Cora / word	61.84	64.18
GOOD-Cora / degree	56.66	54.94
GOOD-Arxiv / time	44.80 ∗	70.39

Appendix

Table 29: Shifted test accuracy (%) under GOOD covariate shift: ERM-trained DSS-Hybrid (GCN encoder, validation-loss-selected configuration) versus the same-protocol ERM GCN. ∗ ERM baseline not converged within the shared budget (see text).

Symbol	Meaning
$\omega$	the single shared standard Gaussian latent; never sampled, integrated out by quadrature
$\Psi_{n}$	normalized Hermite polynomial of order $n$ ; orthonormal basis of $L^{2}(\mathbb{R},\gamma)$
$P$	chaos truncation order; $n,m=0,\dots,P$ index input/output chaos orders
$H_{n}^{(\ell)}\in\mathbb{R}^{N\times d_{\ell}}$	order- $n$ chaos coefficient of the layer- $\ell$ embeddings ( 2 ); the embedding field is $H^{(\ell)}(\omega)=\sum_{n}H_{n}^{(\ell)}\Psi_{n}(\omega)$
$\widetilde{L}_{G}$ , $T_{k}$	rescaled Laplacian $2L_{G}/\lambda_{\max}-I$ and Chebyshev polynomial of degree $k$ (Section 2 )
$G_{\mathrm{lp}},G_{\mathrm{hp}}$	learned low/high-pass filter polynomials of degrees $K_{\mathrm{lp}},K_{\mathrm{hp}}$ with coefficients $c_{k}^{(\mathrm{lp})},c_{k}^{(\mathrm{hp})}$ ( 3 )

Appendix

Table 30: Notation for the DSS layer, Eqs. ( 2 )–( 6 ).

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