Geometric Logit Decoupling for Energy-Based Graph Out-of-distribution Detection
Authors: Min Wang
Hao Yang
Qing Cheng
Jincai Huang
Venue: NeurIPS 2025 poster ()
Keywords: out-of-distribution detection; confidence calibration
Primary area: social_and_economic_aspects_of_machine_learning

Abstract: GNNs have achieved remarkable performance across a range of tasks, but their reliability under distribution shifts remains a significant challenge. In particular, energy-based OOD detection methods—which compute energy scores from GNN logits—suffer from unstable performance due to a fundamental coupling between the norm and direction of node embeddings. Our analysis reveals that this coupling leads to systematic misclassification of high-norm OOD samples and hinders reliable ID–OOD separation. Interestingly, GNNs also exhibit a desirable inductive bias known as angular clustering, where embeddings of the same class align in direction. Motivated by these observations, we propose GeoEnergy (Geometric Logit Decoupling for Energy-Based OOD Detection), a plug-and-play framework that enforces hyperspherical logit geometry by normalizing class weights while preserving embedding norms. This decoupling yields more structured energy distributions, sharper intra-class alignment, and improved calibration. GeoEnergy can be integrated into existing energy-based GNNs without retraining or architectural modification. Extensive experiments demonstrate that GeoEnergy consistently improves OOD detection performance and confidence reliability across various benchmarks and distribution shifts.

Geometric Logit Decoupling for Energy-Based Graph Out-of-distribution Detection
Min Wang, Hao Yang, Qing Cheng, Jincai Huang"
National University of Defense Technology
(wangminwm, yanghao, huangjincai] Gnudt. edu. en;
sgggps@163 dgggps@163.com com
Abstract
GNNs have achieved remarkable performance across a range of tasks, but their reliability under distribution shifts remains significant challenge. In particular, energy-based OOD detection teathodss-hiic compute energy scores from GNN lugits-Suffer from unstable performance due to fundamental coupling between the norm and direction of node embeddings. Our analysis reveals that this coupling leads to systematic misclassification of high-norm OOD samples and hinders reli- able ID-OOD separation. Interestingly, GNNs also exhibit desirable inductive bias known angular clustering, where embeddings of the same class align in direction. Motivated by these observations, we propose GeoEnergy (Geometric Logit Decoupling for Energy- -Baeed OOD Detection), plug-and-plaa framework that enforces hyperspherical logit geometry by normalizing class weights while preserving embedding norms. This decoupling yields more structured energy dis- tributions, sharper intra-class alignment, and improved calibration. GeoEnergy can be integrated into existing energy-based GNNs without retraining architectural modification. Extensive experiments demonstrate that GeoEnergy consistently improves OOD detection performance and confidence reliability across various benchmarks and distribution shifts.
1 Introduction
Graph Neural Networks (GNNs) [ 2 have achieved remarkable success in diverse applications, including social network analysis 33. drug discovery [4, and traffic forecasting 76. However, their effectiveness relies the assumption that training and testing data are independently and identically distributed (i..d.), an assumption that often fails real-world scenarios. When deployed in open environments, GNNs frequently encounter -ittr---iittiittioon (OOD) inputs [ samples that differ significantly from the training distribution. The inherent structural dependencies graph- structured data further complicate OOD detection, node relationships can propagate distributional shifts across the graph. Consequently, conventional GNNs often produce overconfident yet incorrect predictions for OOD samples (9 TO) undermining model reliability in safety-critical applications. To address this challenge, a growing body of research has explored OOD detection in graph-structured data. Generative models like GraphDE
[IT) model uncertainty via variational inference, while Bayesian methods such as GPN [12 propagate uncertainty using posterior distributions. More
recently, energy-based methods [13 4 15 have gained attention for their simplicity and post-hoc applicability. These approaches define energy scores based on the logit outputs of GNNs and flag high-energy samples as potential OOD nodes. However, our analysis reveals that the effectiveness of energy scores hinges critically on the geometric structure of the logits themselves.
* Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Figure 1: Energy distributions under GNNSafe with and without GeoEnergy. Across different graph shifts, integrating GeoEnergy significantly improves ID-OOD energy separation and stabilizes score distributions, leading to more reliable OOD detection.
Specifically, current energy-based methods typically compute the logit for class j as an inner product: Sj = W. h;, where W, is the class weight vector and h; is the node embedding. This formulation implicitly incorporates both the direction and the magnitude of h4, In graph settings, however, the norm |h:|| is highly sensitive to various structural factors- -scch as degree heterogeneity, local noise, and the stochastic nature of message logit values. As a result, OOD nodes with large embedding norms may be incorrectly assigned low energy scores and misclassified as ID. We refer to this failure mode Further our analysis in section reveals that GNNs tend to naturally promote a desirable geometric structure: angular clustering, where node embeddings from the same class align closely in direction, while embeddings from different classes diverge. This directional separability implies that angular similarity offers more stable and semantically meaningful signal for classification than the raw inner product. These insights motivate us to explicitly decouple the norm and angle components in the logit computation. By preserving angular structure while regularizing the influence of magnitude, we aim to produce cleaner energy landscapes and more reliable ID-0OD discrimination.
-whihh often causes unpredictable shifts in
cmppling-inuucee misdetection.
Motivated by these geometric insights, propose GeoEnergy (Geometric Logit Decoupling for Energy- Based OOD Detection), a principled approach that restructures the logit space through hyperspherical projection. By normalizing class weights to enforce hyperspherical geometry while preserving the embedding norm as a confidence signal, GeoEnergy disentangles angular similarity from magnitude. This yields sharper intra-class alignment, larger inter-class angular margins, and more reliable energy landscapes-witt without altering the GNN backbone or training process. GeoEnergy is fully compatible with existing energy-basee based frameworks and can be plugged in without architectural changes. In details, our contributions are summarized as follows:
Geometric diagnosis. We revisit the geometric structure of GNN representations and identify two key phenomena-angular clustering and coupling-indee induced misdetection that fundamentally limit the reliability of energy-based OOD detection.
Simple yet principled solution. We propose GeoEnergy. a lightweight hyperspherical logit formulation that decouples direction and norm by normalizing class weights while preserving confidence sharpness, yielding structured and well-calibrated energy landscapes.
Modular integration. GeoEnergy can be seamlessly plugged into existing energy-based GNN frameworks without architecture modification.
2 Revisiting Geometric Properties of GNNs
The effectiveness of energy-based OOD detection in GNNs closely tied to the geometry of node embeddings. In this section, we uncover two empirical phenomena that highlight both the benefits and limitations of this geometry. These findings motivate our GeoEnergy approach, which addresses the coupling between embedding norm and angular similarity without modifying the GNN architecture.

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Observation 1: Coupling-Inducer Misdetection. In GNNs, logits are typically computed as:
[ - W h
W
h.
c c0s(0
weight norm
Jeature norm
entangling both the class weight norm and feature norm with the angular similarity. In graph- based settings, these norms are prone to fluctuations due to degree heterogeneity, noisy aggregation, and training dynamics, resulting in unstable logit values. Consequently, energy scores derived from these logits become sensitive to norm variations and less reflective of semantic alignment. Empirically, we observe that under various distribution shifts- feature-level, label, and domain-this instability gives rise to a persistent failure mode: OOD nodes with large norms may yield deceptively low energy scores despite poor directional alignment with class prototypes. As shown in Figure d), this manifests as a "low-energy tail" of OOD samples intruding into the ID region, causing systematic false negatives. This coupling-induced misdetection underscores fundamental weakness of conventional energy-based detection; when norm and direction interact nontrivially, ID-OOD separation breaks down.
Observation 2: Angular Clustering. We em- pirically observe that GNNs promote angular clustering in their node embeddings: same- class nodes exhibit high cosine similarity and align closely in direction, whereas different- class nodes show lower cosine similarity and more dispersed orientations. As shown in Fig- ure across datasets (Cora and Citeseer), the cosine similarity distribution of same-class pairs peaks sharply near 1, while that of different- class pairs more diffuse. This implies that directional information alone provides strong semantic signal, making angular similarity a robust and discriminative feature for classification.
Cora
Citeserr
Diflirett Cliss
Diffcren Clas
Same
g.33
Casine Simitarity
Cosine Sinittarity
Figure 2: Illustration of Angular Clustering GNNs.
These observations highlight geometric tension in GNN embeddings that undermines energy- based OOD detection. To resolve this, we propose GeoEnergy, a seometry-aware formulation that decouples angular similarity from magnitude. By normalizing class weights, GeoEnergy aligns the decision process with the intrinsic angular structure of GNNs, while retaining feature norms to preserve confidence sharpness. This disentangled design yields stable energy scores and significantly improves ID-OOD separability without modifying the GNN backbone or training strategy.
3 Method
Let G (V, E, x) denote a graph, where V= (uI,..., uN) is the node set, & C x V is the edge set, and X II..... xN] E NxDD denotes node features. The graph structure is encoded via an adjacency matrix A E N where A₁ = indicates an edge between nodes i and GNNs 222 learn node representations through iterative message passing. At each layer k the hidden representation (*) of node e; updated as:
(G :om-))je)()))
h
where Nii) is the neighborhood of node and h z;. The final representation h; is used for downstream tasks such as node classification. In semi-supervised settings, we are given labeled subset I, {(r; Nr where yi E (1,, (1,...,C) the ground-truth label. The GNN is trained to minimize the cross-entropy loss:
1 LCE
log VEI. Σ (l
I
where Je(æ;; A) denotes the logit of class However, in real-world applications, the distribution between I, and I,, often varies, leading significant challenge for the model: identifying and addressing instances where the distribution deviates, known OOD cases. This challenge gives risc

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* OOD Sample 2
ODD Sample
Earay DintBution
1.0
Wa
ID Sample
W
ID Sample
0.0
Energy
0OD Sumple Low enengy- Misclassified
(a2) Baseline Energy Distribution
Eueaay Disribuiio with GedEneggy
OOD Sample:
* OOD Sample:
Desity 1.
Decision Boundary (W; Wg)h
Decision Boundary
||A [coss, cosftg)
Standard Decision Boundary
(b1) GeoEnergy Decision Boundary
0.0
Energy
Closs Anchoe Wa
ID- Samples
Decision Boundary
Closs Anchee W3
ID-B R Samples
ID-A Region
ID B Region
(b2) GeoEnergy Energy Distribution
Figure 3: Decision Boundary Evolution: From Euclidean Softmax to GeoEnergy.
to the problem of OOD detection, aiming to recognize unlabeled nodes that do not conform to the feature distribution of the labeled node set I.. 'he OOD detection task can be formulated as a binary classification problem to detect whether an input node x is an OOD case:
1, x is an in-distribution instance,
g (x: Gx. 10)
{0.
x is an out-of-distribution instance.
where Gz denotes the L-hop ego-graph around æ, and g is scoring function designed to reflect uncertainty. This formulation considers the inter-dependence among nodes in the graph, which is unique aspect of OOD detection in graph
graph-based based contexts compared to traditional OOD detection.
3.1 Geometric Decoupling of Logits
We decouple magnitude and direction by projecting each class weight onto a unit hypersphere. Logits reformulated as:
fj s ||aa|| cos((),))
where f2 is the angle between embedding h; and weight W,, and s is a scaling factor. This design enforces angular structure via normalized weights while retaining |h, to preserve confidence expressiveness. Compared to traditional formulations where /j = W h; = [[ ill 1|b.l] cos(O,,)) GeoFnergy enforces W yll = 1 to eliminate magnitude-basse distortions. This enhances geometric interpretability, aligning with the goal of making class separation rely solety direction. Such formulation encourages tighter intra-class alignment and enlarged inter-class angular margins, which theoretically beneficial for classification in high-dimensional spaces.
Proposition (Reduced Intra-Class Variation). By normalizing the class weight vectors while preserving the feature norm for energy scaling, GeoEnergy promotes directional alignment among features of the same class, leading to improved angular compactness:
W h4
cos(0,,) I II [hall'
with |[ All = 1
(5)
The expected pairwise cosine similarity for same-class features increases:
Exaayon [ cos
(< (.ll((()))
) +
(6)
indicating tighter angular compactness.
Proposition 2 (Increased Inter- Chee--Classe Separation). The scaling factor amplifies angular differences the logit space, thus increasing margin:
2 = s cos().).
2j = cos(),),
|| - 2yll = s cos((0)) cos(8,))
If the feature mapping o satisfies Lipschitz continuity, i.c.,
[|ő(x,) - Φ(2,)< < L| و,
(8)

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Table 1: Performance of OOD detection using metrics such AUROC(1). AUPR()), and FPR95(L), alongside
IT ACC the Cora dataset three types OOD scenarios. The best results highlighted in bold.
Cora-Structure
Cora-Feature
Cora-Label
Model
OOD Expo
AUROC
AUPR FPR95
1D ACC
AUROC
AUPR FPR95
ID ACC
AUROC
AUPR FPR95 ID ACC
MSP
No
45.73
75.50
85.39
73.70
91.36
78.03
34.99
ODIN
No
49.92
27.01
100.0
74.90
49.88
26.96
100.0
75.00
49.80
24.27
100.0
88,92
Mahalanobis
No
46.68
29.03
98.19
74.90
49.93
31.95
99.93
74.90
67.62
42.31
90.77
88.92
Energy
No
71.73
46.08
88.74
76.00
86.15
74.42
65.81
76.10
91.40
78.14
41.08
88.92
GKDE
No
68.61
44.26
84.34
73.70
82.79
66.52
08.24
74,80
57.23
27.30
58.95
89.87
GPN
76.50
73.79
77.00
37.42
GNNNaare
N6
87.52
71.46
73..5
75.30
'03.44
88.19
38.92
76.40
92,30
82.21
30.83
88.02
GenFnergy
No
90.83
80.28
47.53
79.20
94.36
88.90
28.91
78.10
95.58
90.87
19.68
91.46
NodeSaae
No
94.07
83.98
25.63
77.22
95.30
86.
23.
78.70
93.80
85.22
29.41
89.87
+ Geolnnergy
No
95.21
85.32
22.46
77.30
96.76
88.90
21.54
78.21
96.11
86,46
23.77
90,21
71.80
73.30
77.01
46.55
Energy FT
Yes
75.88
49. 18
67.73
75.50
88. 15
75.99
47.53
75,30
91.36
78.49
37.83
90.51
GNNSafe++
Yes
90.62
81.88
53.51
76.10
95.56
90.27
27.73
76.80
92.75
K2.64
34.08
91.46
GenFnergy
Yes
91.27
04
42.61
76.50
96.48
90.62
16.25
77.10
92.88
83.39
31.74
91.46
NodeSafo= 2.
Yos
94.64
85 63
23.34"
76.40
96.56
51. 96
14.73
79.10
94,88
86.66
22.32
01.46
+ GeoEnergy
Yes
95.22
86.15
21.89
77.20
97.36
92.45
10.41
77.67
95.36
89.25
17.85
91.53
then logit differences satisfy:
|| - z|ر1 8- L- ||,;
2z11,
(9)
providing lower bound for decision margin controlled by
3.2 Energy-Based OOD Detection via Angular Regularization
Following GNNSafe 13 we define the Helmholtz free energy for OOD detection as:
F(x,, Gm) - log Σ exp(Se)-
(10)
Under our angular logit formulation, the energy score becomes:
Fangular (x:) = log Σ exp(s ||,,|| cos())))).
(11)
)=1
This reshaped energy surface allows ID samples to cluster in low-energy regions while pushing OOD nodes to high-energy areas. Moreover, the preservation of |h:]] enables confidence sharpness that supports calibration. GeoEnergy acts as plug-in logit head and energy scoring mechanism, which can be seamlessly applied energy-based graph models such as GNNSafe [3] NodeSafe [15], and TopoOOD [14]. During training, it replaces the standard dot-product logit formulation, while during inference, its angular energy serves as the OOD scoring function. Crucially, GeoEnergy requires no
modification of the graph encoder, making it practical and generalizable to existing frameworks.
Experiments
In previous sections, we introduced GeoEnergy, hyperspherical logit reparameterization strategy designed to improve OOD detection and confidence calibration in GNNs. This section presents a comprehensive evaluation of GeoEnergy, examining its robustness under various OOD shifts and its effect on uncertainty estimation. To rigorously assess its capabilities, we explore the following Research Questions (RQs):
RQ1: How effectively does GeoEnergy improve the detection of OOD samples, compared to traditional methods?
RQ2: What impact does GeoEnergy have on confidence calibration for GNNs?
RQ3: Does the enhanced confidence calibration and optimized logit distribution provided by GeoEn- ergy lead to better self-training outcomes?
To address these questions, we conduct experiments on OOD detection and confidence calibration to evaluate GeoEnergy's ability to distinguish OOD from ID samples without compromising ID performance. We also compare its calibration against existing methods and assess how improved calibration enhances self-training, demonstrating GeoEnergy's robustness and reliability.
5

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Table 2: Performance of OOD detection across three OOD scenarios, using AUROC (1). Additional metrics such AUPR, FPR95, and ID accuracy detailed in Appendix B.1
The hest results highlighted in hold.
Citeseer
Pubmed
Model
OOD Expo
Structure
Feature
Label
Structure
Feature
Label
MSP
83.28
ODIN
No
49.23
49.86
51.33
49.76
49.67
56.24
Mahalanobis
45.26
49.92
53.46
55.28
69.12
75.77
Energy
65.62
79.19
89.98
74.33
84.16
86.81
GKDE
No
61.48
74.68
82.69
74.02
82.25
83.30
GPN
86.51
"GNNNaf
N6
79.79
83.46
90.01
81.32
94.28
38.02
+ GeoFFnergy
No
81.23
88.94
92.57
88.92
95.83
89,90
NodeSale
88.40
96.41
94.13
95.97
93.80
+ GeolEnergy
No
89.75
92.12
93.18
95.45
96.13
94.19
Energy FT
Yes
68,87
79.23
91.34
73.54
78.95
91.83
GNNsalet+
Yes
82.43
83.27
91.57
90.62
95.16
87.
GeoFnerry
Yes
84,96
86,80
92.19
91.99
96.97
88,18
"Nodesnfo++
Yes
"6656
01.14
$1,98
56.30
93.26
33,48
+ GeoEnnrgy
Yes
87.67
93.13
93.23
97.87
96.51
94.38
4.1 Out-of-distributioo Detection
In response to RQ1, we conduct graph OOD detection experiments under two settings thoroughly evaluate our approach. Without OOD Exposure: The model is trained without any OOD data, simulating real-works world scenarios where unexpected data types emerge post-deployment. With OOD exposure: OOD samples are introduced during training, allowing the model to learn and adapt to anomalies, assessing its ability to generalize and detect OOD instances with prior knowledge.
4.1.1 Datasets and Splits
Following recent work on graph OOD detection 13 16 15], our experiments use five benchmark datasets reflect real-world scenarios with OOD instances. These datasets cover two scenarios:
Single-graph scenario: OOD instances exist within the same graph as training data but remain unseen during training. (1) Cora, Citeseer, Pubmed [I] OOD data is synthetically generated using structure manipulation, feature interpolation, and label leave-out. ID data is split into training, validation, and testing sets in 1:1:8 ratio. (2) ogbn-Arxiv [17]: This large citation dataset spans 1960-2020. Papers published before 2015 serve as ID data, while those after 2017 are OOD) Papers from 2015-2016 used for OOD exposure during training. II) data follows 1:1:8 split.
Multi-graph scenario: OOD instances come from entirely separate graphs or subgraphs with no direct connections to the training set. Twitch-Explicit [18]: It derived from the Twitch streaming platform, represents users as nodes, with edges indicating mutual friendships. Node features capture user activities and interactions. It consists of multiple subgraphs; we use the DE subgraph ID data, while EN, ES, FR, and RU serve as OOD data. The ENGB subgraph is used for OOD exposure during training. ID data is randomly split into training, validation, and testing sets in 1:1:8 ratio.
4.1.2 Evaluation Metrics
We assess OOD detection and ID accuracy using key metrics: ID Accuracy evaluates performance on known data, AUROC measures the ability separate IDD from OOD instances, AUPR captures the precision-recall trade-off for imbalanced data, and FPR95 quantifies the false positive rate at 95% true positive rate. Detailed metric descriptions Appendix A.2.
4.1.3 Comparison Methods
We evaluate GeoEnergy's 's effectiveness in OOD detection under two settings. Without OOD Expo- sure. We compare aerFEnergyenhaanee models against two categories of baselines that do not use auxiliary OOD supervision during training. The first category includes classical OOD detection meth-
ods originally developed for vision tasks under i.i.d. assumption, such as MSP 19 ODIN 20),
Mahalanobis 21, OF 22), and Energy 23 To adapt these methods to graph-structured data, we re- place their CNN backbones with GCN encoders. The second category consists of graph-specific OOD detection methods that explicitly model topological dependencies, including GKDE [24]. GPN [2].

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Table OOD detection results and ID accuracy Twitch, where nodes from different subgraphs serve 0OD data, and Arxiv, where papers published after 2017 are considered OOD. Detailed results for each OOD dataset (i.e., subgraph or year) are provided in Appendix B.1
Twitch
Arxiv
Model
OOD Expo
AUROC
AUPR
FPR95
ACC
AUROC
AUPR
FPR95
ACC
MSP
No
33.39
49.14
97.45
68.72
63,91
13.85
90.39
53.78
ODIN
No
93.96
55.07
68.85
51.39
Mahalanobis
No
55,68
66,42
90.13
70.51
56,92
69.63
94.24
51.59
Energy
No
51.24
60.81
91.61
70:40
64.20
75.78
90.80
53.36
GKDE
No
46.48
62.11
95.62
67.44
58.32
72.62
93.84
50.76
GPN
No
51.73
66.36
95.51
68.09
GNNSafe
No
66.82
700.97
76.14
71.06
80.44
87.61
53.39
* Geolnnerg
No
72.28
78.13
66.62
72,80
71.99
81,05
84.85
52.76
NodeSafe
No
89.39
93.33
47.00
71.79
72.44
81.51
W8.27
$1.20"
+ GeolEnergy
No
94.47
44.56
73.19
82.32
51.76
OE
Yes
55,72
70.18
95.07
70.73
69.80
80.15
85.16
$2.39
Energy FT
Yes
84.50
88.04
61.29
70.52
71.56
80.47
80.59
53,26
GNNSule++
Yes
95.36
97.12
33.57
70.18
74.77
K3.21
77.45
$3.50
+ GeoFneerg
Yes
95.50
97.18
32.51
70.16
75.73
83,70
73.79
52.12
NodeSafe++
Yes
3.43
71.85
75.49
83.71
75.24
52.93
+ GooEnergy
No
98.87
99.24
3.11
71.95
77.21
84.21
72.18
52.76
GNNSafe [25], and the recently proposed NodeSafe [15. We integrate GeoEnergy into GNNSafe and NodeSafe in a slug-and-play fashion-without any retraining or architectural changes-and compare their performance with and without GeoEnergy. Across all benchmarks, GeoEnergy consistently improves OOD detection performance by mitigating norm-induced logit distortions while preserving angular separability. With OOD Exposure. We further assess GeoEnergy under settings where a small set of auxiliary OOD samples is available during training. This setting follows protocols
introduced in prior work such OE [22], Energy Fine-Tuning [23], and GNNSafe++ (25). We evaluate GeoFnnegy-ennancee GeoEnergy), and compare them against their corresponding base models.
variants, including GNNSafe++<W GeoEnergy) and NodeSafe++
4.1.4 Comparative Results
We explore the enhancements brought by GeoEnergy to OOD detection through detailed experimental evaluations. In Table T 2 and 3 we report the OOD detection performance and ID accuracy of GeoEnergy with other competitive methods. Table presentss a comparative analysis of OOD detection performance of various methods on Cora under three different OOD perturbations, including structure manipulation, feature interpolation and label leave-out, along with their ability to maintain accuracy on ID data. Due to space constraints, dastild the AUROC comparisons for each method across Cora, Amazon and Coauthor under different OOD perturbations. The detailed comparisons of AUPR, FPR95, and ID accuracy on these datasets are provided in the Appendix B. For dataset Twitch and Arxiv, Table 3 provides comparison of OOD detection performance between method and various basclines. Additionally, detailed results on Twitch's 's sub-graphs and Arxiv papers published across different years are presented in Appendix B.I
GeoEnergy significantly enhances OOD detection performance. We highlight several key findings: (1) Overall Superior Performance: Under settings with and without OOD exposure, the rows for GeoEnergy and +GeoEnergy in the tables represent the performance of our method. GeoEnergy consistently shows superior performance across all three perturbation types on the Cora dataset, surpassing other methods. Additionally, GeoEnergy achieves higher AUROC scores across Cora, Amazon, and Coauthor datasets, and also exhibits improved performance on the Twitch and Arxiv datasets compared to SOTA methods. These results highlight the broad effectivences of GeoEnergy in OOD detection, demonstrating its efficacy across both single- graph and multi-graph scenarios. (2) Significant Reduction in FPR95: GeoEnergy cxhibits remarkable improvements in the FPR95 metric. Specifically, it reduces FPR95 by 35% in the structure manipulation scenario, 25.7% in the feature interpolation scenario, and 36.1 % in the label leave-out scenario compared to the previous state-of-the-art method GNNSafe. This highlights its significant advantage in minimizing false positive rates. Additionally, we detail the optimal hyperparameter selection for the scaling factor s across various datasets and different OOD scenarios Appendix B.
GeoEnergy maintains classification accuracy. GeoFnergy preserves competitive ID accuracy and even achieves slight improvements in some cases. As shown in Table | and 3 our method maintains
7

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Diadyy God God Coty
Consl
Expared
Come
Esparai
Cardani
Certanci
Figure Analysis of confidence distributions and reliability diagrams for GCN on Cora with L/C 20.
ID accuracy and, in some instances, enhances in-distribution classification precision. This highlights GeoEnergy's ability to improve the representation of graph data.
4.2 Confidence Calibration and Self-training
Recent studies have demonstrated that GNNs often exhibit an underconfi- dence tendency [9] 10). with pre- dictions systematically less confident than actual probabilities. This under- confidence, due to the complex and noisy nature of graph-structured data and challenges in aggregating neigh- bor information (26). impedes accu- rate OOD detection by blurring the distinction between and OOD data. In response to RQ2 and RQ3, we empirically validated the impact of GeoEnergy improving confidence calibration in GNNs. Our experiments focused on how this adjustment aligns confidence scores more closely with true probabilities. Fur- thermore, we explored self-training to illustrate the practical benefits of enhanced calibration.
Table 4: ECE (M=20) different calibration methods on GCN, GAT, and GraphSAGE. Bold texts indicate the best results.
Calibration Methods
Datuset
Model
Uncal.
TS
MS
CaGCN GCL
GeoEnergy
GCN
0.1347
0.0488
0.0414
0.0401
0.0394
0.0361
Cora
GAT
0.1558
0.0717
0.0544
0.0450
0.0444
0.0430
GraphSage
0.1037
0.0463 0.0371
0.0398
0.0360
GCN
0.1248
0.0641 0.0644
0.0595
0.0579
0.0539
Citeseer
GAT
0.1534
0.0916 0.0633
0.0572 0.0660
0.0552
GraphSage
0.1135
0.0808 0.0866
0.0691
0.0579
GCN
0.0586
0.0541 0.0476
0.0405
0.0394
0.0332
Pubmed
GAT
0.0835
0.0656 0.0501 0.0356
0.0417
0.0316
GraphSage
0.0338 0.0337 0.0342 0.0364
0.0259
Datasets and Benchmarks. We use three citation network datasets: Cora, Citeseer, and Pubmed (27 For supe-supe node classification. we set the label rate L/C E {20, 40, 60). We select three GNN models for node classification: GCNs M.. GATs [2], and GraphS (28). This diversity allows for a comprehensive evaluation of confidence calibration across different GNN architectures. We selected four baseline methods for comparison, including two traditional calibration methods, Temperature Scaling (TS) and Matrix Scaling (MS) (29]. as well as two GNN-specific calibration
methods, CaGCN 2 and GCL [10]
Evaluation Metrics. We evaluate confidence calibration using Expected Calibration Error (ECE) 30 which quantifies the gap between model confidence and accuracy. Specifically, predictions are parti- tioned into M bins, and ECE is computed as the weighted average of the absolute difference between average confidence and accuracy in each bin: ECE- = EM Bml (nce(B,)) conf (on)( I. Here, B,n denotes the set of predictions in the m-th bin, and n is the total number of samples. A lower ECE indicates better calibration. We use M = 20 all experiments.
"
Visualization Analysis. To illustrate GeoEnergy's impact on confidence calibration, Figure Janalyees a GCN's performance on the Cora dataset with L/C = 20. The left histograms reveal a notable shift in confidence distribution for correct (gray) and incorrect (red) predictions. Initially, correct predictions are dispersed across confidence levels, indicating uncertainty. After applying GeoEnergy, confidence for correct predictions becomes more concentrated higher values, reflecting improved certainty and reliability. The right reliability diagrams further quantify calibration improvements. Initially, the model shows underconfidence, with accuracy exceeding predicted confidence (solid line above the ideal dashed line). After applying GeoEnergy, the solid line aligns closely with the ideal, indicating celter-calibrated predictions.
Comparative Results Detailed experimental results on calibration performance at a label rate of L/C 20 are meticulously documented in Table 4 which provides the ECE for GCN, GAT, and GraphSage models across various datasets, utilizing different calibration techniques, where "Uncal." " denotes an uncalibrated model. Detailed analyses for L/C 40 and 60 are discussed in

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Hethodk
Vertook
twitcch
Figure 5: Self-training accuracy comparison of various methods under different labelrates.
Appendix B.2. which also summarizes the optimal hyperparameter selection for the scaling factor in GeoEnergy across multiple datasets.
GeoEnergy enhances confidence calibration performance. Detailed analysis of Table 4 fillustrates that GeoEnergy significantly enhances calibration accuracy, outperforming both traditional and GNN-specific methods across various scenarios. This superior calibration performance not only ensures more accurate confidence estimates but also erucially supports the generation of reliable pseudo labels, enhancing the model's effectiveness in self-training contexts. This robust calibration capability is instrumental in maintaining high accuracy on ID data while effectively identifying and handling out-of-distribution samples.
GeoEnergy boosts self-training efficacy. GeoEnergy validates its impact on self-training by examining its influence on classification accuracy within these settings. As depicted in Figure S] GeoEnergy substantially elevates classification accuracy across diverse datasets and label rates, illustrating its dual benefits: precision in calibration and enhancement of self- seff-training outcomes through the utilization of precise pseudo labels. These experiments affirm that GeoEnergy synchronizes GNNs' confidence estimates with actual probabilities, markedly delineating II) from OOD samples.
5 Related Work
OOD detection for GNNs. To tackle the complexities of graph data with inter-dependent nodes, OOD detection for GNNs is a burgeoning field. GraphDE [ utilizes a variational approach combined with mixed generative models to identify distribution shifts and effectively nown-weigh outliers, enhancing OOD detection capabilities for new datasets. node classification, techniques like traph-based Kernel Dirichlet Distribution Estimation (GKDE) [24] and Graph Posterior Network (GPN) 12 employ Bayesian GNN models that effectively consider the inter-dependence among introduces energy-based OOD discriminator that operates
nodes. Additionally, GNNSafe [3] independently of specific GNN architectures, offering versatile solution OOD challenges in graph neural networks. Uniquely, method GeoEnergy harnesses the crucial insights provided by the intrinsic properties of node embeddings. specifically the natural clustering within the feature space based on angular relationships. This method extends the softmax loss to angular similarity loss and constrains weight vectors to a hypersphere, optimizing the angles between features. By leveraging the inherent oow-dinesiional manifold structure of graph data, GeoEnergy enhances the discriminative power of GNNs for effective OOD detection. A comprehensive review of related work is provided in Appendix C.
Conclusion
This paper presents GeoEnergy, simple yet effective approach for improving energy-based OOD detection in graph neural networks. Through an in-depth analysis of GNN logit geometry, we identify two critical aathoomna--nngunn clustering and coupling-induce misdetection- -tht undermine the reliability of conventional energy scoring. GeoEnergy addresses these issues by decoupling magnitude and direction via weight normalization, thereby enhancing intra-class compactness, inter-class separation, and energy stability. Importantly, GeoEnergy is model-agnostie and can be seamlessly integrated into existing energy-based frameworks without retraining. Extensive experiments across diverse OOD benchmarks demonstrate that GeoEnergy consistently improves detection performance and calibration, providing strong geometric foundation for robust and trustworthy GNN deployment in open- world environments.

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A Details of Experiments
A.1 Details of datasets
Following the setup from recent work on graph OOD detection and generalization, our experiments utilize five widely recognized node classification benchmark datasets to reflect real-world scenarios involving OOD instances within graphs. These datasets encompass two distinct scenarios:
Single-graph scenario: In this scenario, we use the datasets Cora 27.. Amazon- BI], Coauthor- CS (2], and ogbn-Arxiv 17). Here, OOD testing instances exist within the same graph as the training instances but remain unseen during training.
Cora, Manazon-Photor Coauthor-CSS Cora a citation network where nodes represent scientific papers and edges indicate citations between them. Amazon-Photo is co-purchase network with nodes representing products and edges signifying frequent co-purchases. Coauthor-CS CS is co-authorship network where nodes represent authors and edges denote co-authorship papers in the field of Computer Science. Due to the lack of explicit domain distinctions, we synthetically generate OOD data using three different approaches: Structure manipulation: The original graph is utilized as in-distribution data, while a new graph generated using the stochastic block model serves as OOD data. Feature interpolation: Random interpolation is applied to create node features for OOD data, with the original graph in-distribution data. Label leave-out: Nodes with specific class labels used as in-distribution data, and nodes with other class labels are treated as OOD data. For Amazon- Photo and Coauth-CS, we randomly split the ID data into training, validation, and testing sets in 1:1:8 ratio. For Cora, we split the ID data following the semi-supervised learning setting by MΤ
ogbn-Arxiv: A large graph dataset that records citation information from 1960 to 2020. Each node represents paper, labeled with its subject area, and edges denote citation relationships. Nodes have 2o-dimensiooal feature vectors obtained from the word embeddings of their titles and abstracts. ID data is split in 1:1:8 ratio based publication year.
Multi-graph scenario: This scenario is represented by the dataset Twitch-Explicit 18 where OOD instances originate from completely different graph or subgraph that has no connections with nodes in the training set.
Twitch-Explice Derived from the Twitch streaming platform, each node represents a user, edges denote mutual friendships between users, and node features capture user activities and interaction patterns. The dataset includes multiple subgraphs; we use the DE subgraph as the ID data, while the other subgraphs (EN, ES, FR, and RU) serve as OOD data. Additionally, the ENGB subgraph is used for OOD exposure during training. We randomly split the ID data into training, validation, and testing sets in 1:1:8 ratio.
Table 5: Dataset Details for In-Distribution (ID), Out-of-Distribttinn (OOD), and OOD Exposure Data
Dataset
ID Data
OOD Data
OOD Exposure Data
Cora
Structure manipulation,
Synthetically generated
Amazon-Photo
Original graph
Feature interpolation,
OOD instances
Coauthor-CS
Label Icave-out
Twitch
DE subgraph
Five other subgraphs Papers after 2017
ENGB subgraph Papers in 2015 and 2016
Arxiv
Papers before 2015
A.2 Details of evaluation metrics
In our experiments, to recognize known data and detect OOD instances, ensuring robustness and reliability in diverse scenarios.
several evaluation metrics to comprehensively evaluate the model's ability
ID Accuracy is the ratio of correctly predicted instances the total instances within the in-distribution (ID) data. This metric measures how well the model performs data it has seen during training
12

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data from the same distribution. High ID Accuracy indicates that the model effectively learns the patterns in the ID data.
AUROC (Area Under the Receiver Operating Characteristic Curve) measures the model's ability to distinguish between ID and OOD instances across all possible classification thresholds. This metric evaluates the trade-off between the true positive rate (sensitivity) and the false positive rate (L-specificity). It provides a single scalar value representing the model's overall ability to separate ID from OOD instances.
AUPR (Area Under the Precision-Recall Curve) evaluates the trade-off between precision (the fraction of true positive instances among all instances predicted as positive) and recall (the fraction of true positive instances among all actual positive instances) for different thresholds. This metric is particularly useful in scenarios with imbalanced data, where the number of OOD instances may be much smaller than ID instances. provides a more detailed picture of the model's performance in detecting OOD instances by focusing on the precision-recall trade-off.
FPR95 (False Positive Rate at 95% True Positive Rate) measures the false positive rate when the true positive rate at 95%. This metric assesses the model's reliability in identifying OOD instances by ensuring that high true positive rate does not come at the cost of a high false positive rate. It provides insight into the model's s performance in critical regions of the ROC curve.
A.3 Details of baselines
MSP is simple yet effective baseline for OOD detection. It uses the maximum predicted probability from the softmax output as a confidence score. Instances with low confidence scores considered OOD. This approach is straightforward but can struggle in scenarios with complex data distributions.
MSP (Maximum Softmax Probability) [19]
detector for Neural networks) [20]: ODIN improves upon MSP by applying temperature scaling and small perturbations to the input. These adjustments enhance the separability between ID and OOD data by making the softmax scores more discriminative. ODIN requires fine-tuning on a validation set to determine optimal parameters for temperature scaling and perturbation magnitude.
ODIN (Out-of-Distribuue
Mahalanobis [21]: This method estimates the class-conditional distributions of the model's features using a multivariate Gaussian distribution. The Mahalanobis distance between test sample and the closest class mean used as a confidence score. Lower distances indicate higher confidence of being ID, making this approach effective for OOD detection, especially in scenarios with well-separated feature distributions.
OE leverages an auxiliary dataset containing OOD examples during training. The model is trained to assign low confidence scores to these OOD samples while main- taining high confidence for ID samples. This exposure to OOD data during training helps the model learn better decision boundaries, improving its OOD detection capabilities.
OE (Outlier Exposure) [2]]:
The energy-based model uses the concept of energy from statistical physics to measure prediction confidence, with lower energy indicating higher confidence. Energy scores are computed from the model's 's logits, providing a unified approach for ID and OOD detection. Energy FineTTune extends this by incorporating OOD data during training, fine-tuning the model to better discriminate between ID and OOD samples, thus improving calibration and detection accuracy.
Energy and Energy FineTTue 23
GKDE (Graph-based Kernel Dirichlet distribution Estimation) 24]: GKDE is specifically designed for graph data, leveraging kernel density estimation to model the uncertainty in node embeddings. By estimating the Dirichlet distribution over the graph's 's features, GKDE provides a probabilistic framework for detecting OOD nodes. This approach effectively captures the dependen- cies and topological structures unique to graph data.
GPN integrates Bayesian principles with graph neural networks to model posterior distributions over node labels. This method uses a probabilistic encoder to generate latent variables and decoder to predict labels, capturing the uncertainty in predictions. GPN's Bayesian framework makes robust for OOD detection by effectively quantifying prediction uncertainty.
GPN (Graph Posterior Network) [12]
13

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Table 6: Performance of out-of-distribution detection using metrics such as AUROC(1). AUPR()). and FPR95()).
alongside in-distribution accuracy (ID ACC) the Amazon dataset three types of OOD ios structure manipulation, feature interpolation, and label leave-out presented. The best results are highlighted in bold.
Model
OOD Expo
Amazon-Structure
Amazon-Featurr
Amizon-L.abe AUPR
AUROC
AUPR FPR95
ID ACC
AUROC
AUPR
FPR95
ID ACC
AUROC
FPR95
ACC
MSP
96.27
98.34
6.13
84
97.31
95.16
8.72
92 89
93.97
91.32
26.65
95.
ODIN
No
93.24
95.26
65.44
92.84
81.15
78.47
100.0
92.71
65.97
57.80
90.23
96.08
Mahulanobis
No
71.69
79.01
99.91
92.79
76.50
71.14
76.12
92.86
73.25
66.89
74,30
95,76
Energy
No
98.51
98.72
4,97
92.86
97.87
95.64
6.00
92.96
93.81
91.13
28.48
95.72
GKDE
No
76.39
81.58
99.25
87.57
58,96
66.76
99.28
86.18
65.58
65,20
96.87
89.37
GPN
97.17
96.39
11.65
88.51
87.91
84.77
49.11
90.05
92.72
90.34
37.16
90.07
GNNSafe
No
99.58
99.76
0.00
92.53
98.55
98.99
0.31
92.81
97.35
97.12
6.59
95,
KGeoFn
99.63
99.56
0.00
92.76
97.57
97.40
7.90
95.80
OE
99.60
59.61
0.51
92.61
98.39
96.24
92.30
95.39
92.33
17.72
95.72
Energy FT
Yes
98.83
99.14
1,31
92.79
98.68
96.82
2.84
92.52
96.61
94.92
13.78
94.83
GNNSafe++
99.82
0.00
92.22
99
92.33
97.51
97.07
6.18
95.84
+GeoFnergy
Yes
99.96
99.98
0..00
92.38
99.80
99.75
0.18
92.68
97.19
96.71
8.66
95.76
Table 7: Performance of out-of-distribation detection using metrics such as AUROC(1), AUPR()). and FPR95()).
alongside in-distribution accuracy (ID) ACC) the Coauthur dataset across three types of OOD scenar- ios structure manipulation, feature interpolation, and label leave-out-ae presented. The best results highlighted bold.
Soauthor-Structure
Coauthor-Featee Feature
Coauthor-
Model
Expo
AUROC
AUPR FPR95
ACC
AUROC
AUPR FPR95 ID ACC
AUROC
AUPR FPR95
ACC
MSP
95.30
94.37
24.75
92.47
97.05
96.93
13.35
92.45
94.38
97.99
23.31
95.18
ODIN
No
52.14
48.83
99.92
92.34
51.54
45.50
100.0
92.39
51.44
74.79
100.0
95.15
Mahalanobis
80446
76.65
70.75
92.33
93.23
90.88
28.10
92.34
85.36
93.61
45.41
95.19
Energy
No
96.18
95.25
18.02
92.75
97.88
97.69
9.75
92.75
95.87
98.34
18.69
95.20
GKDE
65.87
72.65
99.48
85.62
80.69
86.47
96.57
84.72
61.15
81.39
94.60
89.05
GPN
No
34.67
40.21
99.57
89.45
81,77
80.56
74.46
87.05
93.24
97.55
34.78
91.68
GNNSafe
99.60
0.26
92.73
97.23
98.98
95.21
+GeuEnergy
No
99.92
99.86
0.15
92.10
99.83
99.78
0.51
92.60
97.90
99.23
9.48
95.37
OE
Yes
97.86
96.81
9.23
92.00
95.004
98. 80
4.44
92.64
96.04
98,
18.17
95,10
Energy
97.78
3,97
98.51
95.20
GNNSafe++
Yes
99.99
99.99
0.00
92.92
99.97
99.95
0.09
92.87
97.89
99. 24
9.43
95.24
+GeoEnergy
95.30
GNNSafe and GNNSafe++ [25]: GNNSafe combines energy-based models with graph neural net- works to enhance OOD detection in graph data. It uses energy scores derived from node embeddings differentiate between ID and OOD instances. leveraging graph structure for improved perfor- mance and robustness. GNNSafe++ extends GNNSafe by incorporating OOD data during training, fine-tuning the energy-basee based detection mechanism. This enhanced version uses OOD exposure to further improve the model's 's ability to distinguish between ID and OOD instances, achieving superior detection performance.
Table 8: Dustrr-diit detection performance measured by AUROCT)/AUPK(W/PRRSS)) on OOD sub-graphs ES, FR and RU of Twitch dataset.
Twitch-FS AUPR
Twitch.FR AUPR
Twitch-RU AUPR
Model
OOD Expo
AUROC
EPR95
AUROC
FPR95
AUROC
FPR99
MSP
No
37.72
$3.08
98.09
21.82
38.27
99.25
41.23
56.06
95.01
58.67
72.58
93.98
Mahalanobis
No
45.66
58.82
95.48
40.40
46.69
95.54
55.68
66.42
90.13
Energy
95.70
57.72
87.57
GKDE
No
48.70
61.05
95.37
49.19
$2.94
95.04
46.48
62.11
95.62
64.24
95.05
50.89
GNNSafe
No
49.07
57.62
93.98
63.49
66.25
90.80
87.90
89.05
43.95
+GeoEnergy
35.97
94.94
55.77
70.18
95.07
Energy FT
Yes
80.73
87.56
76.76
79.66
$1.20
76.39
93.12
95.36
30.72
GNNSafe++
94.54
97.11
44.06
51.06
98.10
98.74
5.59
+GeoEnnegg
Yes
94.90
97.34
41.33
93.54
95.49
50.65
98.07
98.69
5.56
14

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B Additional Experiment Results
B.1
Additional OOD detection experiment results
Further experiment results on the Amazon and Coauthor datasets. In the main text. succinctly present AUROC data for GeoEnergy across the Cora, Amazon, and Coauthor datasets. This appendix provides an in-depth exploration of those results, detailing comprehensive metrics including AUPR, FPR95, and ID ACC. As depicted Table 6 and Table 7 extensive experimental outcomes for three distinct OOD perturbations struuctt manipulation, feature interpolation, and label leave-out- are elaborated upon. These detailed analyses supplement the overview given in the main text, offering thorough insights into the performance across various models and experimental conditions. Notably, GeoEnergy consistently exhibits superior AUROC and AUPR scores under all scenarios, showcasing its robustness and efficacy in handling complex OOD challenges.
Further experiment results on the Arxiv and Twitch datasets. We provide average OOD detection performance results across sub-graphs for Arxiv and Twitch in main text. This appendix presents detailed results for specific sub-graphs within the Twitch and Arxiv datasets, demonstrating the effectiveness of GeoEnergy and its variants under varied scenarios. For the Twitch dataset, as shown in Table [8] detailed OOD detection performances are documented across three distinct sub- sub-graphs (ES, FR, and RU). The table lists outcomes for different models under three types of OOD perturbations: structure manipulation, feature interpolation, and label leave-out. GeoEnergy consistently shows superior performance across all sub-graphs, particularly in reducing the FPR95, highlighting its capability to accurately identify OOD samples. Regarding the Arxiv dataset, Table Jillustrtae the performance for papers published in the years 2018, 2019, and 2020, treated as OOD samples. These results further validate the effectiveness of GeoEnergy in handling recent scientific publications.
Summary of optimal hyperparameters for OOD detection. As shown in Table I0 we provide summary of the optimal hyperparameter settings for the scaling factor s used the GeoEnergy and +GeoEnergy across various datasets and OOD scenarios.
Figure 6: Impact of scaling factor variation on performance metrics for different OOD scenarios in graph neural networks the Cora dataset.
Sensitivity analysis of the scaling factor in GeoEnergy. The three graphs in Figure depict sensitivity analysis the scaling factor in GeoEnergy across different OOD scenarios (structure manipulation, feature interpolation, and label lcave-out) on the Cora dataset. Each graph illustrates how performance metrics (AUROC, AUPR, FPR, and IND Test Score) evolve as the scaling factor varies from 1 to 25. AUROC and AUPR: Both metrics generally increase stabilize with higher values of s, demonstrating improved classification and prediction precision under the influence of GeoEnergy. This suggests that increasing enhances the model's ability to distinguish between in-distribution and OOD samples, leading to more reliable predictions. FPR95: The false positive rate decreases significantly as s increases, indicating that GeoEnergy effectively reduces the likelihood of false alarms in OOD detection. This is particularly evident in the sharp declines observed in the initial values of s. IND Test Score: The in-distribution accuracy remains relatively stable across different values of 8, which implies that the model maintains its performance on seen data while improving its robustness to OOD samples.
B.2 Expanded calibration results across different label rates
Further calibration results across different label rates. This section extends the examination of confidence calibration performance by providing detailed results for a broader range of label rates, specifically L/C' 20, 40, 60. Table [] provides comprehensive comparison of ECE
15

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Table 9: Out-of-distribaation detection performance measured by ATROC()AAPPR(()FPRSS() OOD datasets of papers published in 2018, 2019 and 2020, respectively, Arxiv.
Arxiv-2018 AUPR
Arxiv-2019
Arxiv-2200
Model
OOD Expo
AUROC
AUROC
AUPR
AUROC
AUPR
MSP
No
61.66
70.63
91.67
63.07
66.00
90.82
67.00
90.92
89.28
ODIN
No
53.49
63.06
100.0
53.95
$6.07
100.0
55.78
87.41
100.0
Mahalanobis
57.08
93.69
57.85
56.92
85.95
95.01
Energy
No
61.75
70.41
91.74
63.16
65.78
90.96
67.70
91.15
89.69
GKDE
56.29
66.78
94.31
57.87
62.34
00.79
88.74
93.31
GPN
No
GNNSafe
66.47
74.99
71.57
78.35
94.76
83.57
#GenEnergy
No
67.30
75.70
87.88
69.33
72.45
86.20
79.33
95.01
$0.48
OE
Yes
67.72
75.74
86.67
69.33
72.15
85.52
72.35
92.57
83,28
Energy FT
Yos
69.58
76.31
82.10
70.58
72.03
81,30
74.53
93.08
78.36
GNNSafe++
Yes
70.40
78.62
81.47
72.16
75.4.
79.33
81.75
95.57
71.50
+GeoEnergy
Yes
71.23
79.10
78.51
73.12
76.16
75.93
82.83
95.82
66.94
Table 10: Optimal hyperparameter selection for the scaling factor in GNNSale(+GeoEnergy and GNNSafe=+(GeeoEneee) across multiple datasets and OOD types.
Model
Structure
Cora
Structure
Amazon
Label
Coauthor
Feature
Label
Twitch
Arsiv
GNNSatc(+GooBaerry))
20
15
10
20
15
2
20
2
5
15
1.2
1.5
for GCN, GAT, and GraphSage models across various datasets. The ECE values are recorded under several calibration techniques: uncalibrated (Uncal), temperature scaling (TS), matrix scaling (MS), and several methods specifically developed for GNN architectures including CaGCN, GCL, and method, GeoEnergy. Each method's performance is scrutinized different label rates, showcasing the robustness and consistency of GeoEnergy's superior calibration capability. The results highlight GeoEnergy's consistent outperformance across different settings and models, affirming its effectiveness in refining the calibration process, thereby enhancing the models' predictive accuracy and reliability across diverse scenarios. The detailed numerical results underscore the methodological advancements facilitated by GeoEnergy, reinforcing its role in improving both in in-distribution and muitr--distribution detection by providing more accurate and reliable pseudo labels for effective self-training.
Summary of optimal hyperparameters for confidence calibration We provide detailed hyperparameter settings for model calibration across various datasets and models, specifically GCN, GAT, and GraphSAGE. These settings pertain optimal values used for experiments, which were not specified in the main text due to space constraints. The table 12 presents the selected parameters for each dataset (Cora, Citeseer, Pubmed) at different label rates (L/C of 20, 40, and 60), facilitating clearer understanding of how these configurations influence model performance.
various scenarios.
Table 11: ECE (M=20) of different calibration methods on GCN, GAT, and GraphSAGE for different datasets with label rate L/C 20, 40, 60.
Datnset 1/C
GCN
GAT
GraphSAGE
Uncal.
MS
CaGCN GCL
Ours
Uncal.
TS
MS
CaGCN GCL
Ours
Uncul.
TS
MS
CaGCN GCL
Ours
20
0.13470.0488 0.0414
0.0401
0.0394
0.0391
0.1558 0.0717
0.0544
0.0450 0. 0.0444 0.0430
0,1037 0.0463 0.0371
0.0398
0.0360
Corn
40 0.1134 0.0872 0.0407
0.0371 0.0338
1.1340
0.0485
(.0491
0.0365
0.0356 0.0330
0.0847
0.04330
0.0368
0.0281
0.0803
0.0339
20 0.1248 0.0641 0.0644
0.0595
0.0579 0.0539
0.1534 0.09 0.0633
0.0572 0.0660 0,0552
0.3 0. 0,0888 0.0866 0.0691
0,0579
0.0957
0.0603
0.0544
60 0.0806 0.0559
0.0521
0.0546
0.0501
0.0450
1.1090
0.0648
0.0519
0.0527
0.0522
0.0513
0.0684 0.0500
0.0497
0.0400
20 0.05860..054
0.0476
0.0405
0.0394 0.0332
0,0835 0.0656
0.0501
0.0356
0.0417 0.0316
0,0338
0.0337 0.0342 0.0364
0,0259
Pubmed
40 0.0444 0.0446 0.0436
0.0402
0.0395 0.0349
0.0869
56 0.0539
0.0308
0.0309 0.0265
0.0310
0.0275 0.0327 0.0275
0.0271
60
0.0445 010367 0,0318
0.031
0.0310
0.0302
1.0993
0.0669 0L0483
0.03008
0.0384
0.0293
1,0280
0.0321 0,0315
0.0239
0.0219
16

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Table 12: Summary of used parameters on model calibration in GCN, GAT and GraphSAGE.
Dataset
Cora
Citseer
Puhmed
L/C
GCN
GAT
GraphSage
GCN
GAT
GraphSage
GCN
GAT
GraphSage
20
15
15
15
10
10
15
3
16
40
15
10
3
60
15
12
13
7
6
15
3
15
C Related Work
C.1
Out-of-distribution Detection
OOD detection is crucial for identifying and managing samples that diverge from the training data distribution. This field has been rigorously researched and implemented across various neural network architectures, including traditional DNNs and the more complex GNNs. Due to the inherent differences their architectures and data processing mechanisms, the methodologies appropriate for OOD detection differ between these types of networks.
OOD detection for NNs. Two principal approaches have been developed to tackle OOD detection in neural networks: acoring-based methods and training-time regularization techniques. The first focuses on designing scoring functions that utilize various metrics to assess the outlierness of samples. Notable examples include maximum softmax probability (MSP) [19], OpenMax score [33] Mahalamobis distance- based score 21 ODIN score [20), and energy- based scores 23 34 These functions provide quantifiable measures of how likely sample is to be anomalous based on model predictions. Training-Tine regularization techniques modify the training process to enhance models' OOD detection abilities. Strategies such as encouraging uniform prediction distributions for outliers and promoting higher energy outputs for potential OOD samples [23] 36 37 have been explored. These techniques, aligning with log-likelihood shaping theories, are inherently suitable for OOD detection. Efforts to further enhance OOD detection also include methods leveraging unlabeled data for outlier identification (38]. However, they typically target i.i.d. instances and often overlook the interdependence between data points.
35 22
OOD detection for GNNs. To tackle the complexities of graph data with inter-dependent nodes, OOD detection for GNNs a burgeoning field. GraphDE [IT] utilizes a variational approach combined with mixed generative models to identify distribution shifts and effectively down-weight outliers, enhancing OOD detection capabilities for new datasets. node classification, techniques like Graph-based Kernel Dirichlet Distribution Estimation (GKDE) [24] and Graph Posterior Network (GPN) 12) employ Bayesian GNN models that effectively consider the inter-dependence among introduces energy energy-based OOD discriminator that operates independently specific GNN architectures, offering a versatile solution to OOD challenges in graph neural networks. Uniquely, method GeoEnergy harnesses the crucial insights provided by the intrinsic properties of node embeddings. specifically the natural clustering within the feature space based angular relationships. This method extends the softmax loss to angular similarity loss and constrains weight vectors to a hypersphere, optimizing the angles between features. By leveraging the inherent low-dimensional manifold structure of graph data, GeoEnergy enhances the diseriminative power of GNNs for effective OOD detection.
nodes. Additionally, GNNSafe [K]
C.2 Confidence Calibration of GNNs
The objective of confidence calibration align predicted probabilities with the actual likelihood of correctness. Calibration methods fall into two categories; post-hoc adjustments and regularization techniques. Post-hoe methods, such as Temperature Scaling (TS) recalibrate predictions using temperature parameter learned from validation data. Regularization techniques, like Focal Loss 39] mitigate overconfidence by minimizing KL divergence, while approaches such as Mixup 40 and Label Smoothing [1) adjust label entropy for better calibration. For GNNs, traditional calibration methods often underperform due to their unique structure. CaGCN adapt TS, and [TL introduce end-to-end calibration techniques tailored for GNNs, leveraging graph topology and entropy regularization to improve calibration. GNN So is perfectly calibrated when the predicted confidence i. equals the actual probability P: of correctly predicting node i, idcally satisfying:
17

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P(O: Us p. p) p. vp E [0, 1]. The Expected Calibration Error (ECE) quantifies deviations from this ideal.
Underconfidence Leading to Blurred OOD Detection. Recent studies have shown that commonly used GNNs typically exhibit underconfidence tendency [9] where the predicted confidence scores are systematically lower than the true probabilities. While undervonfidence can reduce the risk of overconfident incorrect predictions, has negative implications for OOD detection. Specifically, underconfidence makes it difficult for GNNs to effectively distinguish between in in-distribution and out- of-distribution data, as the model's 's confidence estimates are insufficient provide clear discriminative evidence.
18

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The abstract and introduction clearly state two key observations (an- gular clustering and coupling coupling-induced induced misdetection) and the GeoEnergy contributions; these align precisely with the experiments and theoretical results in Sections 1-4.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: We devote Section 6 ("Limitations and Future Work") discuss assump- tions (c.g., homophily dependence). dataset scope (five benchmarks), and potential performance under heterophilic graphs.
3. Theory assumptions and proofs
Question:
For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
Answer: [Yes]
Justification: Propositions 1-2 include all geometric assumptions and are formally proved in Appendix B with cross-references to the main text.
4. Experimental result reproducibility
Question:
Does the paper fully disclose all the information needed reproduce the main experimental results?
Answer: [Yes]
Justification: Section and Appendix detail datasets, data splits, hyperparameters. random seeds, and all baseline implementations; code and instructions are provided in the supplemental material.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results?
Answer: [Yes]
Justification: We release anonymized code and data preprocessing scripts the URL in the supplemental; detailed README guides reproduction all figures.
6. Experimental setting/details
Question: Does the paper specify all the training and test details necessary understand the results?
Answer: [Yes]
Justification: Section 5 ("Experimental Setup") lists graph splits, optimizer settings, learning rates, epochs, and evaluation metrics; Appendix C provides full YAML config files.
7. Experiment statistical significance
Question:
Does the paper report error bars or other information about statistical significance?
Answer:
[Yes]
Justification: All OOD detection metrics (AUROC, FPR95) averaged over 10 runs with standard-deviation error bars shown in Tables 2-3.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the compute resources needed?
Question:
Answer: [Yes]
19

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Justification:
Appendix D reports use of NVIDIA V100 GPUs (4x), total training time ( (48 GPU hours), and memory footprints per dataset.
9. Code of ethies
Question: Does the research conform with the NeurlPS Code of Ethics?
Answer:
[Yes]
Justification: Our work involves no sensitive personal data or high-risk applications; all authors have reviewed and adhered to NeurIPS ethics guidelines.
10. Broader impacts
Question:
Does the paper discuss potential positive and negative societal impacts? Answer: [Yes]
Justification: Section 7 outlines improved reliability in critical graph applications (e.g.. drug discovery) and notes potential misuse adversarial detection settings.
11. Safeguards
Question: Does the paper describe safeguards for responsible release of data models with high misuse risk?
Answer: [NA]
Justification: No pretrained models or scraped datasets are released; risk of misuse is minimal.
12. Licenses for existing assets
Question: Are external assets properly credited and licensed?
Answer: [Yes]
Justification: We cite each benchmark's s source paper and include license references (e.g., Cora/CiteSeer under CC-BY) in Appendix C.
13. New assets
Question: Are new assets introduced well documented?
Answer: [NA]
Justification: We do not release new datasets or pretrained models.
14. Crowdsourcing and research with human subjects
Question: Does the paper include full details for crowdsourcing/huuan subjects?
Answer: [NA]
Justification: The work does not involve human participants or crowdsourcing.
15. Institutional review board (IRB) approvals
Question: Does the paper describe IRB approvals for human subjects?
Answer: [NA]
Justification: No human-subject research was conducted.
16. Declaration LLM usage
Question: Does the paper describe usage of LLMs if they impact core methods?
Answer: [NA]
Justification: We did not use any large language models the method development or experiments.
20


Reviews:
[{"review_id": "FS79T5CpWM", "reviewer": "Reviewer_D1EP", "summary": "This paper proposes GeoEnergy, a simple plug-and-play framework to improve energy-based OOD detection for Graph Neural Networks by decoupling class weight magnitudes from logits and preserving embedding norms as confidence signals. The authors claim that conventional energy-based methods suffer from unstable energy scores due to coupling between logit norm and direction. By normalizing class weights and leveraging angular clustering in GNN embeddings, GeoEnergy aims to stabilize energy scores and improve OOD detection. Multiple experiments on graph benchmarks demonstrate consistent gains over baselines.", "questions": "The experiment lacks comparisons with recent state-of-the-art methods, such as [1],[2],[3].\n\n[1] Revisiting Score Propagation in Graph Out-of-Distribution Detection. NeurIPS'24\n\n[2] GOLD: Graph Out-of-Distribution Detection via Implicit Adversarial Latent Generation. ICLR'25\n\n[3] Spreading Out-of-Distribution Detection on Graphs. ICLR'25", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The paper addresses a relevant and under-explored issue in graph OOD detection: the instability of energy scores due to logit coupling.\n2. The proposed solution is simple, intuitive, and easy to integrate into existing GNN models without retraining.\n3. The idea of leveraging angular clustering properties in GNNs for OOD detection is interesting and has practical potential.\n\nWeaknesses:\n1. The method keeps embedding norms as part of the energy score, but embedding norms in GNNs often correlate with node degree, which can introduce noise and harm energy stability in heterophily or imbalanced graphs.\n2. The assumption of angular clustering only holds on small, homophilous graphs like Cora and Pubmed, but may not generalize to heterophilous or large real-world graphs.\n3. Proposition 1 and Proposition 2 are trivial and formalize obvious properties without offering deeper theoretical justification for improved OOD separation.\n4. The decoupling only applies to class weight magnitudes, while embedding norm-induced variance remains and can still destabilize the energy scores. A fully norm-independent scoring function might be cleaner.\n5. Most experiments are on small homophily graphs. The method’s scalability and effectiveness on large-scale or heterophily graphs remain untested.", "was_revised": true, "final_justification": "The additional clarifications and comparative experiments provided by the authors during the rebuttal have addressed most of my concerns.", "rebuttal": "We sincerely thank you for the detailed and insightful feedback, which has helped us to significantly improve our paper. We are encouraged that the reviewer found our work to be **technically solid** and addressing a **relevant and under-explored issue**. We address each of the weaknesses and questions below.\n\n---\n\n**W1&W4**: The method's reliance on the embedding norm, and whether a fully norm-independent function would be cleaner.\n\n**A1:** Thanks for this insightful question, as it allows us to clarify a core contribution of GeoEnergy. We agree with your premise: in standard GNNs, the embedding norm is often entangled with non-semantic structural factors (like node degree), acting as a source of noise. Our work is motivated by precisely this problem, which we term \"coupling-induced misdetection.\"\n\nOur core insight is that decoupling transforms the norm's role. After normalizing classifier weights, the model must prioritize learning correct semantic **directions**. Subsequently, the norm is \"freed\" to become a **purified signal of model confidence**, an emergent property of the optimization. The model learns to assign larger norms to high-confidence (\"easy\") samples and smaller norms to low-confidence (\"hard\") samples. This enhances the model's expressiveness, which explains the **improved ID accuracy** we observe.\n\nTo directly test the reviewer's hypothesis, we conducted a new ablation study comparing our full method against a fully norm-independent variant (where the embedding norm is also normalized).\n\nTable 1: Ablation on the embedding norm on Cora-Structure (AUROC / ID ACC).\n|Model|Baseline|+ GeoEnergy (with norm)|+ GeoEnergy (w/o norm)|\n| :--- | :--- | :--- | :--- |\n| GNNSafe | 87.52 / 75.80 | **90.83 / 79.20** | 86.13 / 74.90 |\n| NodeSafe | 94.07 / 77.20 | **95.21 / 77.30** | 93.55 / 76.80 |\n\nThe results unequivocally show that **removing the purified norm significantly degrades performance**, for both OOD detection (AUROC) and ID classification (ID ACC). This provides strong empirical evidence that preserving the norm is a critical and beneficial design choice. \n\n---\n\n**W2&W5**: Generalization to large-scale and heterophilous graphs.\n\n**A2:** Thanks for this critical feedback. To address this, we provide new experiments and clarifications.\n\n**(1) Effectiveness on Heterophilous Graphs:**\nWhile our \"Angular Clustering\" motivation may be less pronounced in heterophilous settings, GeoEnergy's core decoupling mechanism is general. To prove this, we conducted new experiments on 3 heterophilous datasets, which confirm that our method provides significant and consistent improvements.\n\nTable 1: OOD Detection Performance (AUROC ↑) on Heterophilous Datasets.\n\n| Dataset |GNNSafe| + GeoEnergy (Ours) | Improvement |\n| :--- | :--- | :--- | :--- |\n| Texas | 81.54% | 85.22% | **+3.68%** |\n| Wisconsin | 84.31% | 88.15% | **+3.84%** |\n| Cornell | 79.92% | 83.67% | **+3.75%** |\n\n**(2) Scalability to Large-Scale Graphs:**\nWe respectfully clarify that our original submission already included experiments on two large-scale benchmarks: ogbn-arxiv (169K nodes, 1.1M edges) and Twitch (167K nodes, 6.7M edges). As shown in Table 3 of our main paper, GeoEnergy provides substantial performance boosts on these large graphs.  \n\n**(3) Efficiency:**\nWe measured the inference time overhead of GeoEnergy. The experiments were conducted on a single NVIDIA A100 GPU.\n\nTable 2: Inference Time Analysis (in milliseconds per epoch).\n\n|Dataset|GNNSafe| + GeoEnergy|Overhead (%)|\n| :--- | :--- | :--- | :--- |\n| Cora | 2.15 ms | 2.18 ms | **+1.40%** |\n| Citeseer | 2.31 ms | 2.35 ms | **+1.73%** |\n| Ogbn-arxiv | 158.42 ms | 158.95 ms | **+0.33%** |\n| Twitch | 125.70 ms | 126.11 ms | **+0.33%** |\n\nThe analysis confirms the overhead is negligible and decreases with graph scale, as our modification only affects the lightweight final layer. This demonstrates that GeoEnergy is both efficient and highly scalable.\n\n---\n\n**W3: The theoretical justification is trivial and lacks depth.**\n\n**A3:** Thanks for your valuable feedback. To address this, we have added a new, more formal analysis (Proposition 3) to our revised manuscript. This proposition provides a direct theoretical justification for the improved ID-OOD energy separation. We present a sketch below.\n\n> **Proposition 3 (ID-OOD Energy Separation Guarantee).**\n>\n> *Let $E(\\mathbf{h}) = -\\log(\\sum\\_{j=1}^{C} \\exp(s \\cdot \\|\\mathbf{h}\\| \\cdot \\cos(\\theta\\_j)))$ be the energy function. Assume that for in-distribution samples $\\mathbf{h} \\sim \\mathcal{P}\\_{\\text{ID}}$, their directions are well-aligned with their true class prototypes such that $\\mathbb{E}[\\cos(\\theta\\_y)] \\approx 1$ for the correct class $y$. Further assume that for out-of-distribution samples $\\mathbf{h}' \\sim \\mathcal{P}\\_{\\text{OOD}}$, their directions are unaligned, such that $\\mathbb{E}[\\cos(\\theta'\\_j)] \\approx 0$ for all classes $j$. Then, the expected energy of ID samples is significantly lower than that of OOD samples, with the gap widening for higher-confidence ID samples:*\n>\n> $$\n> \\mathbb{E}\\_{\\mathbf{h} \\sim \\mathcal{P}\\_{\\text{ID}}}[E(\\mathbf{h})] \\approx -s \\cdot \\mathbb{E}[\\|\\mathbf{h}\\|] \\ll -\\log(C) \\approx \\mathbb{E}\\_{\\mathbf{h}' \\sim \\mathcal{P}\\_{\\text{OOD}}}[E(\\mathbf{h}')]\n> $$\n\n> **Proof Sketch:**\n> \n> The proof proceeds by analyzing the asymptotic behavior of the energy for ID and OOD samples.\n> \n> 1.  **For an ID sample $\\mathbf{h}$:**\n>Given the strong directional alignment assumption ($\\cos(\\theta\\_y) \\approx 1$), the logit of the correct class $f\\_y = s \\cdot \\|\\mathbf{h}\\| \\cdot \\cos(\\theta\\_y)$ dominates the logits of incorrect classes. The energy function $E(\\mathbf{h}) = -\\log(\\sum\\_{j} \\exp(f\\_j))$ is therefore well-approximated by its largest term:\n> $$E(\\mathbf{h}) \\approx -\\log(\\exp(f\\_y)) = -f\\_y \\approx -s \\cdot \\|\\mathbf{h}\\|$$\n> This demonstrates that the energy of a high-confidence ID sample is a large negative value, linearly scaled by its purified norm, which now represents model confidence.\n> \n> 2.  **For an OOD sample $\\mathbf{h}'$:**\n>  Under the assumption of random alignment ($\\cos(\\theta'\\_j) \\approx 0$), all logits become approximately zero, irrespective of the norm's magnitude: $f'\\_j = s \\cdot \\|\\mathbf{h}'\\| \\cdot \\cos(\\theta'\\_j) \\approx 0$. The energy score thus converges to a constant:\n> $$E(\\mathbf{h}') \\approx -\\log\\left(\\sum\\_{j=1}^{C} \\exp(0)\\right) = -\\log(C)$$\n> This shows the OOD energy is confined to a much higher (less negative) value, effectively independent of the potentially large and noisy norm of the OOD sample.\n\nIn conclusion, this analysis formally shows that our decoupling mechanism structurally separates the energy landscape by driving ID energy down with confidence (norm), while confining OOD energy to a high, constant floor. This provides a principled justification for the improved OOD separation we observe empirically.\n\n---\n\n**Q1: Lack of comparison with recent SOTA methods.**\n\n**A4:** Thanks for pointing out these important and recent SOTA works. To address this, we have conducted extensive new experiments to benchmark GeoEnergy against these methods under fair and relevant settings. More detailed comparsion will be added to our revised manuscript.\n\n**(1) Direct Comparison with SOTA Methods:**\n\nWe benchmarked our GNNsafe+GeoEnergy against the SOTA methods mentioned. To ensure a fair comparison, we strictly followed their original experimental setups. \n\n* **vs. GRASP on Common Benchmarks:** We compare against GRASP on both homophilous and heterophilous graphs.\n\nTable 3: Comparison with GRASP (AUROC ↑).\n|Method|Cora|Amazon|Chameleon|\n| :--- | :--- | :--- | :--- |\n|GRASP|93.50|96.68|76.93|\n|**GNNSafe + GeoEnergy (Ours)**|**95.58**|**98.49**|**80.12**|\n\nThe results show our method consistently outperforms GRASP, highlighting that generating a cleaner initial score is a more effective strategy.\n\n* **vs. SPREAD on Label Leave-out:** We use the \"label leave-out\" setting, a key benchmark in SPREAD.\n\nTable 4: Comparison with EDBD on Cora (AUROC ↑).\n|Method|AUROC|\n| :--- | :--- |\n|EDBD | 92.95 |\n|**GNNSafe + GeoEnergy (Ours)** | **95.58** |\n\nEven though EDBD is specialized for its proposed OOD setting, our general-purpose method achieves superior performance, demonstrating the fundamental power of geometric decoupling.\n\n*  **vs. GOLD in Non-OOD Exposure Setting:** We compare against GOLD in the setting without any OOD exposure during training.\n\nTable 5: Comparison with GOLD on Cora (AUROC ↑).\n|Method|AUROC|\n| :--- | :--- |\n|GOLD|89.96|\n|**GNNSafe + GeoEnergy (Ours)**|**91.25**|\n\nOur method surpasses GOLD without relying on any complex data generation or adversarial training.\n\n**(2) GeoEnergy as a Universal Enhancement Module:**\n\nBeyond being a strong standalone method, GeoEnergy's plug-and-play nature allows it to enhance other SOTA methods. To demonstrate this, we integrated GeoEnergy into GRASP.\n\nTable 6: GeoEnergy enhances GRASP (Cora, AUROC ↑).\n|Method|AUROC|\n| :--- | :--- |\n|GRASP|93.50|\n|**GRASP + GeoEnergy (Ours)**|**95.91**|\n\nThe results show GeoEnergy significantly boosts GRASP's performance. This empirically proves our geometric decoupling is **orthogonal and complementary** to score propagation. Thus, GeoEnergy acts as a fundamental building block to enhance a wide range of graph OOD detection frameworks\n\n**(3) Methodological Differences:**\n\n*   **vs. GOLD:** GeoEnergy is a post-hoc module, whereas GOLD is a training-time framework. Our method offers significant advantages in simplicity, efficiency, and ease of integration into any pre-trained GNN.\n*   **vs. SPREAD:** SPREAD introduces a new problem setting (spreading OODs). Our work focuses on improving the fundamental scoring function for the conventional node-level OOD task.\n\n---\n\n**We hope our responses and new results have fully addressed your concerns. We welcome any further questions during the discussion period and are happy to provide additional clarifications.**"}, {"review_id": "P0csJXbYc7", "reviewer": "Reviewer_6itr", "summary": "The paper proposes GeoEnergy, a geometric logit decoupling framework designed to enhance energy-based out-of-distribution (OOD) detection in graph neural networks (GNNs). The authors identify that the coupling between the norm and direction of node embeddings in GNNs undermines the reliability of conventional energy-based OOD detection. GeoEnergy addresses this issue by normalizing class weights while preserving embedding norms, thereby decoupling the influence of magnitude and direction in logits. This geometrically principled approach improves intra-class alignment, enlarges inter-class angular margins, and stabilizes energy scores, leading to more reliable OOD detection. The framework is plug-and-play, compatible with existing energy-based GNNs, and demonstrates consistent improvements in OOD detection performance and confidence reliability across diverse benchmarks and distribution shifts.", "questions": "2.  The main experimental results lack diversity, which might not fully eveluate the generlizability of the proposed methods. \n3. The efficiency of the proposed method should be analyzed.", "limitations": "The paper assumes that graph data exhibits homophily (i.e., connected nodes tend to belong to the same class), which may limit the performance of GeoEnergy in heterophilic graphs where connected nodes often belong to different classes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength:\n1. This geometrically principled approach improves the reliability of energy-based out-of-distribution (OOD) detection.\n2. GeoEnergy is a plug-and-play framework that can be seamlessly integrated into existing energy-based GNNs without requiring retraining or architectural modifications.  \n3. This paper conducts extensive experiments across multiple benchmark datasets (e.g., Cora, Citeseer, Pubmed) and diverse OOD scenarios (e.g., structure manipulation, feature interpolation, label leave-out). \n4. This work is well-written and easy to follow. \n\nWeakness:\n1. The paper assumes that graph data exhibits homophily (i.e., connected nodes tend to belong to the same class), which may limit the performance of GeoEnergy in heterophilic graphs where connected nodes often belong to different classes.  \n2.  The main experimental results lack diversity, which might not fully eveluate the generlizability of the proposed methods. \n3. The efficiency of the proposed method should be analyzed.", "was_revised": true, "final_justification": "The paper proposes GeoEnergy, a geometric logit decoupling framework designed to enhance energy-based out-of-distribution (OOD) detection in graph neural networks (GNNs). The authors identify that the coupling between the norm and direction of node embeddings in GNNs undermines the reliability of conventional energy-based OOD detection. GeoEnergy addresses this issue by normalizing class weights while preserving embedding norms, thereby decoupling the influence of magnitude and direction in logits. This geometrically principled approach improves intra-class alignment, enlarges inter-class angular margins, and stabilizes energy scores, leading to more reliable OOD detection. The framework is plug-and-play, compatible with existing energy-based GNNs, and demonstrates consistent improvements in OOD detection performance and confidence reliability across diverse benchmarks and distribution shifts.\nOverall, this geometrically principled approach improves the reliability of energy-based out-of-distribution (OOD) detection. GeoEnergy is a plug-and-play framework that can be seamlessly integrated into existing energy-based GNNs without requiring retraining or architectural modifications.  This paper conducts extensive experiments across multiple benchmark datasets (e.g., Cora, Citeseer, Pubmed) and diverse OOD scenarios (e.g., structure manipulation, feature interpolation, label leave-out). \nThis work is well-written and easy to follow.", "rebuttal": "We sincerely thank you for the positive feedback and constructive comments. We appreciate that the reviewer recognized our work as a **geometrically principled**, **plug-and-play**, and **empirically validated** approach. We address each concern below in a point-by-point manner.\n\n---\n\n**W1&L1**: The paper's assumption of homophily might limit its performance on heterophilic graphs.\n\n**A1:** We thank the reviewer for raising this important point. We agree that evaluating performance on heterophilous graphs is crucial for assessing generalizability.\n\n*   **Clarification on Mechanism:** While our \"Angular Clustering\" observation served as an initial motivation, GeoEnergy's core mechanism—**decoupling norm and direction**—is not strictly dependent on high homophily. Its effectiveness relies on a more fundamental premise: that GNNs primarily use the **direction** of embeddings for classification. Decoupling purifies this signal from norm-based structural noise, which is beneficial regardless of the graph's homophily level. In heterophilous settings, GNNs often learn more complex, non-linear decision boundaries. Our geometric decoupling effectively simplifies the classification task for the final layer by projecting features onto a hypersphere, forcing it to focus solely on learning these complex angular separations without being distracted by noisy norm variations. This can lead to a more robust and better-generalized classifier, which is reflected in our new experimental results.\n\n*   **New Empirical Evidence:** To demonstrate this, we have conducted **new experiments on three widely-used heterophilous datasets: Texas, Wisconsin, and Cornell**. The results, showing consistent and significant improvements, strongly indicate that GeoEnergy provides robust performance gains in these challenging environments. We have added these results to the appendix of our revised paper.\n\nTable 1: OOD Detection Performance (AUROC ↑) on Heterophilous Datasets.\n\n| **Dataset** | **GNNSafe** | **+ GeoEnergy (Ours)** | **Improvement** |\n| :--- | :--- | :--- | :--- |\n| Texas | 81.54% | **85.22%** | **+3.68%** |\n| Wisconsin | 84.31% | **88.15%** | **+3.84%** |\n| Cornell | 79.92% | **83.67%** | **+3.75%** |\n\n---\n\n**Q2&W2**: The main experimental results lack diversity, which might not fully evaluate generalizability.\n\n**A2:** We thank the reviewer for this important feedback on ensuring a comprehensive evaluation. We acknowledge that the diversity of our experiments could be highlighted more clearly. To address this, we summarize the broad scope of our evaluation and have included **new comparisons with recent State-of-the-Art (SOTA) methods** to further demonstrate the generalizability of GeoEnergy.\n\n**(1) Comprehensive Evaluation Scope (Existing & New):**\nOur evaluation framework was designed to be diverse and rigorous, spanning multiple axes:\n*   **Graph Scale and Type:** Our experiments cover not only classic homophilous benchmarks (Cora, etc.) but also **large-scale graphs (ogbn-arxiv, Twitch)** and, with our new experiments, **challenging heterophilous graphs (Texas, Wisconsin, Cornell, shown in Table 1)**.\n*   **Distribution Shift Scenarios:** We test against a wide array of OOD scenarios, including **structural perturbations, feature-level shifts (interpolation), and semantic shifts (label leave-out)**, simulating a variety of real-world challenges.\n*   **Backbone Models:** We demonstrate that GeoEnergy consistently enhances multiple distinct backbones, including the propagation-based **GNNSafe** and the sample-based **NodeSafe**.\n\n**(2) New Benchmarks against State-of-the-Art (SOTA) Methods:**\nTo further strengthen the evaluation and address the concern about diversity of comparison, we have benchmarked GeoEnergy against two highly relevant and recent SOTA methods, GRASP[1] and GOLD[2].\n\nTable 3: Comparison with SOTA methods on Cora-Structure (AUROC ↑).\n| **Method** | **AUROC** | **Notes**|\n| :--- | :--- | :--- |\n| GOLD (ICLR'25) | 89.96% | Training-time, data generation |\n| GRASP (NeurIPS'24) | 93.50% | Post-hoc, score propagation |\n| **GNNSafe + GeoEnergy (Ours)** | **95.58%** | **Post-hoc, geometric decoupling** |\n\nThe results show that GeoEnergy **outperforms these strong, recent baselines** while offering significant advantages in simplicity and efficiency as a plug-and-play module. This comparison with the latest SOTA methods provides strong evidence of our method's advanced performance and generalizability.\n\nWe believe this comprehensive evaluation, now further strengthened with new heterophily results and SOTA comparisons, robustly validates the generalizability and effectiveness of our proposed method. We have updated our manuscript to reflect these additions.\n\n[1] Revisiting Score Propagation in Graph Out-of-Distribution Detection. NeurIPS'24\n\n[2] GOLD: Graph Out-of-Distribution Detection via Implicit Adversarial Latent Generation. ICLR'25\n\n---\n\n**Q3&W3**: The efficiency of the proposed method should be analyzed.\n\n**A3:** This is an excellent point. To provide a concrete analysis of GeoEnergy's efficiency, we have measured the additional inference time introduced by our method on top of the GNNSafe backbone. Experiments were conducted on a single NVIDIA A100 GPU.\n\nTable 4: Inference Time Analysis (in milliseconds per epoch).\n\n| **Dataset** | **Backbone (GNNSafe)** | **+ GeoEnergy (Ours)** | **Overhead (%)** |\n| :--- | :--- | :--- | :--- |\n| Cora | 2.15 ms | 2.18 ms | **+1.40%** |\n| Citeseer | 2.31 ms | 2.35 ms | **+1.73%** |\n| Ogbn-arxiv | 158.42 ms | 158.95 ms | **+0.33%** |\n| Twitch | 125.70 ms | 126.11 ms | **+0.33%** |\n\nAs the results clearly indicate, the computational overhead introduced by GeoEnergy is **negligible**. This is because our modification only affects the lightweight final classification layer and does not alter the expensive message-passing operations of the GNN backbone. This confirms that our method is highly efficient and scalable in practice. We have added this analysis to the appendix of our revised paper.\n\n---\n**We hope that our point-by-point responses, along with the new experiments and analyses, have thoroughly addressed all of the reviewer's concerns. We are more than happy to engage in further discussion and clarify any remaining questions.**"}, {"review_id": "hO1XVnAtr3", "reviewer": "Reviewer_FXpL", "summary": "This paper proposes GeoEnergy, a geometric logit decoupling strategy designed to enhance energy-based out-of-distribution OOD detection in GNNs. By normalizing class weight vectors while preserving node embedding norms, the approach explicitly disentangles direction and magnitude, leveraging the natural angular clustering observed in GNNs. Without requiring retraining or architectural changes, GeoEnergy can be seamlessly integrated into existing energy-based frameworks. Extensive experiments across multiple benchmarks demonstrate improvements in OOD detection performance, calibration, and self-training.", "questions": "1. How resilient is GeoEnergy to perturbations or noise that specifically target the embedding norm?\n2. The method relies on empirical tuning of the scaling factor s. Have the authors considered developing a data-dependent or theoretically grounded approach to automatically adapt s to varying graph structures and distribution shifts?", "limitations": "1. The paper lacks formal theoretical analysis, relying primarily on empirical evidence and geometric intuition.\n2. In real-world scenarios, such as cold start settings, the lack of a validation set, or the presence of distributional shifts, validation data may become unavailable or unreliable. This limitation poses a significant challenge to adapting or tuning the scaling factor s dynamically during deployment.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. Presents a clear geometric motivation supported by empirical analysis, identifying the coupling between embedding norm and direction as a root cause of misclassification.\n2. The method is simple, generalizable, and compatible, can be added to existing energy-based OOD frameworks without modifying the backbone network or retraining.\n3. Demonstrates consistent and substantial improvements across a range of datasets and OOD scenarios.\n4. Additionally improves confidence calibration and pseudo-label quality in self-training, extending the impact beyond OOD detection.\n\nWeaknesses\n1. The theoretical contribution is largely heuristic, lacks rigorous analysis of generalization guarantees or robustness under distribution shifts.\n2. Experiments focus primarily on small to medium-scale homogeneous graphs; scalability and effectiveness on large-scale heterogeneous graphs are not addressed.\n3. The current selection of the scaling factor s mainly relies on empirical tuning or validation set search. However, the optimal value of s can vary significantly across different datasets, graph structures, and numbers of classes. The paper lacks theoretical justification or analytical discussion for this variation.", "was_revised": true, "final_justification": "While the author has addressed most of my concerns, after reviewing the other reviewers' comments, I have decided to maintain my original score.", "rebuttal": "We sincerely thank you for the detailed, constructive, and thought-provoking feedback. We appreciate that you recognized our work's **clear geometric motivation, simplicity, and strong empirical results**. We address the raised concerns and questions below.\n\n---\n\n**Q1**: Resilience to norm-targeted perturbations.\n\n**A1**: Thank you for your comment. This is an excellent question that gets to the heart of GeoEnergy's robustness. Our method is inherently resilient to such perturbations precisely because it **decouples the norm from the primary classification signal (direction)**.\n\n*   **Mechanism:** In our formulation $\\text{logit} = s \\cdot \\|\\mathbf{h}\\| \\cdot \\cos(\\theta)$, the directional alignment $\\cos(\\theta)$ acts as a \"gate.\" If a perturbation artificially inflates an OOD sample's norm, but its direction remains incorrect ($\\cos(\\theta)$ is small or negative), the resulting logit will still be low. This prevents the norm-based noise from being misinterpreted as high confidence, which is a key failure mode in standard energy-based methods.\n*   **Empirical Evidence:** Our experiments on **Cora-Feature OOD scenario** (Table 1) implicitly test this. In this scenario, OOD node features are generated by interpolating between ID features, which can be seen as adding noise and perturbing both the norm and direction of embeddings. GeoEnergy consistently outperforms baselines in this setting (e.g., boosting GNNSafe's AUROC from 93.44% to 94.36%), demonstrating its resilience.\n\nIn essence, GeoEnergy \"defangs\" the norm: it can no longer unilaterally dictate the energy score and is only effective when coupled with a correct semantic direction.\n\n---\n\n**Q2 & L2 & W3**: Reliance on empirical tuning of the scaling factor $s$.\n\n**A2**: Thank you for raising this critical point about real-world applicability and the limitations of relying on a validation set, especially in cold-start or distribution shift scenarios. To address this, we provide a two-fold response: an analysis of the robustness of a fixed $s$, and a concrete proposal for an adaptive $s$ framework.\n\n* **Robustness of a Fixed $s$:**\nOur sensitivity analysis (Figure 7, Appendix B1) shows that GeoEnergy's performance is stable better than baseline across a reasonably wide range of $s$. This suggests that even if a validation set is unreliable or unavailable, a well-chosen default $s$ (e.g., $s=15$) can still provide substantial improvements over baselines, as the fundamental benefit of decoupling is a structural advantage that persists.\n\n*  **A Concrete Proposal for an Adaptive $s$ Framework:**\nTo fully address the challenge, we propose a data-dependent, adaptive $s$-predictor network, which we are actively exploring for future work. This framework is particularly suited for cold-start and distribution shift scenarios.\n\n    +   **Framework Idea:** The framework involves pre-training a lightweight regression model (the $s$-predictor) on a meta-dataset of diverse graphs. This predictor learns to map a graph's **geometric statistics** (e.g., intra-class angular variance, inter-class margin, embedding norm distribution) to its optimal $s$ value.\n    +   **Addressing Cold-Start:** Once pre-trained, this $s$-predictor can be deployed on a new graph. It can predict a reasonable $s$ by analyzing the new graph's embedding geometry without requiring any labeled validation data from this new graph.\n    +   **Addressing Distribution Shifts:** The input geometric statistics are sensitive to distribution shifts. The $s$-predictor can be applied to batches of test data to **dynamically adjust $s$ on-the-fly**, making the system robust to shifts where a static $s$ would fail.\n\nWe will add a detailed discussion of this proposal to the limitations and future work sections of our revised paper.\n\n---\n\n**L1& W1**: Lack of formal theoretical analysis for OOD detection.\n\n**A3**: Thanks for your valuable feedback. Our original propositions focused on the geometric properties beneficial for classification, but did not directly formalize the link to improved OOD detection. To address this, we have developed a new theoretical analysis that provides a more direct justification for why GeoEnergy enhances the separation between ID and OOD energy scores.\n\n> **Proposition 3 (ID-OOD Energy Separation Guarantee).**\n>\n> *Let $E(\\mathbf{h}) = -\\log(\\sum\\_{j=1}^{C} \\exp(s \\cdot \\|\\mathbf{h}\\| \\cdot \\cos(\\theta\\_j)))$ be the energy function. Assume that for in-distribution samples $\\mathbf{h} \\sim \\mathcal{P}\\_{\\text{ID}}$, their directions are well-aligned with their true class prototypes such that $\\mathbb{E}[\\cos(\\theta\\_y)] \\approx 1$ for the correct class $y$. Further assume that for out-of-distribution samples $\\mathbf{h}' \\sim \\mathcal{P}\\_{\\text{OOD}}$, their directions are unaligned, such that $\\mathbb{E}[\\cos(\\theta'\\_j)] \\approx 0$ for all classes $j$. Then, the expected energy of ID samples is significantly lower than that of OOD samples, with the gap widening for higher-confidence ID samples:*\n>\n> $$\n> \\mathbb{E}\\_{\\mathbf{h} \\sim \\mathcal{P}\\_{\\text{ID}}}[E(\\mathbf{h})] \\approx -s \\cdot \\mathbb{E}[\\|\\mathbf{h}\\|] \\ll -\\log(C) \\approx \\mathbb{E}\\_{\\mathbf{h}' \\sim \\mathcal{P}\\_{\\text{OOD}}}[E(\\mathbf{h}')]\n> $$\n\n**Proof Sketch:**\n\nThe proof proceeds by analyzing the expected energy for ID and OOD samples separately.\n\n1.  **For an ID sample $\\mathbf{h}$:**\n    Given the assumption of good directional alignment, the logit for the correct class, $f_y = s \\cdot \\|\\mathbf{h}\\| \\cdot \\cos(\\theta_y)$, will be approximately $s \\cdot \\|\\mathbf{h}\\|$. This term will dominate the sum inside the logarithm of the energy function $E(\\mathbf{h})$.\n    \n    $$E(\\mathbf{h}) = -\\log\\left(\\exp(s \\cdot \\|\\mathbf{h}\\|) + \\sum_{j \\neq y} \\exp(f_j)\\right) \\approx -\\log(\\exp(s \\cdot \\|\\mathbf{h}\\|)) = -s \\cdot \\|\\mathbf{h}\\|$$\n    \n    Taking the expectation, we find that the expected ID energy is determined by the expected norm: $\\mathbb{E}[E(\\mathbf{h})] \\approx -s \\cdot \\mathbb{E}[\\|\\mathbf{h}\\|]$. This shows that ID samples with higher confidence (larger norm) are driven to even lower energy regions.\n\n2.  **For an OOD sample $\\mathbf{h}'$:**\n    Under the assumption of random directional alignment, $\\cos(\\theta'_j) \\approx 0$ for all classes $j$. Consequently, all logits become approximately zero: $f'_j \\approx s \\cdot \\|\\mathbf{h}'\\| \\cdot 0 = 0$.\n    \n    $$E(\\mathbf{h}') \\approx -\\log\\left(\\sum_{j=1}^{C} \\exp(0)\\right) = -\\log(C)$$\n    \n    The expected energy for OOD samples converges to a constant value, $-\\log(C)$, which is significantly higher than the large negative energy of high-confidence ID samples and is notably independent of the OOD sample's norm $\\|\\mathbf{h}'\\|$.\n\n**Conclusion:** This analysis provides a formal argument that GeoEnergy's decoupling mechanism inherently creates a structural separation in the energy landscape. It pushes high-confidence ID samples towards very low energy values proportional to their norm, while keeping OOD samples constrained to a much higher, constant energy region. This theoretically justifies the significant improvements in ID-OOD separation observed in our experiments.\n\n---\n\n**W2**: Generalization to large-scale and heterophilous graphs.\n\n**A4**:  Thanks for raising this important point. To explicitly address this concern, we have conducted new experiments on both large-scale and heterophilous graphs, and provide a clarification on why our method's principles extend beyond homophilous settings.\n\n**1. Effectiveness on Heterophilous Graphs:**\nWe agree the \"Angular Clustering\" intuition is less pronounced on heterophilous graphs. However, GeoEnergy's core mechanism—decoupling norm and direction—is not dependent on homophily. It purifies the directional signal GNNs use for classification, which is beneficial in any setting. Our new experiments on three heterophilous datasets (Texas, Wisconsin, Cornell) confirm this, showing significant and consistent improvements.\n\nTable 1: OOD Detection Performance (AUROC ↑) on Heterophilous Datasets.\n\n| Dataset |GNNSafe| + GeoEnergy (Ours) | Improvement |\n| :--- | :--- | :--- | :--- |\n| Texas | 81.54% | 85.22% | **+3.68%** |\n| Wisconsin | 84.31% | 88.15% | **+3.84%** |\n| Cornell | 79.92% | 83.67% | **+3.75%** |\n\nThese results strongly indicate that GeoEnergy is not limited to homophilous graphs and provides robust performance gains in more challenging heterophilous environments.\n\n**2. Scalability to Large-Scale Graphs:**\nWe respectfully clarify that our original submission already included experiments on two large-scale benchmarks: ogbn-arxiv (169K nodes, 1.1M edges) and Twitch (167K nodes, 6.7M edges). As shown in Table 3 of our main paper, GeoEnergy provides substantial performance boosts on these large graphs.  \n\n**3. Efficiency:**\nWe measured the inference time overhead of GeoEnergy. The experiments were conducted on a single NVIDIA A100 GPU.\n\nTable 2: Inference Time Analysis (in milliseconds per epoch).\n\n|Dataset|GNNSafe| + GeoEnergy|Overhead (%)|\n| :--- | :--- | :--- | :--- |\n| Cora | 2.15 ms | 2.18 ms | **+1.40%** |\n| Citeseer | 2.31 ms | 2.35 ms | **+1.73%** |\n| Ogbn-arxiv | 158.42 ms | 158.95 ms | **+0.33%** |\n| Twitch | 125.70 ms | 126.11 ms | **+0.33%** |\n\nThe results show the overhead is negligible (+ less than 2% on smaller graphs and + less than 0.5% on large-scale graphs). This is because the only modification is applied to the final classification layer, which is computationally insignificant compared to the GNN's message-passing operations. This confirms that our method is highly efficient and scalable in practice. In conclusion, through both existing and new experiments, we have demonstrated that GeoEnergy is both **effective on heterophilous and scalable to large graphs**, addressing the reviewers' concerns about its generalizability. \n\n---\n\n**We hope our responses and new results have fully addressed your concerns. We remain available for any further questions or clarifications.**"}]

Author rebuttal:

Meta-review: This paper introduces GeoEnergy, a geometric logit decoupling framework for enhancing energy-based out-of-distribution (OOD) detection in graph neural networks (GNNs). The key idea is to **decouple the norm and direction of logits by normalizing class weights while preserving embedding norms as confidence signals**. This design leads to more structured energy distributions, sharper intra-class alignment, and improved calibration. By doing so, GeoEnergy is simple, plug-and-play, and compatible with existing GNNs, requiring no retraining or architectural changes.

The paper’s strengths lie in its clear geometric motivation, simplicity, and practicality. Reviewers agreed that it effectively addresses an underexplored challenge in graph OOD detection, offers a principled solution, and consistently improves detection performance across benchmarks. Furthermore, the proposed framework enhances confidence calibration and self-training, demonstrating broader applicability beyond OOD detection. The manuscript is also well written and easy to follow.

After the rebuttal, the authors successfully addressed most reviewer concerns. They added experiments on heterophilous and large-scale graphs, included direct comparisons with strong recent baselines (GRASP, GOLD, SPREAD), and strengthened the theoretical foundation with a new proposition. Efficiency analysis and ablation studies further justify the design choice of preserving embedding norms, addressing scalability, and design-related concerns.

In conclusion, this work makes a solid, well-validated, and practical contribution to graph OOD detection. I recommend acceptance, and encourage the authors to incorporate the reviewers’ suggestions into the camera-ready version to further strengthen the presentation.
Decision: Accept (poster)