WATS: Wavelet-Aware Temperature Scaling for Reliable Graph Neural Networks
Authors: Xiaoyang Li
Linwei Tao
Haohui Lu
Minjing Dong
Junbin Gao
Chang Xu
Venue: ICLR 2026 Poster (2026)
Keywords: Graph Neural Network
Calibration
Primary area: learning on graphs and other geometries & topologies
TL;DR: Calibrating Graph Neural Networks with Graph Wavelet

Abstract: Graph Neural Networks (GNNs) have demonstrated strong predictive performance on relational data; however, their confidence estimates often misalign with actual predictive correctness, posing significant limitations for deployment in safety-critical settings. While existing graph-aware calibration methods seek to mitigate this limitation, they primarily depend on coarse one-hop statistics, such as neighbor-predicted confidence, or latent node embeddings, thereby neglecting the fine-grained structural heterogeneity inherent in graph topology. In this work, we propose Wavelet-Aware Temperature Scaling (WATS), a post-hoc calibration framework for node classification that assigns node-specific temperatures based on tunable heat-kernel graph wavelet features. Specifically, WATS harnesses the scalability and topology sensitivity of graph wavelets to refine confidence estimates, all without necessitating model retraining or access to neighboring logits or predictions. Extensive evaluations across nine benchmark datasets with varying graph structures and three GNN backbones demonstrate that WATS achieves the lowest Expected Calibration Error (ECE) among most of the compared methods, outperforming both classical and graph-specific baselines by up to 41.2\% in ECE and reducing calibration variance by 15.84\% on average compared with graph-specific methods. Moreover, WATS remains computationally efficient, scaling well across graphs of diverse sizes and densities. The implementation is available at \url{https://github.com/lxy1134/WATS.git}

Published as a conference paper at ICLR 2026
WATS: WAVELET-AWARD TEMPERATURE SCALING FOR RELIABLE GRAPH NEURAL NETWORKS
Xiaoyang Li Independent Researcher
Linwei Tao
University of Sydney
Lxy2896924855 gma i com
1inwei, tace sydney edu.
Haohui Lu
Minjing Dong
Charles Darwin University haohui.. luucdu.
City University of Hong Kong
minjdong@ci tyuutci.ttcuuo edd.uu.uuuuuuuuuu.uuuououuuuuu.. edu. hk
Junbin Gao
Chang Xu
University of Sydney jurbin.gao@sydaie gwo.sydney.....ooooo...a......e edu.au
University of Sydney
c. xu@sydney, edu, au
ABSTRACT
Graph Neural Networks (GNNs) have demonstrated strong predictive performance relational data; however, their confidence estimates often misalign with actual predictive correctness, posing significant limitations for deployment in safety- critical settings. While existing graph-aware calibration methods seek to miti- gate this limitation, they primarily depend on coarse one-hop statistics, such as neighbor- predicted confidence, or latent node embeddings, thereby neglecting the fine-gran rreeggaieed structural heterogeneity inherent in graph topology. In this work, we propose Wavelet-Awaad Temperature Scaling (WATS), post-hoc calibration framework for node classification that assigns node-specific temperatures based tunable heat-kernel graph wavelet features. Specifically, WATS harnesses the scalability and topology sensitivity of graph wavelets to refine confidence esti- mates, all without necessitating model retraining or access to neighboring logits or predictions. Extensive evaluations across nine benchmark datasets with varying graph structures and three GNN backbones demonstrate that WATS achieves the lowest Expected Calibration Error (ECH) among most of the compared methods, outperforming both classical and graph-specine baselines by up to 41.2% ECE and reducing calibration variance by 15.84% average compared with graph- specific methods. Moreover, WATS remains computationally efficient, scaling well across graphs of diverse sizes and densities. The implementation is available https: https:://ithiii com
ceighbor-predicted
wwt.cccc..ccc.ce 1xy 1wT.//TT//AAA.... WWTS.aISEE.
INTRODUCTION
GNNs offer a principled approach for learning over structured data and have achieved strong empirical performance across a wide range of domains, including social network modeling (Fan et al. 2019).
traffic forecasting (Sharma et al., 2023), and healthcare applications (Gao al. 2024 Lu & Uddin
2021
They support key tasks such as node classification (Zhao et al., 2021 Sun et al. 2022). link
prediction (Zhang& & Chen 2018 Luo al. 2023), and graph-level inference (Zhang et al.) [02T]]
Godwin et al. (021). J. While GNNs are widely adopted for their representational power, their output confidence often fails to reflect true predictive reliability, which is an issue of growing concern in high-stakes domains, for instance, medical diagnosis and financial risk assessment.
Model calibration, which measures the alignment between a model's predicted confidence and its true correctness likelihood (Guo et al. 2017), is key aspect of model reliability. A well-calibrated model expected to produce predictions whose confidence scores accurately reflect the observed accuracy. For example, prediction made with 70% confidence should be correct 70% of the time. Recent findings highlight that GNNs behave differently from standard independent and identically
distributed (i.i.d.) trained models such CNNs and transformers (Melotti al. 2022 Tao al.

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2025
Wen et al. 2024). Unlike CNNs and transformers, which often suffer from overconfidence, GNNs tend to be systematically underconfident: their predicted confidence scores are consistently
lower than their true accuracy (Wang et al., 2022 Hsu et al., 2022 Liu et al. 2022).
Several recent approaches aim to address this issue through graph-aware calibration strategies, including Graph Attention Temperature Scaling (GATS) (Hsu et al.
(022). CaGCN (Wang et al.
2021). Graph Ensemble Temperature Scaling (GETS) (Zhuang et al. 2024). and SimCalib (Tang
al. 2024). These approaches typically enhance node-level calibration by integrating neighbour structural cues with their predictive status, such as confidence level.
However, they predominantly rely shallow neighborhood statistics opaque latent representations, which may result in unstable and inaccurate uncertainty estimation. As they capture only limited, local information and fail reflect the broader structural context, leading to unreliable calibration, especially for low-degree scenarios. For illustration, we provide empirical evidence that nodes with similar local statistics can exhibit different miscalibration levels across graphs. Based on the above limitation, we aim to build method that: (i) flexibly incorporates neighborhood information without relying on additional explicit pretraining status (ii) maintains high calibration performance across diverse graph domains, while remaining lightweight and post-hoc. (iii) performs temperature scaling at the node level to allow identical correction based on multi-hop structural information.
Therefore, in this work, we propose calibration framework for node classification, called WAVELET- AWARE TEMPERATURE SCALING (WATS), which introduces flexibly scaled structural features through graph wavelets. We incorporate graph wavelets because they offer principled way to capture structural information at multiple scales (Donnat et al. 2018). Our methods, wavelets spatial localization controlled by the scale parameters and k; enabling fine-grained capture of multi-hop dependencies. Differs from graph wavelet algorithms used in neural network (Donnat et al., 2018 Behmanesh et al., 2022) in that does not aim to reconstruct or smooth node features for node classification. Instead, it uses wavelet coefficients as structural signatures to indicate the node uncertainty, which allows WATS adaptively calibrate predictions based the underlying structure. helping to correct confidence errors where standard methods fall short. Our main contributions are summarized as follows:
WATS: We propose a novel post-hoc calibration framework that performs node-wiss wide temperature scaling using flexibly scaled graph wavelet features.
We show that graph wavelet features are stable and geometry-aware, encoding local graph structure without relying on potentially noisy signals such as neighboring logits or distances to labeled nodes.
Robust, interpretable structural features:
Extensive empirical validation: Across multiple graph benchmarks and GNN architectures. WATS consistently improves calibration quality, reducing ECE and outperforming both classical and graph- specific calibration baselines.
2
RELATED WORK
2.1
UNCERTAINTY CALIBRATION FOR NEURAL NETWORKS
Uncertainty calibration aims to align a model's predicted confidence with the true likelihood of correctness. Existing approaches are typically divided into two categories: in-training and post-hoc
In-training approaches incorporate uncertainty estimation within the model optimization process. For example, Bayesian Neural Networks (BNNs) and variational inference methods achieve this by imposing probabilistic distribution over model parameters (Gal & Ghahramani] 2016 MacKay| 1995 Springenberg et al. 2016). Alternative frequentist strategies, such as quantile regression (Romano) al. 2019), are also employed to generate calibrated probability estimates.
Post-hoc methods, in contrast, calibrate a pre-trained model without modifying its internal parameters. These include non-parametric techniques such as histogram binning (Zadrozny & Elkan 2001) and isotonic regression (Zadrozny & Elkan, 2002). parametric approaches that assume a specific transformation form, including temperature scaling (TS) (Guo et al.) (017) and Beta Calibration
(Kull et al. 2017), as well as distribution- free frameworks like conformal prediction (Tibshirani et al. 2019) which provides valid coverage guarantees for model outputs.
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2.2
GIRAP-SPECIHHE CALIBRATION METHODS
While post-hoc calibration methods perform well on Euclidean data with CNNs, their effectiveness declines on graph-structured data due the lack of relational modeling (Hsu al. 2022 Wang et al.
2021
To address this, several graph-aware approaches have been proposed. CaGCN Wang et al.
2021
uses GCN-based temperature predictor to incorporate structure, while GATS (Hsu al.
2022) applies attention over neighborhoods for node-specific temperatures. GETS (Zhuang et al.
2024)
introduces sparse miixure-of-exppee using degree, features, and confidence. SimCalib (Tang
et al. (224)) adds similarity- tumilarity-preeerving regularization, and et al. al.2(333) use reinforcement learning
to adapt calibration to graph structure. Beyond post-hoc methods. Yang et al. (2024a) reweight edges
during training to improve calibration, and Yang et al. (2024b) propose calibration-awar loss
targeting underconfidence caused by shallow GNNs. These methods collectively integrate graph structure and node-level signals to enhance calibration.
2.3 GRAPH WAVELET
Graph wavelets provide compact, spatially localized bases that are well suited to graph signal processing and structural representation learning. Whereas classical wavelets such Haar and Daubechies are defined on Euclidean lattices (Bruce et al. 1996; Resnikoff & Wells] 1992), graph wavelets extend these ideas non- -un-ddclid domains by leveraging the spectral properties of the graph Laplacian (Das) 2004 Shuman et al. 2013 Xu et al. 2019).
particularly versatile construction is the lifting scheme (Sweldens 1998), which can be transferred to graphs without any data-driven training. Hammond et al. (2011) further improved practicality by replacing the costly Laplacian elgendecomposition with Chebyshev polynomial approximations, enabling efficient wavelet transforms on large graphs. Since then, graph wavelets have supported a variety of downstream tasks, including graph convolutional architectures (Xu et al.) 2019) Deb et al. 2024), multimodal wavelet networks (Behmanesh et al.] 2022), community detection via scale- adaptive filtering (Tremblay & Borgnat| (014)), and diffusion- based node embeddings that capture multi-scale structural patterns (Donnat et al. 2018). All these methods and application indicates the importance of Graph wavelet in both theoretical and empirically practice
METHOD
3.1 PRELIMINARY STUDY
We address the problem of uncertainty calibration in semi-supervised node classification tasks over graphs. Let G = (V, E) denote a graph, where V is the set of nodes and & is the set of edges. The adjacency matrix is denoted by A E IR NxN. where N VI. Each node i/ V has feature I; E X, and for a subset of labeled nodes L C V. the true label V/ E {1...., K} is provided. Let X [ [. be the feature matrix and Y [y,,.... yN the label vector, GNN fe performs node classification via predicting node- wise class probabilities pi(y)) as
i arg max (1), è max (v),
where yA denotes the predicted label and e denotes its confidence. In the field of model calibration, a well-calibrated model provides confidence that aligns with the true accuracy well as
Fly = 0: = c) = Vc [0,
The measurement of model calibration can be computed via Expected Calibration Error (ECE) (Guo ct al., 2017) as [E[IP(>; 9. (2.) c). however, ECE cannot be easily computed due to limited samples. Thus, an estimation of ECE is introduced by grouping samples into M bins with equal confidence intervals as B,, { A m-1
M < ê₁ ]. where N is the subset of node that
used in evaluation A C V. Given the bin accuracy Ace(
[... Σier.,. 1( y:) and bin
confidence Conf Con) B_. EieB., R. é. the approximation can be achieved by computing the expected difference between bin accuracy and confidence
M
Bm N
ECE = Σ
AAco(Bm) - Conl(Bm))
(1)
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3.2 UNCERTAINTY ESTIMATION IN GNNS
Calibrate via One-hop Statistics - is widely adopted in GNNS, including GCN (Kipf & Welling 2016) and GAT (Veličković al. 2017), which can be simplified via degree- normalized mean aggregator as
h (1+1)
1 d, +++1
n))), d. == JEN(S)
Σ
where d; excludes the node itself, so d; + accounts for the self-loop. The final embedding h (L) induces the confidence č₁ This local aggregation implicitly determines the final prediction and the associated confidence é of node i. Although GATS confines all structural operations, including neighbor temperature aggregation, attention weights, neighbor confidence averaging 1-hop, and CaGCN and GETS stack two GCN layers to nominally reach 2-hop, each layer itself still performs only 1 1-hop aggregation. As result, these methods are unable to adaptively capture longer-range dependencies. Although these calibration techniques show effectiveness, argue that one-hop statistics only cannot provide an accurate estimation of node uncertainty in GNNs. Considering a simplified one-hop estimator of confidence:
Σ Vj.
d +1
JE(I)(N(()
where yj E (0,1) is the true label indicator for node j. Then the per-node calibration bias is
bias; = 1/2 1(: = y:)l x yi
Σ JEN(()
(3)
d +1
As shown in Eq. 3 for example, when d, 2 and the neighbor labels are [0, 1). the average is 1/3 regardless of the true label Vt, making the estimate uninformative, which means high uncertainty. In sparse or low-homophily regions, one-hop neighborhoods may carry weak or misleading signals, resulting in poor bias approximation. This motivates structure-aware calibration that aggregates richer signals beyond immediate neighbors.
A complementary critical insight into this problem was articulated by Wang et al. (2022) who uncovered a paradoxical phenomenon: as GNN depth increases, predictive accuracy diminishes, yet model confidence paradoxically rises, as shown in Figure [ Highlighting that calibration errors may not merely from local neighbor information, but from multi-scale structural effects spanning across the graph. This observation reveals the weakness of local- naialleeigbbbbbbbe based calibration: while shallow cues correlate with uncertainty, they cannot capture the non-local dependencies that drive systematic confidence misalignment in deeper GNNs.
These findings underscore the necessity of calibration approaches that go beyond local neighbor statisties. Prior works (Wang al.) 2022) Hsu et al.) 2022 Tang et al. (224] have emphasized some graph features, such as homophily and node similarity. GETS (Zhuang et al. 2024) partially
addresses this by incorporating degree embeddings into mixture-of-experts framework. While all these methods lack the ability to capture multi-hop structural patterns or incorporate unstable confidence. Therefore, addressing the above limitation and improving the calibration requires leveraging multi-hop context and structure-aware features, enabling the model to better handle both under- and over-confidence in structurally diverse regions of the graph.
3.3 WAVELET-AWARE TEMPERATURE SCALING
We propose WATS, a lightweight and effective node- wise calibration framework that can be seamlessly applied to any pretrained GNN with scalability to large graphs. Unlike conventional or graph-specific post-hoc methods that rely on global or one-hop features, WATS introduces structural perspective by leveraging graph wavelet features with tunable scales.
These wavelet representations capture rich, scalable structural signals (Hammond et al. 2011 Crovella Kolaczyk 2003), often neglected in calibration. By learning a temperature for each node based on its structural embedding, WATS aligns confidence with correctness in fine- graiiedd node-specific manner. In addition to its strong empirical performance, WATS is also architecture-agnosee making broadly applicable across diverse graph types and calibration scenarios.
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Core: Layers us ACC & Confidence
Pubmedd: Layers vs ACC & Confidence
Citeseer: Layers vs ACC & Confidence
100
100
ACC Cort
90
85
4g.
valu (66
763
24
667
ACC
52
Cord
14
LB
Layers
Figure 1: Test accuracy (ACC) and average predictive confidence of GCNs with increasing depth on Cora, Pubmed, and Citeseer. In all three datasets, deeper models exhibit decreasing accuracy while confidence increases, indicating depth- depth-induced induced miscalibration.
3.3.1 GRAPH WAVELET TRANSFORM
Traditional graph signal processing often relies on graph Fourier transform, which projects signals into the spectral domain using the eigenvectors of the normalized graph Laplacian Lym =
D-1/2AD 10DD-1-- 1/22 as orthonormal bases. Given a signal x E TR- its Fourier transform defined as x UT x and the inverse as x Už, where U contains the eigenvectors of Lym (Shuman
al. 2013). While this formulation enables spectral filtering via Uge U x, it suffers from several limitations (Hammond et al. 2011: Xu et al. 2019 Zheng et al. (02T): (1) The eigendecomposition of Lyym has high computational cost (O(N)) )) (2) generally dense, making the transform costly for large graphs; (3) The resulting filters lack localization in the vertex domain, limiting their ability
to capture localized structural patterns.
To overcome these issues, we adopt the graph wavelet transform, which retains the spectral benefits of Fourier analysis while introducing localization and sparsity. Graph wavelet bases are constructed using a heat kernel scaling function g(sA) e where > 0 is scale parameter controlling the diffusion extent. The wavelet operator defined as:
V. U
where A; are the eigenvalues of Laym The inverse transform uses g(-s)), yielding efficient localized filtering analogous to diffusion. Direct computation of W is still impractical for large graphs. To address this, we adopt the Chebyshev polynomial approximation to avoid explicit eigendecomposition.
following (Hammond et al. 2011 Xu et al. 2019). We first rescale Lym as: L Amaa Lyy
I. Amax R 2 and define Chebyshev polynomials (T k== via the recurrence: To Xo, Tı LX,. Тr 22T,T, x- Tk-2, for where X, is the initial input signal.
In our setting, we choose Xo as the log-degree to preserve structural properties while mitigating skewed degree distributions. Degree encodes a node's connectivity and its potential for information aggregation in message- -passing GNNs, and in previous work (Zhuang et al. 2024). it is shown to be correlated with miscalibration.
The wavelet scaling function g(sA) = - is approximated using K-order Chebyshev series:
((sA)
co
1
+ Σ ck (x(A)
2
with the Chebyshev coefficients c given by:
2 f cos(kf)g Amax (cos 0 +1)) dB
As stated in Hammond et al. 1).11). these are computable constants before training. The final
wavelet-transformet feature matrix S is constructed by applying this polynomial filter to the input signal:
K
1 To
=
+
T
(5)
2
1=1
This is followed by row-wise & normalization:
S
H, =
Vi E {1,
[I.
(6)
|.
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The hyper- ppper--arameter K sets the maximum receptive-field size (i.e., the number of hops considered), while the scale parameter $ governs the extent of diffusion. A small s restricts diffusion and thus accentuates local structure, whereas a large allows more extensive diffusion, leading stronger smoothing and the integration of broader, long-range context. In practice, selecting appropriate values for k and enables control over the locality and granularity of the wavelet features. This flexibility is crucial for capturing diverse structural patterns across graphs of varying density and topology.
3.3.2 NODE-WISH TEMPERATURE SCALING
Based on the extracted wavelet features, we predict node-specific temperature parameter to rescale the logits produced by the original GNN. Given the feature matrix E RN+x(K+1), we employ two-layer multilayer perceptron (MIP) capture the non-linear relationship and predict the temperatures:
T; (ofPplus(MLP(HH))
(7)
where h; is the wavelet feature vector for node i, and Softplus ensures the positivity of the predicted temperatures. This design provides flexible and efficient mechanism for uncertainty calibration across the graph. The calibrated logits are obtained via post-hoc temperature scaling:
where 2; is the original output logit from the GNN, and is the rescaled logit after calibration. The temperature predictor is trained by minimizing the cross-entropy loss on the validation set using the rescaled logits.
EXPERIMENT
4.1 EXPERIMENT SETTING
We evaluate the calibration performance of our proposed WATS method on nine widely-used graph datasets: Cora (McCallum et al. 2000), Citeseer (Giles et al. 1998), Pubmed (Sen et al. 2008). Cora- Full (Bojehevski & Günnemann 2017), Computers (Shchur al. 2018), Photo (Shchur et al. 2018). Reddit (Hamilton et af.] 2017). Roman and Tolokers(Platonov et al. (023). These datasets cover a range of graph sizes, homophily, and label complexities, providing a comprehensive benchmark for calibration analysis, detailed graph summary is shown in Appendix.
Following previous practice /Wang et al. 2021 Hsu et al. 2022 Tang al. 2024), adopt three commonly used GNN architectures as base models, which are GCN (Kipf & Welling 2016), GAT (Veličković
2017) and GCNII (Chen al.] 2020). The models are trained under semi- supervised node classification setting. After training, we perform post-hoc calibration using different methods without modifying the model parameters. Detailed training settingS of these based models
are shown in Appendix.
Follow the experiment settings (Hsu et al.) [2022] Tang et al. 2024 Zhuang et al. 2024). We randomly use 20% of nodes for training, 10% for validation and calibration training, and 70% for testing. For each method, calibration parameters are learned on the validation set and evaluated on the test set. Calibration performance is measured using the ECE with 10 bins.
We compare several post-hoc calibration methods. TS applies global temperature to all logits (Guo et al. 2017), while ETS averages predictions from multiple remperature-tuned models (Zhang et al. 2020). CaGCN uses a lightweight GCN to learn node-specific temperatures (Wang et al. 2021) and GATS employs attention-baue aggregation introduces a sparse mixture-of-er fixturr--of-exprtts that combines degree, features, and logits (Zhunng et al. 2024). WATS, our proposed method, predicts temperatures using tunable graph wavelet features and rescale logits. The detailed experiment setting are displayed in detail in Appendix.
one-hop neighbors (Hsu et al. 2022p. GETS
4.2 EVALUATION AND ANALYSTS
We evaluate the calibration effectiveness of WATS across nine benchmark datasets and three repre- sentative GNN architectures, with results summarized in Table Empirical findings demonstrate

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Table 1: Each result is reported as the mean + standard deviation over 10 runs. 'Uncalib' refers uncalibrated outputs, and 'oom' indicates out-of-memory failures where the method could not complete. Best performance on ECE are highlighted for each configuration.
Dataset
Model
Uncalib
ETS
CAGCN
GAIS
GETS
WATS
23.20 3.21
2.57 0.78
+
4.44 1.47
2.38 0.65
4.09 1.36
2,11 0.43
Citeseer
GAT
15.61
1.14
3.22 0.29
3.55 0.41
3.35 0.41
3.22
3.80
2.05
3.13 0.23
GCNII
13.32 # 10.99
7.39 4.38
7.33 = 4.48
8.65 # 1.77
8.66 1.60:22 2.62
3.42 6,68
7.27 3.49
Computers
GCN
5.94 +: 0.52
3,88 0,70
3.91 +: 0.49
2.04 + 0.34
3.34 1:34::00 0.61
2.94 +: 1.26
1.20 0.19
5.86 1.26
2.12 0.19
2.11
2.99 0.64
2.01 0.17
3.95
3.73
2.17 0.16
GONI
10.30 1.37
10.30 1 0.67
6.91 0.87
5.69 0.47
5.62 0.56
2.89 1.26
3.89 0.68
Cora
2.25
2,98
2,96
GAT
0.38 17.26
2,03 0.31
1.92 1: 0.31
2.56 1: 0.38
2,15 4.30
2.97
0.47
2,02 + 0.30
GCNII
3.28
3,38 0.92
4.35
3,43 1.07
6.76 4.94
3.23
Cora-full
GCN
27.79 + 0.22
5.06 0.10
5,00 + 0.09
3.87 # 0.22
5.13 0.10
3.11 1.95
1.94 - 0.11
2.50
0.26
2.16
GCNH
9.66 1:27
3,51 0.62
3.50 1. 0,61
3.28 1. 0.97
3.50 1: 0.59
3.01
0.88
2,92 0.98
GCN
0.23
2,45 0.22
2.47 + 0.20
1,72 0.22
2:22 0.19
3.25 1.63
1.64 0.31
Photo
GAT
3.21 0.47
181 = 0.43
2.34 + 0.50
1,71 0.10
1.80 : 0.43
1.67 3,05
1.63
0.18
GCNI
1.27
1.65
Puhmed
GCN
14.33 1: 1.20
2.55 0.38
2.81 1: 0.47
1,82 1: 0.36
2,30 0.52
2.34 0.51
1.12 0.09
GAT
10.67
+ 0.30
0.88 0.09
0.88 +
0.91 +: 0.11
0.89 0.10
0.22 0.90
0.84
0.08
GCNII
12,94 + 1,18
3.21 = 0.91
3.65 = 0.91
2.02 = 1.67
2,42 2: 0.93
2.23
0.31
2. 0.34
GCN
3.36 + 0.12
2,99 1.59
3,44 + 0.67
2.54 + 0.38
3.67 + 2.64
2.54
1.08
2,45
0,22
Tolokers
GAT
3.48 1: 0.37
3,03 0.21
3,10 +: 0.19
1.69 +: 0.15
2.891: 0.27
2.51
0.77
2,16 0.22
GONI
6.40 1: 0.62
4.39 0.61
4,02 1: 0.33
4.38 0.61
4,02 0.33
4.41 0.48
3.34 0.20
0.40
4.02 0.27
+
3.95
4.61
Roman
GAT
16.43 1.66
3,91 0.89
4.61 = 0.94
4,51 = 0.68
3.62 0.84
4.48 + 1.44
3,31 0.59
GUNI
21.00 0.42
3.61 0.65
3.61 + 0.65
4.62 0.96
4.38 0.84
4.34 1.18
2.92 1.27
Reddit
GCN
6.69 + 0.12
1.64 = 0.05
1.64 + 0.05
1.45 # 0.08
00m
2.20 0.36
0.90 + 0.05
0.16
3.29 0.08
+
GCNII
17.73 + 1.10
1.41 0.36
1.44 + 0.34
1,20 = 1.20
2.99 - 0.53
0.88 0.28
that WATS consistently achieves the lowest ECE in most of scenarios, highlighting its efficacy in leveraging localized, flexibly scaled structural information for post-hoc uncertainty calibration. For architectures like GCNII, While their initial residual connections effectively mitigate over-smoothing. the strong connections force the model to rely heavily on the original node features. In contrast, our results prove that graph wavelets are able to capture sufficient local topology information to correct these confidence levels, effectively addressing the limitations of the base models. Beyond achieving superior average ECE scores, WATS also exhibits reduced standard deviations across runs, indicating improved robustness and stability compared to existing methods. These evidence prove that graph wavelet is able to capture sufficient local topology information to correct the confidence level. Moreover, even when the base model is already reasonably well calibrated, for example, on the Photo and Computers, WATS consistently delivers further reductions in calibration error, demonstrating its ability to adaptively refine predictive confidence across a range of baseline reliability levels.
To illustrate this effect, we visualize WATS on Citeseer in Figure The reliability diagram in Figure 2a shows that the uncalibrated model systematically under confident, with predicted probabilities below empirical accuracy across bins. After calibration, the curve aligns closely with the diagonal, indicating improved confidence-acee accuracy agreement. The degree stratified analysis in Figure 2b shows that under confidence is strongest for nodes with low degree; calibration restores agreement across all degree ranges and reduces variance. Overall, WATS improves calibration and robustness, especially in structurally sparse regions. Full visualizations for the main experiments provided in the Appendix.
Furthermore, GATS's reliance full attention node's neighborhood leads to poor memory scalability and resulting in out-of -eteoo-ymoory failures on large graphs such as Reddit, while WATS remains efficient, which improve the scalability of WATS.
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Chasser Rolabilty Dagram
Calibration analysis on
1.2
Lical Gaf
2.55
0.5
0.5
0.4
0.4
0.3
Cal Cont
02-
Uncal Acc
Uncal Can
0.0
0.1
02
03
0.5
0.5
0.7
09
1.0
[3.2] [2.2]
[2.2] 22.31 [3.3]
(3.4]
14.50
[3. (6,300)
Confieecce
Degret Ranges
(a) Reliability diagram on citeseer.
(b) Degree-binned binned calibration analysis on citeseer.
Figure 2: "Uncali" refers the uncalibrated result and "Cali" refers to the calibrated result. (a) shows the reliability diagram comparing calibrated and uncalibrated outputs. The diagonal dashed line indicates perfect calibration (b) presents a degree- binnedd analysis of accuracy and confidence. Solid and dashed lines represent calibrated accuracy and confidence respectively.
4.3 ABLATION STUDY
4.3.1 DIFFERENT BASE SIGNAL
We ablate the base signal used by the WATS with log-degree, raw degree, and identity metrix GCN, while keeping all other components and hyperparameters fixed. T ECE (L) on nine datasets. Log-degree attains the best or tied-best ECE the majority of datasets, consistently improving over raw degree and matching or approaching the identity baseline. The only exception Pubmed, where the identity feature yields a marginally lower ECE. These results support the use of logarithmic transform to compress extreme degrees while preserving the connectivity ordering, thereby stabilizing learning and improving generalization from low-degree high-degree regions.
Table 2: ECE (L) comparison between log- degree, raw degree, and identity matrix base signal.
This comparison isolates the effect of the base structural signal used by the temperature regressor.
Citeseer
Computers
Cora
Photo
Cora: Cora-Full
Pubmed
Reddit
Roman
Tolokens
Log-degree
2.11 0.43
1,20 + 0.19
1,82
01.77
1.41 + 0.31
1.94 0.11
1.12 = 0.09
0.90 0.05
3.42 0.77
2,45 + 0.22 2.73 : 0.18
Degrec
3,77
1.12
Mdnniiy Matrix
2,16 1:
1.31 + 0:20
2.21 0.8
1,73 0.17
2.99 0.39
0.12
1.20
0.08
3.60 1.07
4.3.2 DIFFERENT GRAPII FEATURES
To assess the effectiveness graph wavelet features in post-hoc calibration, we conduct comparative analysis against several widely used structural descriptors, including log-degree, betweenness central- ity, clustering coefficient, and their various combinations, all evaluated under consistent framework. As summarized in Table wavelet-based representations consistently yield superior calibration performance across most datasetss While certain individual features or their combinations may perform competitively on specific datasets, they tend to exhibit limited generalizability and often result in higher calibration error overall. This highlights the insufficiency isolated structural indicators and underscores the necessity of incorporating rich, multiscale topological signals. In contrast, graph wavelet features demonstrate both effectiveness and robustness across diverse graph structures, suggesting that the information they encode captures nuanced patterns that cannot be fully replicated by aggregating conventional structural features.
4.3.3 SENSITIVITY ANALYSIS OF GRAPH WAVELET HYPER-PARAMENT PARAMETERS.
To assess the robustness of WATS, we perform an exhaustive grid search over the Chebyshev order k {2, 4} and the heat-kernel scale
{ 0.4, 0.8. 1.2. 1.6, 2.0, 2.5, 3.0, 4.0} on nine node- classification benchmarks, we visualize the changes of ECE for varying and s for Cora-full, Cora, Computers and Roman Figure (full results about hyperparameters are in Appendix).
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Table 3: ECE (L) comparison between graph wavelet and alternative structural features, where "Deg" denote log transformed degree, "Cen" denote betweenness centrality, "Clus" denote clustering coefficient, and 'oom' indicates out-of-memory failures where the method could not complete. Graph wavelet consistently outperforms other variants across most datasets.
Dataset
Graph wavelet
Deg
Clus
Deg, Cen
Cen,
Deg, Clus
Deg, Clus, Cen
Citesoor
2.11 + 0.43
3.53 + 1.16
3,10
+ 1.05
6,75
= 1.44
7.24
+ 1.80
7,11
+ 1.80
7.10
+ 1.80
7.12 = 1.69
Computers
1.20
0.19
161
0.33
+
2.75
0.82
182
0.22
2.78
2.60
0.71
2.72 0.66
Cora
1.82 1 0.27
2.42
0.72
1.86
0.32
4.77
1 0.40
4,43
1 0.63
4,51
1 0.48
4.51
0.48
4.55
1 0.40
Cora-full full
1.94
0.11
3.08
1.34
5.32
5.16
5.19 0.26
Photo
1,41 0.31
0.29
2.23
0,40
1.85 0.38
1.96
0.30
1.93
19330030
1.87
+
1.89 0.35
Pubmed
1.12 0.09
0.4
2.90
- 0.38
2.30
0.29
183
0.20
1.22 0.22
193
193 0.22
1.94 0.20
Tolokers
2.45
2.35
3.02
1.46
2.79
0.34
2.29
0.39
2.71 0.27
3,63
+ 0.88
2.58
0.21
Roman
3 0.77
4.27 2 0.68
4.04
+ 0.50
3.88
= 0.65
3.60
0.77
3.71 = 0.78
4.03
+ 0.39
3.89 = 0.58
Reddit
0.90 - 0.05
1.58
0.17
ocm
00m
00m
ocm
0om
oom
hyperpsrameter anclysis for Cors-full
Hyperparameter analyds for Corn.
Hypersaromete analvsia for Computiors
Hyper anelysls for Rom
120
2200
175
380
Figure 3: Sensitivity analysis of wavelet pyper-parameters. Each plot shows the ECE scores on different datasets with varying wavelet scale parameter (x-axis) and polynomial order k. Each line represents a different Chebyshev order k: blue for k = 2. orange for k = 3, green for k = 4 and grey for previous SOTA.
On highly homophilous graphs, WATS exhibits reduced hyperparameter sensitivity: variations in the Chebyshev order l and the diffusion scale s induce only minor changes in ECE, yielding flatter performance curves and stronger robustness when > 1.2 on high-homophily graphs, such as Cora and Computers. In contrast, on low-homophily graphs. like Cora-full, calibration is more sensitive to s and k. On heterophilous graphs, calibration quality depends on both i: and s: small orders constrain the receptive field and miss meale structure, whereas large orders amplify noise propagation and degrade calibration. These observations yield practical guidance that favors moderate spectral scale and mid-level polynomial order. A sensible default for new graphs is k = 3 with 2.0.
However, WATS surpasses the previous SOTA across a broad and practical range even away from the optimum; for example, k E {3, 4) with moderate already delivers consistently lower ECE, underscoring the robustness and generality of wavelet- based structural signals for calibration.
4.4 COMPLEXITY ANALYSIS
We further compare the complexity with other post-hoc calibration method to prove the computational efficiency. Our method consists of two main components: graph wavelet feature extraction and two- layer MIP for temperature prediction. Let K' be the Chebyshev polynomial order. Fach Chehyshey term requires a sparse matrix multiplication, leading to a total time complexity of O [[[KE|= + /I where [g' and |V] denote the number of edges and nodes, respectively. The first term accounts for K sparse multiplications over the Laplacian, while the second accounts for the intermediate tensor concatenation and normalization steps. The wavelet features of each node (dimension K + 1) are passed through a two-layer MLP with hidden size h. The per-node computation costs O((K + 1)h). and thus the total cost over all nodes is: O(V Combining the above, the overall time complexity of our method is:
O El + DIk + [4 = O (V(k)).
Compared to CaGCN Wang et al. 1 with O + F) and GATS (Velčkovii et al. 2017 with O(E + [== where F' stands for the dimension of the node hidden features and // is the number of independent attention heads, our model is significantly more efficient, especially when F is large or multi-head attention is used. GETS (Zhuang et al.) 2024) incurs higher cost due expert selectionmechanism. which introduces additional complexity by selecting the top-k experts per node. This results in an overall complexity of O + F2 + |V] M F), where k & M.
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In practice, the wavelet transformation can be precomputed and reused static input. The wall- clock time and memory usage of WATS and other baseline methods across graph datasets of varying complexity reported in the Appendix.
LIMITATIONS AND FUTURE WORK
Limitations: The scope of our current study is primarily confined to node classification. Furthermore, WAT' relies on the assumption that topological signals inherently correlate with model logits. In scenarios where this correlation is weak spurious, relying wavelet-derived temperatures could potentially degrade calibration reliability
Future work: We aim explore the integration of structurally similar, albeit spatially distant, neighborhoods to incorporate a broader global structural context. Specifically, this entails capturing and aggregating nodes that share similar structural embeddings regardless of their multi-hop distance. We hypothesize that this approach could further enhance both calibration performance and model robustness, provided that the incorporation of global information carefully regularized to avoid introducing extraneous noise. Additionally, we plan to investigate the generalizability of graph wavelets in improving calibration and predictive performance across wider array of graph-learning tasks, including edge prediction and dynamic graph modeling.
CONCLUSION
We introduce WATS, lightweight post-hoc calibration framework that assigns node-specific tem- peratures from graph wavelet features. By leveraging structural representations, WATS captures diverse structural patterns and implicitly broadens each node's 's receptive field, improving post-hoc information use with minimal overhead. Across nine benchmarks and three GNN backbones, WATS consistently attains the lowest ECE and markedly stabilizes calibration, yielding more reliable predictions, especially in high-risk settings.
7 ACKNOWLEDGEMENT
This work was supported by Sichuan Science and Technology Program Y025Y1H20162.
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A USAGE OF LARGE LANGUAGE MODEL
In writing this paper, we used a Large Language Model (LLM) solely as a writing assistant to enhance linguistic quality, not generate substantive content. The L.M was employed improve readability, concision, and academic tone: to correct grammar, spelling, and punctuation; and to strengthen logical flow and transitions throughout the text.
B EXPERIMENT SETTING
We randomly conduct the train test split 10 times for each dataset with identical random seed. We employed the GATS, GETS and CaGCN based on their paper and code. The hyperparameters for backbone GNNs training are based on the complexity of graph data. The detail is given below Table and
Table 4: Summary of training parameters of GCN and GAT
Dataset
Hidden Dim.
Dropout
Epochs
Learning Rate
Weight Decay
Citeseer
0.5
200
1 10 2
x 10
Computers
64
0.8
200
1x 10
X 10-3
Cora-full
64
0.8
200
1x 10
1 10-3
Cora
0.5
200
1 10
X 10
Photo
64
0.8
200
1 10
1x 10-3
Pubmed
16
0.5
200
1x 10
5 X 10
Tolokers
16
0.2
200
1 10-2
X 10-4
Roman
0.2
200
1 10-2
10-4
Reddit
16
0.5
200
1x 10
10-4
Table 5: Summary of training parameters of GCNII
Dataset
Layers
Hidden Dim.
Dropout
Epochs
Learning Rate
Weight Decay
Citeseer
16
0.5
200
X 10 2
X 10 -4
Computers
16
64
0.8
200
1 X 10-2
1 10
Cora-full
16
64
0.8
200
1 1 10
1 X 10
Cora
16
0.5
200
1 1 10
x 10 -4
Photo
16
64
0.8
200
1 1 10-2
1 10-3
Pubmed
16
16
0.5
200
1 X 10-2
X 10
Tolokers
16
16
0.2
200
1 X 10-2
X 10 -4
Roman
16
16
0.2
200
1 X 10-2
X 10 44
Reddit 16
16
0.5
200
1 X 10-2
X 10
Full details of WATS in calibration is given be on Table 0 Hidden dimension and drop out are chosen based the data complexity. The hyperparameter of graph wavelet and chosen based on the Ablation study.
Full details of the chosen datasets is given Table It reports the number of nodes, edges, average node degree, input feature dimensions, and number of classes for each dataset. These datasets cover diverse range of graph sizes. densities, and classification tasks. This diversity ensures comprehensive evaluation of the proposed method under varying structural and semantic conditions.
Computational Environment. All experiments conducted using the following environment with PyTorch 2.4.0 (Python 3.11, CUDA 12.4.1), Hardware: NVIDIA RTX 4090 GPU with GB RAM Runpod cloud service (Ubuntu 22.04)
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Table 6: Calibration settings for WATS
Dataset
Hidden Dim.
Dropout
k
Citeseer
32
0.4
3
0.8
Computers
64
0.4
3
2.5
Cora-full
128
0.2
4
0.4
Cora
16
0.95
4
0.4
Photo
32
0.4
4
0.8
Pubmed
32
0.4
4
0.4
Tolokers
32
0.6
4
Roman
32
0.2
4
2.5
Reddit
64
0.4
4
3
Table Summary of selected datasets
Dataset
#Nodes
#Edges
Avg. Degree
#Features
#Classes
Citeseer
3,327
12,431
7.4
3,703
Computers
13,381
491,556
73.4
767
10
Cora
2,708
13,264
9.7
1,433
Cora-full
18,800
144,170
15.3
8,710
70
Photo
7,487
238,087
63.6
745
Pubmed
19,717
108,365
10.9
500
Tolokers
11,758
1,038,000
176.6
10
Roman
22,662
65,854
5.8
300
Reddit
232,965
1 114,848,857
98.5
602
41
C TIME AND PEAK MEMORY USAGE
We report the wall-clock time and memory usage WATS and other baseline methods across graph datasets of varying complexity in Table and Table respectively.
Table 8: Comparison of Computation Time (seconds)
WATS Feat.
WATS Calib Time (s)
GETS Calib
GATS Calib Time (s)
TS Calib
CaGCN Calib Time (s)
Dataset
Time (s)
Time (s)
Time (s)
Cora
0.0624
1.0134
22.2517
2.1201
0.3941
4.1283
Computer
0.2374
0.7716
10.8242
5.0126
0.2408
4.9372
Cora-Full
0.3134
1.0461
9.2106
5,2835
0.4754
4.7652
Reddit
42.6636
1.3149
45.2529
NAN
1.1054
20.8815
Table 9: Comparison of Memory Usage
WATS Feat.
WATS Calib
GETS Calib
GATS Calib
TS Calib
CaGCN Calib
Dataset
Memory
Memory
Memory
Memory
Memory
Memory
Cora
207.31 MB
207.92 MB
97.39 MB
95.00 MB
94.65 MB
94.65 MB
Computer
207.83 MB
211.62 MB
195.91 MB
313.41 MB
172.45 MB
172,46 MB
Cora-Full
1955.27 MB
1963,90 MB
1401.74 MB
1349.16 MB
1344.47 MB
1343.71 MB
Reddit
12456.00 MB
4322.00 MB
5938.96 MB
>17.5 GiB
3855.37 MB
3857.81 MB
D HYPERPARAMETER ANALYSIS RESULTS
Here the full result for the experiment on hyperparameter analysis. Tables 10 to report the cali- bration performance measured by ECE of WATS under varying graph wavelet hyperparameters, specif-
16

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Published as a conference paper at ICLR 2026
ically the Chebyshev order & 3,, 4] and diffusion scale E {0.4, 1.2, 1.6, 2.0, 3.0, 4.0). For each dataset, ECE values are presented across a range of s values.
Table 10: ECE (4) for different diffusion scales with Chebyshev order K-2.
s.4.4
s.0.8
s.1.2
1.6
2.0.
2.2.5
sw3
Citeseer
2.54 0.86
2.50
0.86
2.46 + 0.88
2.47
0.89
2.47
0.91
2,48
0.92
2.50 + 0.93
2.51
0.94
Cora
2.00 0.26
1.99
0.26
1.99 + 0.27
2.02
20000.2 0.27
2.01
0.28
2.01
0.27
2.02 - 0.28
2.02 = 0.28
Computers
0.26
Pubmed
1.17 0.19
1.19 0.19
1.20 + 0.20
1.20=0.17 0.17
1.21
0.188
1.20-0.: 0.17
1.18 + 0.13
1.17 0.13
Reddit
1.19 1.19 1:9:::0.000 0.09
1.15 0.08
1,10 0.11
1,88 0.12
1.07
0.13
1.03
0.12
0.99 1 0.08
0.96 0.11
Cora-full
1.95 0.21
2.25
0.25
2.46 + 0.24
2.51
0.29
2.58
+ 0.35
2.55-
0.37
2.72 +
2.68 0.41
Photo
1.75 1:2:::0..3
1.76 0.22
1.71 + 0.24
-.22
1.70
0.25
1.71 0.22
1.69 0.22
1.71 0.23
Roman
4.32
-18=
3.08
3.73= 0.70
3.89
3.80
0.57
3.80 0.56
3.97 0.36
Tolokers
0.11
2,71 0.17
2.06
2 0.17
2.73
0.13
0.10
2.76
0.17
2.70 2: 0.21
2.74
Table 11: ECE (4) for different diffusion scales with Chebyshev order K333
s=0.4
s=0.8
s=1.2
1.6
2=2.0
2=2.5
8m3
Citeseer
2.21 2: 0.49
2.11
0.43
2.18
+ 0.52
2.25 0.59
2.25
0.61
2.24
0.62
2.25 4: 0.61
2.30 = 0.65
Cora
2,22 0.87
2,25
1.00
2.00
0.28
2.26
1.99
0.24
1.99
0.24
1.99 1: 0.24
2.00 0.26
Computers
1888 1 0.39
1.45
1.39
-.30=
1.25
0.199
1.19
0.20
Pubmed
1.14
0.18
1.18 1.18
1.17-012
1.18
1.18=0.13
1.18
0.12
1.19
Reddit
1.08
=
1.22
1.10=0.5
0.009
0.16=0.10
0.93
0.06
0.00
Cora-full
1.98
2.24=0.24
2.25 0.14
2:3=
0.16
2.36= 0.33
2.24 0.16
0.31
Photo
1.29
1.79
1.72
181=011
1.79
1.70=030
1.79
0.22
1
Roman
0.14
3.82
3.95
-.50
3.79
-.94
Tolokers
2.67
0016
2.76 0.12
2.71
2.61
=.25
2.50 = 0.27
2.66 0
Table 12: ECE ({) for different diffusion seales & with Chebyshev order K=4.
s=0.4
s=0.8
8=122
1.6
2=0
2=2.5
8=3
8=1
Citeseer
2.77
1.03
2.68
1.02
2.63
1.05
2.56
0.96
2.73
1.08
2.73
1.02
2.74
1.04
2.74
1.02
Cora
1.98 0.26
0.26
1.99
1.98
0.25
Computers
1.93 +: 0.50
1.44
0.25
1,30
+ 0.20
0.25 1.28
1.23
8 0.21
1.30
0.25
1.26 + 0.22
1.29 = 0.22
Pubmed
1.12
1.18 0.12
1.18 2 0.12
1.19=0.13
1.21
0.13
1.19=
1.19 2
0.11
Reddit
1.17=:0000
1.19 0.11
1.18
0.12
1.13
0.18
1.02
0 0.17
0.93
0.16
0.90
1 0.05
0.91 0.14
Cora-full
0.11
2.03=0.1
2.18 0.23
2.29-0.23 0.23
2.262:222
2.23
2,22 2.21 0.21
1.64
0.25
Roman
4.33
1.09-0.26
1.08 - 0.25
-.21
1.08
3.43
3.60
3.60 0.70
Tolokers
2.67
2.74 0.18
2.69 - 0.22
2.78
0.22
2.72 2.72 0.27
2.51
0.22
2.45 2 0.22
2.50 0.23
E FULL WATS VISUALIZATIONS
We provide the full visualizations of the calibration performance for WATS. Figures to [1 illustrate the calibration performance WATS on the other datasets. Each figure includes (a) a reliability diagram showing the alignment between predicted confidence and actual accuracy, and (b) degree- binned analysis comparing confidence and accuracy before and after calibration. Results show that WATS significantly improves calibration and reduces the discrepancy between accuracy and confidence across all degree ranges and all confidence levels. Error bars indicate standard deviation over 10 runs.
17

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Published as a conference paper at ICLR 2026
Car Ralabiliy Diogram
analysis
2.0
Lical
0.5
0.4
Krated Eranec Canseanta
0.1
02
03
0.5 Confidecca
0.5
0.7
09
1.0
Degrne Rangee
(a) Reliability diagram on Cora.
(b) Degree-binned calibration analysis on Cora.
Figure 4: Calibration performance of Cora dataset.
Calibration analysis on computers
Computers Reliabliity Dagram
Cal
0.3
0.7
0.5
20.6
EXStA AUCAAAY
USCA
(2649)
(7--000)
0.3
02
1.0
(b) Degree-binned calibration analysis Comput- ers.
(a) Reliability diagram on Computers.
Figure 5: Calibration performance of Computers dataset.
Photo Diagram
analysis on photo
3,0
Uncall
0.55
0.4
0.2
Beoted Confeerce
Docod besiee
13:997
p
1o0417
10417
0.1
02
03
0.5
0.5
0.
0.0
1.0
Cenfidence
Degine Ra Abe
(a) Reliability diagram on Photo.
(b) Degree-binned binned calibration analysis on Photo.
Figure Calibration performance of Photo dataset.
18

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Published as a conference paper at ICLR 2026
analysis
Roliabilty Dagram
1.0
Calbreted
10
0.3
Lncal
Unodireete Accurecy
2.4
0.3
0.-
0.3
(75-38)
0.0
011
02
03
0.4
0.5 Confidence
0.5
0.7
08
0.9
1.0
Degree Rarges
(h) Degree-binned calibration analysis on Cora- full.
(a) Reliability diagram Cora-full.
Figure 7: Calibration performance of Cora-full dataset.
Reddit
analysis
1.0
Lncal
0.5
2.6
0.6
0.5
0.4
0.4
0.3
0.2
brated ACCURCC
Uncill bratcs ACCUTEDY
00
DI.M
the 11m1l (190ua00))
0.1
02
03
0.5 Confidesca
0.5
0.7
0.5
1.0
Degine Rampee
(a) Reliability diagram on Reddit.
(b) Degree-binned calibration analysis on Reddit.
Figure 8: Calibration performance of Reddit dataset.
Pubmed Rellabllty Diagram
Callbration
Call
0.7
0.5
0.5
0.8
0.6
ACCPECY Bestee
Dicod Beviee Coshaoon
0252
12-4
23:m
Deg Rar 935
(a) Reliability diagram on Pubmed.
(b) Degree- calibration analysis on Pubmed.
Figure 9: Calibration performance of Pubmed dataset.
19

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Published as a conference paper at ICLR 2026
analysis
Utccal Call
1.0
Calbreted
Unsdirrete Accuney
0.6
Ro0002
14
Prodicted probaaiiltt
Degree Rangee
(a) Reliability diagram on Roman.
(b) Degree-binned calibration analysis on Roman.
Figure 10: Calibration performance Roman dataset.
Tolokers Rellability Diagram
Calibration analysis on tolakers
A00LD0)
0.5
0.4
0.4
Ca Reotee Co-fecece
0.2
Dscod Beaiee Contaranr
12-4
12:47
(744227
1222 1961
641
04
08
probablityy
(a) Reliahility diagram on Tolokers.
(b) Degree-binned calibration analysis Tolokers.
Figure 11: Calibration performance of Tolokers dataset.
20


Reviews:
[{"review_id": "EvFgKI3K8M", "reviewer": "Reviewer_Cr2Z", "summary": "The authors aim to address the calibration issue in node classification. Existing methods only consider local topology when performing calibration, which limits their effectiveness. To overcome this issue, the author proposes using graph wavelets, which can capture information from more distant nodes, to determine the temperature for calibration. Experimental results demonstrate that the proposed method outperforms existing approaches on both GCN and GAT models.", "questions": "Please see the Weakness.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Proposes a method that can consider beyond one-hop neighborhood information\n- Demonstrates the effectiveness of the proposed method across datasets with diverse characteristics\n- The approach is lightweight, as it only requires running a 2-layer MLP on the validation set", "weaknesses": "- The method novelty of the proposed approach appears limited. Graph wavelets are already well-established, and in this work, they are applied to the calibration setting with only minor engineering and no specialized adaptation, which diminishes the originality of the method.\n- The method involves training a MLP on the validation set, where the current split allocates 10% to validation and 20% to training. This is a relatively large validation portion, and in practical scenarios, such a large validation set may not be feasible. Consequently, the performance might degrade as the validation size decreases.\n- Experiments are conducted only on relatively old architectures such as GCN and GAT, with no evaluation on recent GNN architectures, particularly those that include skip connections. Thus, it remains unclear whether the proposed method generalizes well to modern GNNs.\n- Although the paper emphasizes the advantage of capturing beyond one-hop information, the evaluation is limited to 2-layer GCN and GAT, making it uncertain whether the method performs equally well on deeper architectures.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lCEQzRzJrM", "reviewer": "Reviewer_md66", "summary": "This paper focuses on the problem of calibrating model outputs for GNN architectures. Recent work shows that GNNs tend to be underconfident, which sets them apart from a lot of the NN literature and calls for bespoke strategies. A few strategies have already been suggested, that essentially try to predict the temperature parameter based on graph features. The central contribution of this paper is to replace this by a graph wavelet transform. The authors then show that their method performs well (where performance is measured using ECE) in several graphs, and outperforms competitors.", "questions": "(1) The characterization of conformal prediction as an \"in-training approach [that]  uncertainty estimation within the model optimization process\" seems off. CP acts as a wrapper, with no need to train any algorithm, so it doesnt interfer with the optimization process at all. It belongs to the post hoc methods.\n\n(2) Could you explain this sentence: \"GNNs tend to be systematically underconfident: their predicted confidence scores are consistently\nlower than their true accuracy\" --- what does \"predicted confidence scores\" mean, computed how?\n\n(3) Tables 7-9 seem to show that the performance of the method is independent of the temperature parameter $s$ and $k$ -> any insight from the theory perspective?\n\n4) For the ablation study, why not use the features? (Im not necessarily asking for more experiments, just curious to hear why it was not considered)? \n\n\nNotes: Line 68: \".Differs\" --- the full stop there seems to be an error.\nLine 269: \"The hyper-parameter k sets the maximum receptive-field size\" -> k should probably be K?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1) The paper's exposition is clear.\n2) The results on experimental data look convincing.", "weaknesses": "As a disclaimer, I am not an expert on calibration (let alone on the calibration of GNNs), so here are a couple of points that I was hoping the authors could comment on.\n\n1) Overall, the idea of using the graph structure to calibrate the uncertainty is fairly natural, but I wonder if the authors could try to phrase mathematically what they are trying to achieve.\nFor instance: I could imagine a setting where the GNN doesnt account perfectly for the signal on the graph, and as a consequence, the residuals are correlated.  For instance, let's simplify things by assuming that you're doing regression to predict $y_i = (\\alpha X_i + \\beta (\\sum_{j \\sim i} X_i/ d_i)  + \\epsilon_i$  but that you're not using the graph structure, so $\\hat{y}_i = \\hat{\\alpha} X_i$.\n\nConsequently, using the wavelets that you are using is a much better idea than using predictors (like degree), as the wavelets will encode some deeper structural information (e.g where the errors are relative to one another) that will probably allow to calibrate more accurately the error, without having the overhead of expensive computations like GAT, etc.  \nAnother scenario could be that you've explained the graph structure away using your GNN, but the errors $\\epsilon_i$ are not iid (network effects --- e.g. friends talk to one another and exert mutual influence on each other). In this case, maybe a way of formulating the problem is that each node has a neighborhood effect: \n$y_i = (\\alpha X_i + \\beta (\\sum_{j \\sim i} X_i/ d_i)  +  u_i + \\epsilon_i$ \nwhere $u_i \\sim N(0, \\Sigma)$ where $\\Sigma$ encodes dependencies between nodes. It is probably the case that the architecture you're suggesting is able to find dependencies that allow you to account for this random effect.\nIs any of these close to what you were envisioning for the effect of the wavelets? Can you explain why the wavelet is a better idea?\n\n2) The authors discuss the computational complexity of the method. It would have been insightful to compare the running time of each methods as well, on top of reporting the ECE.\n\n3) It looks like the model doesnt crucially depend on K or s. What if, instead of the spectral wavelets, the authors used the Laplacian embedding of the graph --- that would prevent them from computing huge svd (just keep the top K) --- would we have similar results?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IbCPwpIXKN", "reviewer": "Reviewer_jtMi", "summary": "This paper introduces WATS, a post-hoc calibration framework for GNNs that learns node-wise learnable temperatures from graph wavelet-based structural features. By incorporating hierarchical structural signals, WATS improve miscalibration of GNNs compared to one-hop-based temperature scaling approaches.", "questions": "- Does similar trends of accuracy and confidence according to different numbers of layers in Figure 1 persist under heterophilous graphs?\n- Could the authors show the wall-clock time analysis and memory consumption of the proposed method compared with baselines?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The application of graph wavelet transform in calibration domain is interesting.\n- WATS substantially surpasses prior work in both small- and large-scale datasets.", "weaknesses": "- The discussion in Section 3.2 is a bit confusing. Even if model confidence exhibit high uncertainty in cases of disagreeing neighbors, a single node-level bias itself may not be problematic, since calibration error is inherently a population-level quantity, not measurable at a single-node level in practice. Could the authors provide additional explanation on this?\n- According to Figure 1, deeper GCNs become more confident but less correct, which seems that increasing the receptive field rather worsens calibration. Furthermore, such phenomenon is likely attributable to over-smoothing rather than multi-scale connectivity or structural differences. This interpretation appears sometwhat misaligned with the design philosophy of WATS, which assumes that calibration errors arise when the calibration network fails to capture higher-order context. Could the authors clarify this?\n- Minor) Found a typo in line 128.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6TVy5ngrLX", "reviewer": "Reviewer_Wzpk", "summary": "WATS proposes a post-hoc, label-free calibration method for GNNs using graph wavelet features. A small MLP predicts a per-node temperature, guided by local spectral characteristics computed via heat-kernel wavelets. This allows finer calibration than global temperature scaling and mitigates overconfidence in deep GNNs.", "questions": "1. How robust is WATS to heterophily or noisy topology?\n2. Can wavelet-based calibration transfer to edge or graph classification tasks?\n3. Does precomputing wavelets limit adaptability to dynamic graphs?\n4. Could you briefly explain the intuition using wavelets to design a calibration algorithm in GNN?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Simple yet effective node-wise temperature scaling without retraining the base GNN.\n2. The wavelet embedding captures multi-scale structural cues for confidence correction.\n3. Extensive experiments across datasets and backbones with consistent ECE reduction.\n4. Clear complexity and sensitivity analysis, including practical hyperparameter guidance.", "weaknesses": "1. Relies on the correlation between structure and calibration error, which may not hold for heterophilous graphs.\n2. The wavelet scale parameter tuning could be expensive for a large graph", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)