Decoupled Graph Energy-based Model for Node Out-of-Distribution Detection on Heterophilic Graphs
Authors: Yuhan Chen
Yihong Luo
Yifan Song
Pengwen Dai
Jing Tang
Xiaochun Cao
Venue: ICLR 2025 Poster (2025)
Keywords: Node OOD detection; Energy-based Models; Graph Neural Network
Primary area: learning on graphs and other geometries & topologies

Abstract: Despite extensive research efforts focused on Out-of-Distribution (OOD) detection on images, OOD detection on nodes in graph learning remains underexplored. The dependence among graph nodes hinders the trivial adaptation of existing approaches on images that assume inputs to be i.i.d. sampled, since many unique features and challenges specific to graphs are not considered, such as the heterophily issue. Recently, GNNSafe, which considers node dependence, adapted energy-based detection to the graph domain with state-of-the-art performance, however, it has two serious issues: 1) it derives node energy from classification logits without specifically tailored training for modeling data distribution, making it less effective at recognizing OOD data; 2) it highly relies on energy propagation, which is based on homophily assumption and will cause significant performance degradation on heterophilic graphs, where the node tends to have dissimilar distribution with its neighbors. To address the above issues, we suggest training Energy-based Models (EBMs) by Maximum Likelihood Estimation (MLE) to enhance data distribution modeling and removing energy propagation to overcome the heterophily issues. However, training EBMs via MLE requires performing Markov Chain Monte Carlo (MCMC) sampling on both node feature and node neighbors, which is challenging due to the node interdependence and discrete graph topology. To tackle the sampling challenge, we introduce Decoupled Graph Energy-based Model (DeGEM), which decomposes the learning process into two parts—a graph encoder that leverages topology information for node representations and an energy head that operates in latent space. Additionally, we propose a Multi-Hop Graph encoder (MH) and Energy Readout (ERo) to enhance node representation learning, Conditional Energy (CE) for improved EBM training, and Recurrent Update for the graph encoder and energy head to promote each other. This approach avoids sampling adjacency matrices and removes the need for energy propagation to extract graph topology information. Extensive experiments validate that DeGEM, without OOD exposure during training, surpasses previous state-of-the-art methods, achieving an average AUROC improvement of 6.71% on *homophilic* graphs and 20.29% on *heterophilic* graphs, and even outperform methods trained with OOD exposure. Our code is available at: [https://github.com/draym28/DeGEM](https://github.com/draym28/DeGEM).

Published as a conference paper at ICLR 2025
DECOUPLED GRAPH ENERGY-BASED MODEL FOR NODE DUT-OF-DISTRIBUTION DETECTION ON HET- EROPHILIC GRAPHS
Yuhan Chen Yihong Luo"; Yifan Song", Pengwen Dai', Jing Tang" Xiaochun Cao 41
The School of Computer Science and Engineering. Sun Yat-sen University
2 The Hong Kong University of Science and Technology
3 The Hong Kong University of Science and Technology (Guangzhou)
The School of Cyber Science and Technology, Shenzhen Campus of Sun Yat-sen University
ABSTRACT
Despite extensive research efforts focused on on images, OOD detection on nodes in graph learning remains underexplored. The dependence among graph nodes hinders the trivial adaptation of existing ap- proaches on images that assume inputs to be i.i.d. sampled, since many unique features and challenges specific to graphs are not considered. such as the het- erophily issue. Recently, GNNSafe, which considers node dependence, adapted energy-based detection to the graph domain with state-of-the-arr performance, how- ever, it has two serious issues: derives node energy from classification logits without specifically tailored training for modeling data distribution, making it less effective at recognizing OOD data; 2) it highly relies on energy propagation, which is based homophily assumption and will cause significant performance degrada- tion on heterophilic graphs, where the node tends to have dissimilar distribution with its neighbors. To address the above issues, suggest training Energy- based Models (EBMs) by Maximum Likelihood Estimation (MLE) to enhance data dis- tribution modeling and removing energy propagation overcome the heterophily issues. However, training EBMs via MLE requires performing Markov Chain Monte Carlo (MCMC) sampling on both node feature and node neighbors, which is challenging due to the node interdependence and discrete graph topology. To tackle the sampling challenge, we introduce Decoupled Graph Energy- based Model (DeGEM), which decomposes the learning process into two parts-a graph encoder that leverages topology information for node representations and an energy head that operates in latent space. Additionally, we propose Multi-Hou Graph encoder (MH) and Energy Readout (ERo) to enhance node representation learning, Condi- tional Energy (CE) for improved EBM training, and Recurrent Update for the graph encoder and energy head to promote each other. This approach avoids sampling adjacency matrices and removes the need for energy propagation to extract graph topology information. Extensive experiments validate that DeGEM, without OOD exposure during training, surpasses previous ofate-of-the-art methods, achieving an average AUROC improvement of 6.711% 1% on homophilic graphs and 20.29% on heterophilic graphs, and even outperform methods trained with OOD exposure. Our code is available at: https:: //gi thub. www.coo//o/oooooo htmm/ddaam22///11 8/DGGM
(OOD) detection
INTRODUCTION
Detecting nutrof-Distribution (OOD) data is crucial for enhancing Al models' robustness, reliability, and safety in real-world scenarios, where input data may deviate from the training data distribution. Many works (Hendrycks & Gimpel, 2016 Liang et al. [0188] Lee et al. 2018 Hendrycks et al. 2018 Liu et al. 2020 have been proposed for OOD detection tasks i.i.d. data, e.g., images. Such an
"Co-First Authors: Yuhan Chen Straym@qq.com and Yihong Luo (yluocg conggeCooonnn.s.h..)
'Correponding Authors: Jing Tang (jingtang @ hk) and Xianchun Cao (caoxiaochun @ mail. sys..cu.us.....

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urgent need also exists in the domains that apply graph data, such medical-diagnosii (Kukar 2003) and autonomous driving (Dai & Van Gool 2018). However, there has been a notable lack of work focusing graph OOD node detection, Lc.. detecting OOD nodes on which the model is expected to have low confidence (Amodei et al. 2016 Liang et al. 2018). Due to the interdependence among graph nodes, it is hard apply the methods designed for i.i.d. inputs directly.
Recently, GNNSafe (Wu et al. 2023a adapted energy-based
OOD detection for graph data mainly by using GNNs as the backbone, and output energy for nodes as confidence, achiev- ing current state-of-the-eee performance. However, it trained by classification loss and constructs node energy by classifica- tion logits without specifically designed training for modeling data distribution, limiting the performance of identifying OOD nodes. Moreover, the energy propagation technique, used in GNNSafe, heavily relies on the homophily assumption, i.e.,
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Twitch
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Amazon
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neighbors often belong to the same class. This will lead to significant performance degradation where neighbors do not share a similar distribution. To mitigate these issues, we suggest training Energy-based Models (EBMs) to detect OOD instances via Maximum Likelihood Estimation (MLE) to obtain better energy, and removing energy propagation to address the heterophily issue. However, training EBMs via MLE requires performing Markov Chain Monte Carlo (MCMC) sampling in training, which is notorious for graphs due to the complexity of graph topology.
Cornell
heterophilic graphs,
Figure 1: AUROC across graphs.
In this paper, our proposed Decoupled Graph Energy-based Model (DeGEM) alleviates both the heterophily issue and sampling challenges of leamning EBMs for large graphs. The key insight in our work is: GNNs can extract the topology information, forming latent space without interdependence. Hence, we can conduct MCMC sampling on latent space to train the energy head. This approach is computationally efficient, as it avoids sampling the adjacency matrix, and eliminates the need for energy propagation techniques, which could degrade performance on heterophilic graphs. Specifically, DeGEM decomposes the EBM into two components: graph encoder and energy head. First, the graph encoder is trained by the Graph Contrastive Learning (GCL) algorithm and classification loss for obtaining informative node representations, then the energy head, trained over latent space by MLE, outputs the energy scores of nodes, which are used as nodes' OOD scores. By doing so, DeGEM transposes operations inherently dependent on the graph structure into the latent representation domain, thereby decoupling subsequent steps from the graph structure's dependency. The benefits are twofold: 1) MCMC sampling can be efficiently conducted on low-dimensional latent space to sample node representations only. dramatically decreasing the computation cost; 2) the Energy head does not require a propagation operation to further extract topology information, avoiding performance degradation on heterophilic graphs.
Moreover, better unleash the effectiveness of DeGEM in node OOD detection, we propose several principled training designs: Multi-Hop Graph encoder (MH) and Energy Readout (ERo) to enhance node representation learning, Conditional Energy (CE) to improve EBM training, and Recurrent Update to effectively update the CE and ERo jointly, enabling the graph encoder and energy head promote each other. Furthermore, we found that existing node OOD detection methods are evaluated only on homophilic graphs, with heterophilic graphs being overlooked. We therefore conduct comprehensive evaluation of existing methods across homophilic and heterophilic graphs. The results show that existing graph-based methods deliver even worse performance on heterophilic graphs, compared to those graph-agnostic node OOD detection methods (Fig. D. Thanks to the powerful constructed DeGEM, our method can achieve state-of-the-art performance on OOD detection across both homophilic and heterophilic graphs without OOD exposure.
Our key contributions can be summarized as follows:
Our proposed DeGEM decomposes EBM into two components, graph encoder for extracting topology information and an energy head for estimating density, which avoids the notorious challenges of sampling the adjacency matrix when training via MLE and prevents serious performance degradation on heterophilic graphs
We are the first to evaluate node OOD detection performance on both homophilic and heterophilic graphs and provide a comprehensive assessment of existing graph- based methods.
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Augmented Node Representations
Graph Contrastive
Learning
MCMC
Energy Function
Augmented Graph
Sampling
Sampled Energies
Graph Encoder
Energy Readout
Contrastive Loss
Original Graph
Sampled Nodes
Original Node
Readout
Traditional MCMC sampling on graphs: Difficutt
Original Graph
Representations
Condition
Classification Loss
Graph Encoder
MCMC Sampling
Energy Function
Original
Sampled Node Energies
Representations
Original Node Energies
Original Graph
Original Node Representations
Sampled Node Representations
Sampled Node Representations
Energy Head
EBM
Loss
Our proposed MCMC sampling graphs: Easy!
Sampling
Hisg Stap-gradiie
Recument Update
EBM Training
5it
Sampled Node Energies
Figure 2: A & The detailed graph contrastive learning and EBM training process. The readout of original node representations in A participates the Conditional Energy (CE) in B, and the original node energies in B are delivered to A for Energy Readout (ERo). We propose Recurrent Update mechanism to jointly train the CE and ERo effectively. The comparison between traditional MCMC sampling and our proposed MCMC sampling.
We conduct extensive experiments to demonstrate the superiority of DeGEM. The results show that DeGEM, trained without OOD exposure, outperforms state-of-the-arr methods -whthhhswwhehher trained with or without OOD exposure-on on both homophilic and heterophilic graphs
whether
2 PRELIMINARY
Notations. We denote an undirected graph without self-loops as G (X, AJ, where X = { E N is the initial node feature matrix, do is the feature dimension, and E NxN is the adjacency matrix. (a,,) is the feature of neighbors of node D.. D is diagonal matrix standing for the degree matrix such that DR = Σ Aij- () = (h. in the f-th layer, where d is the hidden dimension. We use Y fy: J. R denote the ground-truth node label matrix, where C is the number of classes and y; is one-hot vector.
( Ji=1 E Nxd is the representation matrix
Graph Dut-of-Distribution Node Detection. OOD detection refers to identifying data samples that do not conform the distribution of the training data, while keeping the classification capability of in-distribution (ID) data. Formally, an OOD detection score function S(.,) should be defined to map the node and its neighbors a scalar score, such that S(x))()))) yields a higher value for OOD nodes than for ID nodes. It can be seen that in the context of graph data, OOD detection becomes challenging due the complex interplay between node features and graph topology. The OOD score for each node depends on itself and also the relational context provided by other nodes within the graph, which is distinct from OOD detection in the vision domain.
Heterophily Issue. Most traditional GNNs are designed based on the homophily assumption (Kipf Welling 2016 Velickovic et al. 2018). where linked nodes tend to be similar. However, heterophilic graphs-whers where linked nodes often belong to different categories- are prevalent in real-world appli- cations, on which the traditional GCNs perform poorly (Pei et al. 2020). While extensive works have been proposed to address the heterophily issue (Abu-EI-Haija et al.) 2019 Bo et al., 202T] Zhu et al. 2020a) Chien et al., 2021 Li et al. 2022a Chen al. [2023] Luo et al., (024a) in node classification tasks, it has not received sufficient attention in the context of node OOD detection, resulting in performance degradation when applying models designed for node OOD detection to heterophily graphs.
Energy-based Model. A deep EBM models (Du & Mordatch, 2019) the data distribution pa(æ)
using Boltzmann distribution pa pa( (a) exp (opp--ane [a]] with the energy function Ex(x), where Zo is the
Zo
corresponding normalizing constant. EBM is trained by minimizing the negative log-likelihood (NLL)
Lo of pa (x), such that Lg = -Em=pa(a) log Po log)o(z))
The EBM loss can be reformulated as follows:
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N
M
L = (x))
Ee Fa(
- Bg-ro
Eg (sg(())))
1
Σ
Eg
(z:)
1 M
Σ
Eg
[g(g((0))) (1)
N
j=1
where sg(:) denotes the stop-gradient operation. Sampling ø from po (+) can be achieved by a K-step Markov chain Monte Carlo (MCMC) sampling:
T (b) (( FV
+ ((
where ( is given initial samples, À is the step size, and ((k) ~ _NO0, o2) with a given noise o². More details of derivation can be found in the Appendix
METHODOL.OGY
3.1 REVISITING GRAPH EBM FOR OOD DETECTION
Different from image data, the graph data is neither continuous nor i.d.. The inherent challenge how to define and train an EBM over (x,,(())). Since (x) is discrete, potentially huge, and includes arbitrary neighbor nodes, it is hard to sample ((xx) by MCMC for training EBM.
To avoid such challenge, GNNSafe (Wu al. 2023b) proposes to train the EBMs by maximizing the conditional likelihood expí (x
Rext Serppho
Corate
log (y(()x En exp(- Eg (m.y))) via node classification, where (x v) = -V (xx)) and fo(:,-) is classifier. Finally, energy score of each node can regarded as OOD score, i.e., log E, exp(-Ee ( y)).
ESN (CC)
cat 20121
S(x,N(x)) Ex(x)
The EBMs trained in this way exhibit limitations in terms of capa- bility and heavy reliance on sufficient labeled data. In particular, it has not been trained for modeling the marginal data distribution, resulting in an inferior ability to capture the data distribution. For instance, may falter even when tasked with handling simple 2D dataset, i.e., Gaussians (Fig. 3). This indicates that the energy constructed by the classification logits does not effectively capture the underlying data distribution. Consequently, such methods intuitively face challenges in addressing OOD detection for graph data, particularly when labeled data are scarce.
Figure 3: Visualization com- parison on 2D data.
3.2
OVERVIEW OF OUR METHOD: DEGEM
The framework of proposed method is shown in Fig. DeGEM first utilizes graph coder trained by GCL algorithm (Veličković et al.) 2018 to extract node representations h = ga (a=)) E IRd. In this way, the topology information of the original graph well-eneoded into h, such that the follow-up steps can be free from A. Next, a K K-step MCMC sampling is applied over the low-dimensional latent space h ~ q (h) to learn an energy function fw, which is defined as an MLP. Please Appendix E.1 for detailed training algorithm.
3.3
OUR DESIGN
The powerful capability of EBM comes from its minimal restrictions modeling, but at the same time, this is also a double-edged sword that leads to difficult learning. To address the difficulty of defining EBMs on graphs, propose to restrict EBM modeling formulation to some extent. Specifically, we suggest decomposing EBM into two parts: the first part focuses on extracting graph structural information, and the second part is dedicated learning the energy.
For node æ and its neighbor (xx), we can define the EBM as Ex(x) fw o Ja (x, (xx)) where Ia (x,, = h IRd is graph encoder that focuses extracting the graph structural information, and fo E R is an energy function that outputs the energy score.
Suppose we have fixed ga that learns graph structural information well, following Eq. ( the EBM loss for learning fa can be reformulated as:
Leom = E Eps(ar (ssl (2;
(mr( (a (sg(/h)) - Byoch) [L((g(hi)) à (I (b)((ll))))
Ep_
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where (h) f (a())) (x))) distribution over latent space of real data and EBM samples, respectively.
and (h(h) = po(z)0 9a (x)) are the aggregated
It can be observed that the optimal solution is achieved Pa (h) qe (h),, and optimizing the objective actually equals minimizing the KL divergence between two distributions over latent space. Hence formulate surrogate optimization objective for directly learning latent EBM fw:
KT (pa (h))
= (p))
[bw) - (a (h))
(2)
where qu (h) expl ((1))) Sampling from qu (h) can be achieved by running K-step MCMC Zu
sampling. By the surrogate objective, we can move the MCMC sampling from (ar,N(x)) space to the informative latent space, avoiding the awkward and challenging scenarios of needing to sample nodes under high-dimension and their neighbors.
A left challenge is obtaining flexible ga to extract graph topology information. We suggest utilizing Graph Contrastive Learning (GCL) algorithm for training the graph encoder to extract node representations. This can not only extract informative features but also deal with the limited labeled data case. Among graph contrastive learning methods, we propose using DGI (Veličković et al. 2018) the learning algorithm for Ja- Intuitively, DGI contrasts between nodes and the whole graph, while the OOD detection task requires detecting the outliner nodes in a whole graph, the contrast between the local features and the global can bring more benefits. The effect of different GCL algorithms will be shown in Sec. 4.4
DGI first shuffles the rows of node features matrix to get augmented graph {X', A'), then obtains node representations by h; = Da (V:,,((r:)) and h2y = ga (aa)) respectively. The contrastive loss for learning Ja given by:
N
N
La 8) = 2N
Σ
log Do (hi, 8) +
Σ
log(1 -
(3)
where 8 = Σ h1 E Rd is the summary readout of all the node representations, which can be regarded as a global view the whole graph. Following DGI (Veličković et al. 2018), we use bilinear scoring function for the discriminator Dw(-,-), which is also learned by minimizing Let.
Additionally, the node representations h2 can be further enhanced by ID nodes classification with prediction head Is (e.g., MLP) as D; (h,), and the cross-entropy loss Les is used for learning the prediction head Ig (-) and also graph encoder 9a ..
Multi- Hop (MH) Graph Feature Encoder. Simply using a one-layer GCN, as in DGI (Veličkovid et al. 2018), cannot fully extract informative representations from original graph features. Addi- tionally, y, similar nodes often lie in the distance in heterophilic graphs, so combining long-range information could bring positive effects in addressing heterophily issue. Therefore, L-layer GNN is employed to learn graph node features in our proposed method. Furthermore, the weighted self- loops technique is used simple enhanced method for graph filter (Bo al. 202T Chen al. 2023). Finally, all the intermediate features (the output of all the layers) are fused to obtain the final representations.
First, symmetric normalized adjacency with weighted self-loops is used for extracting node features:
(+)
[II D- ADAAAD ((-1), x (0) X,
where 3 is hyper-pauee parameter. Note that this operation for the positive samples can be done in preprocessing since it does not involve any learnable parameters, which greatly reduce the running time. Then the intermediate features are transformed and fused to obtain the final representations:
(4)
Wene + Denes ( =
(()
(5)
where W and are learnable weights and bias. || is concatenation operation. The combination of Eqs. and (5) is denoted by H Ja (X, A).
Conditional Energy (CE). The readout summary s of ID dataset can be used as supervised infor- mation in energy function fu, to compare between node representations and global representations
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when computing energy. For instance, fw (h,, s) = psh + fw 8) h Ws. and etc., where p is a hhpee--paaaet parameter,, W and are learned parameters. The benefit of the global information introduced by the proposed CE is twofold: 1) It can guide MCMC sampling, reducing the sampling difficulty; 2) can act as global prior information when detecting OOD nodes.
Energy Readout (ERo). In DGI, the readout summary is obtained by simply averaging all the
node representations & N Σx= h4. However, the importance of cach node may not be the same. Considering that the density of node can represent the importance of the node some extent and energy is un-normalized density, we propose to use energy for conducting the weighted readout:
N
N
1-7) his N
8p =
P.h; + (1 -
h
YP: +
N
where P; softmax( - fww (h:)) ش expl - fw (h,))
R is the weight, and is hyper-parameter for trade-off between weight average and direct average.
exp(- (A,))
3.4 RECURRENT UPDATE FOR LEARNING DEGEM
To apply CE and ERo simultaneously. the energy readout &p and conditional energy fw (h; 8) should be updated recurrently. The Recurrent Update make energy function fw and graph encoder Sa promote each other, leading to better learning. Next, we introduce the Recurrent Update in details. The weights of conditional energy are initialized as uniform distribution pO)) = [N. N In the e-th epoch, node representations of original graph (X, A} and shuffled graph (X' A') are first obtained by h; = Ia (N...((<:) and h = Ia (V())) respectively. and h, ~ qw (h) is sampled by K-step MCMC sampling, then the final learning each iteration is conducted follows:
N. E
Step Learning energy head fw given graph encoder Ja: The energy head fw is learned by MLE over informative latent space by surrogate objective given the well-trained graph encoder Go and according global representation (
N
1 1- N
Pi
+
sg(h:],
1 M M (sglhal))
Lu Leban
Σ
(agg 5(0))
Σ
N
Step 2: Learning graph encoder Ja given energy head fw: The Graph Encoder Ja is learned by the mix of graph contrastive learning and node classification given well-trained energy head fw for weighted readout as follows:
(- (-fu(sa(hh), s(()) 8()))
p softmax
N
(c) 1 1-7) hi +
N
==1
La (h, ((6)) (L)
where is hyper-parameter, Les is the cross-entropy loss. See Algorithm for the detailed learning process.
Inference. For an unknown node æe, its representation obtained by h, Ia (x (xx)), then the energy score fa (h,,, (F) serves as the OOD score, where (FF)) is the summary readout of ID dataset in the final training epoch, and its label prediction is computed by ÿ. T (hw).
3.5 UNDERSTANDING OF USING DGI FOR LEARNING GRAPH ENCODER
By comparing Eq. 3 and Eq. 2 it can be found that there are some similarities between these two objectives. Actually, the learning of DGI can be understood learning an EBM with global condition by noise contrastive estimation (Aneja et al. 2021 Gutmann & Hyvärinen, 2010) as the following form:

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Table 1: OOD detection performance on four homophilic datasets (Cora, Amazon, Twitch and Arxiv) measured by AUC (1) and Acc (1) with three OOD types (Structure manipulation, Feature interpolation, Label leave-out). Other results for AUPR, FPR95 are deferred to Appendix D
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E ((u(e (roee, - [
Lat
where g(<,,N())) [ ((z)) s)pa(a M( )d g(r. (e 8) denotes the density of augmented samples given data samples, and r rxx, = (Da (x. (xx)) s).
After perfectly training, the optimal has following form (=z)))((z))s)) = V(ale
patm_N(sr\ss)
The inherent connection between DGI and EBM enables the effectiveness of constructing energy function by the learned 9a- However, the optimal DGI learns the density ratio between data distribu- tion and randomly augmented data distribution, which is unsuitable for OOD detection, hence it still needs to learn fo for output energy.
4 EXPERIMENTS
4.1 SETUP
Dataset and Evaluation Metrics. We evaluate DeGEM on seven benchmark datasets for node classification tasks (Yang et al.
et al.) 2018 Rozemberczki et al. 2021] Wang et al. 2020 Pei et al. 2020). including four homophily datasets (Cora, Amazon- Twitch and ogbn- Arx i v) and three heterophily datasets (Chame leon, Actor, and Cornell), We mainly follow (Wu et al. 2027 2023b) to adopt two established methods to simulate OOD scenarios. In the multi-graph context, OOD samples stem from distinct graphs or subgraphs not linked to the training nodes. Conversely, in the single-graph setting, OOD samples are part of the same graph as the training data but remain unseen during training. For Twi tch, we treat one subgraph as in-distribution (ID) and others as OOD, using one for OOD exposure during training. In ogbn- v, we split the nodes by publication year for ID, OOD, and OOD exposure sets. For Cora, Amazon, Chame leon, Actor, and Cornell 1 that have no clear domain information, we synthesize OOD data in three ways: i) Structure manipulation (S); ii) Feature interpolation (F); iii) Label leave-out (L). See detailed settings and splits in Appendix E.2 For the assessment of OOD detection performance, we employ standard metrics: Area Under the Receiver Operating Characteristic curve (AUC), Area Under the Precision-Recal curve (AUPR), and the False Positive Rate at 95% True Positive Rate (FPR95). In-distribution (ID) performance is quantified using the accuracy (Acc) metric on the testing nodes. In the following text, we use AUC to denote AUC with a little bit of abuse. Due to the space limits, we mainly present the results of AUC and Acc, the full results with AUPR and FPR95 can be found in Appendix
Baseline Comparisons. Our model is benchmarked against two categories of baseline methods. The first category comprises models that handle i..dd inputs, which are predominantly used in computer vision. These include MSP (Hendrycks & Gimpel| 2016). OD IN (Liang et al., 2018). Mahalanobis (Lee et al., 2018), OE (Hendrycks et al. (018)) and Energy (-FT) (Liu al. 2020). We also include Res prssidalFli capable of modeling data distribution. For a fair comparison, we substitute the CNN backbones in these models with a GCN encoder. The second category consists of methods tailored for graph- structured data, such as GKDE (Zhao et al., 2020b), GPN (Stadler et al. 2021b), OODGAT (Song & Wang 2022) and GNNSafe (++) (Wu et al.) 2023a). It is noteworthy that OE, Energy-FT and GNNSa fet+ incorporate OOD samples during the training phase, i.e., trained with OOD exposure.
(Zisselman & Tamar 2020). density- -bsed method
Implemetation Details. We implement our model by PyTorch and conduct experiments on 24GB RTX-3090ti. Epoch number E = 200, MH layer number L 5, hidden dimension d = 512, MCMC steps K = 20. We use Optuna (Akiba et al.] 2019) to search
for our proposed
model and baselines (see Appendix E.3 for detailed search space).
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Table 2: OOD detection performance three heterophilic datasets (Chame eon, Actor, Cornell) measured by AUC (1) and Acc (1) on with three OOD types (Structure manipulation. Feature interpolation, Label leave-out). Other results for AUPR, FPR95 are deferred to Appendix
Chancion Feature
Actoe Peature
Cocncll Feature
Methed
Straciure
Label
Sirucuure
Lated
Stracture
Lakd
MSP
59.78
33.0
Th.82
$2.22
53.5
13.99
11.38
31 64
50.1
34.54
56.33
37.37
$1.50
41 50
71.03
$1.52
68 50
63.81
7h
23.02
ODN
60:99
50
55.00
13.34
08.0 07
37.1
43.92
52.57
65.10
15.78
67.89
T3
$5.26
38.10
1616
65.81
50.60
35-75
Eneny
91:54
37.00
67:35
37.58
59.9
4:00
54.35
25 21
13.70
24.55
55.00
18.71
$3.09
38.00
$5.00
18.10
09.30
02.05
11601
31.90
48:24
34.50
55.82
34.30
56.16
34.50
40.22
27.85
58.07
27.65
50.64
27.03
67.80
43.54
65.33
43.54
73.09
43.54
57.32
35.22
OKDE
30.12
00.22
25.65
56.08
19.50
13.60
$0.53
14.97
51.18
71.76
GPN
$2.90
20.41
66.90
33.99
22.68
34.71
18.58
18.07
62.13
33:21
15.04
38.08
10.08
43.5 5/
$3.35
43.85
$2.03
63.81
76.32
3481
OODGAT
54.89
25 10
53.86
23.73
03.33
4069
51.35
25 65
$2.00
25.77
65.39
36.30
67.16
42 18
69.43
43.85
08.52
16.75
00087
35.00
GNNSafc
33333
57.46
33.07
31.76
50.65
13.02
74.66
76.23
55.33
DeGEM (Owrs)
59.39
67.82
99.70
57.00
89.68
64.06
99.76
31.97
99.98
33.87
100.0
36,03
97:07
48.30
100000
65.31
93.00
77.14
97.55
$3.53
Table 3: OOD detection results on varying the available label ratio, measured by the average AUCT
AUPRT/ / FPR95
00D
Cora ALTFTT
Twltch AUFRI
Chomeleon AUTPRI
Carnell AUPRR
Arg ACTPRT
Expe
AUCT
F79055
AUCC
FRR055
AUCCT
FFROEI
AUUCT
FRRO5)
AUCC
FPROE]
Latel Rate
OE
Von
65.35
42.01
15.60
53
71.
91.87
67.29
744 96
89: 12:13
55
55.55
94.25
63.52
64.26
91.12
Eacrg-FT
Yoa
72.05
55.02
83.04
66. 14
90:54
70:87
56. 25
62: 77
85.88 63
50 56
70 75
99.12
68.09
69.28
BA.00
GINNSafee
Yes
79.74
64.55
79.40
77.75
55.01
64.30
55.21
47.20
U5:77
59 52
70:51
04.34
(6.50
IV
$7.95
DeGEM
07.87
18:00
94.67
51.41
96.64
14.04
11:25
96.99
97.30
21.70
Libel Rate
OE
Ven
78.63
60.63
70.02
58 15
58.81
91.38
76 52
79.30
67.40 17.45
13.00 94
79.33 09
90.25
31.26
12:15
80.00
Excrg-FF
Yea
77.58
56.54
63.36
87 30
30.23
$3.09
68.90
74.14 14
83. 53
57.00
15.34
04.37
75.35
14.16
75. 25.40
GINNSafee
Yes
86.07
72.80
27.39
55. 02
36.15 14
41.13
62.67
23
11. 00.05
73. 12
$3.00 01
85.25
78.02
77.56
19:12
DeGEM
97.53
91.95
0.57
09.51
97.03
98.29
13.00
96.30
16.37
96.27
95.91
21.96
Libel Rate
1001
OE
Vea
K3.35
68.91
67.19
62.5 55
13.53
83.43
51.07
$2.83
52.23
31 71
12.47
74.05
34.63
74.44
70.45
Escrg-FF
Von
89.05
74.79
64.34
87:3 35
90.43
06.97
79.89
$2.72
$4.00
78.81
65.00
64.65
$3.75
$3.46
$7.99
GINSSafee
Yes
92.69
53.02
40.63
96.9 58
98.03
34.76
08:33
63.37
05:07
79. 10
55.0 16
52.65
79:15
$2.79
00:03
DeGEM
No
09.18
90.78
20:11
13.38
33.75
96.89
17:62
4.2 EVALUATION RESULTS
We report the results of the proposed DeGEM and competing baselines on different datasets in T and Our finding indicates that the DeGEM consistently outperforms competing baselines without OOD exposure terms of AUC and Acc across both homophilic and heterophilic datasets. Specifically, DeGEM increases the average AUC by 4.26% (resp. 20.63% on homophilic (resp. heterophilic) graphs and improves the average Acc for homophilic (resp. heterophilic) graphs by 2.37% (resp. 13.95%). On the contrary, although the baselines tailored for graph inputs surpass those designed for i.idd data on homophilic graphs, they show a lower performance on heterophilic graphs. Additionally, after adding the Energy Propagation technique to Energy (i.c., GNNSafe). the average AUC decreases significantly from 70.01 % to 55.23% and 56.42%, indicating that the Energy Propagation technique will cause severe performance decrease heterophilic graphs.
The high performance of DeGEM is primarily attributed to the enhancement brought by the robust representation learned by the GCL algorithm and the classification loss, as well as the powerful data modeling capabilities of EBM trained via the MLE approach. Overall, these findings underscore the superiority of DeGEM over baseline approaches, on both homophilic and heterophilic graphs.
4.3 EVALUATION ON LIMITED LABELS
DeGEM (Ours) GNNSafe +
Given the time-consuming and labor-intensive na- ture of obtaining node labels on graphs in the real
Average Performance
100
Energy.FT
OE
world, an interesting evaluation task is to evaluate the OOD detection performance of a model with
50-
limited category labels. We follow the data splits
used in the experiments of Sec. 4.2 as a foundation and progressively reduce the proportion of available labels in the training set: from 100% to 50%, and
25-
100%
50%
10%
Relative Labeled Ratio (%)
then to 10%. To highlight the performance of our method, we select three basclines OE, Ene Energy-FT and GNNSa fe++ which incorporate OOD samples during the training phase, i.e., trained with OOD exposure. Note that we do not reduce the number of nodes used for OOD exposure training. but only decrease the number of category labels available within the distribution. We present the average results of each dataset in Tab. 3 and the average performance of each method in Fig. 4 for more intuitive comparison. We report the full results in Appendix[]] The results show that our proposed DeGEM maintains high performance under different label proportions, with no significant decline; in contrast, all baselines exhibit a marked performance drop as the labels ratio decrease. In particular, when the available label rate is 100%, method still outperforms baselines, highlighting the
Figure 4: Performance across labeled ratios.
8

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Published as a conference paper at ICLR 2025
Table 4: Ablation study measured by the average AUC1. 'Eprop' stands for energy propagation. MMLE-Eneryy' indicates whether the energy head is trained via maximum likelihood estimation, otherwise, the node energy is obtained by classification logits Classify-Eneryy 'GCL' indicates whether graph contrastive learning algorithm is employed when training the graph encoder.
Components
Homophily
Heterophily
Eprop
Classify-
MLE-
GCL
CE
ERo
Cora
Twitch
Avg
Cham.
Cornell
Avg
Avg
Energy
Energy
Energy GNNSafe
87.36
67.07
73.20
79.83
76.51
76.87
90.84
71.00
80.92
48,00
73.02
60.51
70.72
47.61
73.90
69.11
74.43
71.77
77.51
67.91
72.71
72.24
88.87
84.77
90.63
88.82
87.82
88.82
69.81
79.32
92.57
84.95
88.76
84,04
92.34
83.12
87.73
93.57
86.31
89.94
92.80
84.77
88.78
94,46
85.35
89.91
89.35
97.47
93.61
95.54
95.23
95.
95.57
95.55
98.03
94.96
96.50
96.24
95.49
95.87
96
96.24
92.41
94,32 32
95.80
93.99
DeGEM
99.12
95.65
97.38
96.46
96.29
96.37
96.88
superiority of our method. These results demonstrate that our method, benefiting from its effective design, can consistently and effectively extract graph topology information and learn the data density well, thereby significantly outperforming previous state-of-the-art graph OOD detection methods in the more practical scenario of limited labels.
4.4 ABLATION STUDY
The Impact of Proposed Techniques. In what follows, we evaluate the impact of training the energy head via MLE (MLE-Energy) versus deriving node energy directly from classification logits (Classiff) ferfff--Eeeeee -y-iferryErrr)) Additionally, we assess the use of DGI algorithm during graph encoder training (GCL., and explore the effectiveness of the Energy- Prergy-Propaggtion technique (Eprop), the Multi-Hop Graph Feature Encoder (MH)). Conditional Energy (CE), and Energy Readout (BRo). We present the results in Tab. The baseline (Row 1) utilizes a GCN for node classification and obtains the node energy by the classification logits. Since Eprop is based on homophily assumption, the performance increases slightly (Avg +3.7%) on homophilic graphs but shows a dramatic degradation (Avg -16%) on heterophilic graphs (Row 2). Compared to the baseline (Avg 76.87%) solely utilizing the MLE for training the energy head (Avg 72.24%) or employing GCL algorithm for training the graph encoder (Avg 73.90%) does not work. It is the combination of MLE and GCL algorithm that brings positive effects for OOD detection tasks (Avg 87. 87.82%). More detailed discussion can be found Observation 1-4 below. With MH, DeGEM achieves significant performance boost (Avg +11 %, Row 9), demonstrating that combining local and global information simultaneously enhances performance on both homophilic and heterophilic graphs. CE alone can improve the average performance (Avg +0.63% Row 10); however, using ERo on its own does not yield similar benefits (Avg -0.94%, Row 11). This because CE takes the global view into account, producing better node energy, which enables ERc to generate a more optimized readout summary. Therefore combining them together via Recurrent Update can bring more positive effects (Avg + +1.33% Row 12).
Observation 1: combining DGI with classification does not work. As shown in Tab, 4 although the wwwssiifer-Eeeeeee variant shows 5.93% improvement of AUC on heterophilic graphs after adding DGI loss (GCL. into the classification task (Row 3), its performance on homophilic graphs dramatically decrease from 77.22% to 65.37%. This indicates that the integration of DGI algorithm and Classifier" www.fffererrrggggreeeee has an overall negative impact on its performance.
Observation 2: naively training Energy Head via MLE does not work. As shown in Tab, 4 when decoupling the model into two parts (a graph encoder and an energy head), if we train the energy head via MLE but only employ classification loss for training the graph encoder (Row 4), the model will show worse performance compared to the illasifier-Enet
variant (Row 1).
Observation 3: combining DGI and MIEE works well As shown in Tab. 4 after adding DGI loss (GCL), the performance of MLE-Energy variant dramatically improves from 72.74% 87.82% on average in terms of AUC, outperforming previous state-of-the-an method GNNSafe.
Observation 4: combining other GCL and MLE not works well We try replace the DGI algorithm with other widely used GCL algorithms: GRACE (Zhu et al.) (0205)) and SUGRL (Mo)
al. 2022) (without MH, CE, and ERo). From Tab. DGI consistently outperforms its counterparties.

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Published as a conference paper at ICLR 2025
Table 5: The performance of different GCL algorithms, measured by the average AUC1.
Cora
Twitch
Cham.
Cornell
Avg
GCN
68.25
53.94
34.65
75.75
68.15
GRACE
79.38
SUGRL
82.41
72.15
82.71
63.67
75.24
DGI
88.87
84.77
90.63
87.00
87.82
From the above observation, we can conclude that the key reason behind the success of proposed DeGEM is due to the symbiosis between DGI and EBM. We extensively evaluate the potential variants, however all variants show significantly worse performance. This also aligns with our theoretical analysis in Sec. 3.5] which said that the learning of DGI can be viewed as learning EBM via variational learning. The property enables us to use DGI for learning the graph encoder as part of the final constructed EBMs.
RELATED WORK
EBMs on Graph. Some previous works (Hataya et al.] 2021 Liu al. 2021 apply EBMs on graphs that are trained with MLE have been proposed for graph generation. However, their works are performed on small graphs, where the over-computational issue of the sampling adjacency matrix is neglected, making them non-scalable. In contrast, our proposed CLEBM focuses on OOD detection, decomposing the EBM networks into informative representation extraction and output energy score based on given latent. The design enables to move MCMC to latent space, which does not suffer from sampling adjacency matrix and therefore has excellent scalability.
Graph OOD Detection. OOD detection for non-graph data by neural networks has garnered considerable attention in the literature (Hendrycks & Gimpel, 2016 Hendrycks et al., 2019 Bevandić al. 2018 Liu et al. 2020 Mohseni al.] 2020: Ren et al. 2019). However, these methods typically assume that instances (such as images) are i.i.d., overlooking scenarios with inter-dependent data contrast, Graph OOD Detection that inherently includes inter-dependent structures has not been explored well. Recently, some works (Li et al. 2022b; Bazhenov et al., 2022) focus on Graph OOD Detection on graph-level, i.e., detecting OOD graphs. These works treat each graph an independent instance, while OOD detection on node-level presents unique challenges given the non-negligible inter-dependence between instances. To this end, Bayesian GNN models have been proposed that can detect OOD nodes within graph by incorporating the inherent uncertainty in such inter-dependent data (Zhao et al., 2020a Stadler et al., 2021a). OODGAT (Song& & Wang, 2022) emphasizes the importance of node connection patterns for outlier detection, explicitly modeling node interactions and separating inliers from outliers during feature propagation. Energy-based Detection on graphs has been explored in GNNSafe (Wu et 2023b), by directly combing GNNs and Energy-based Detection on Images (Liu et al.) (020)) However, their energy score is directly construed by classification, which is less effective. Moreover, they use energy propagation to enhance performance, which highly relies on the homophily assumption. Additionally, some methods train the model with OOD Exposure, i.e., training with both a known ID dataset and known OOD dataset (Hendryeks al. 2018 Liu et al. 2020 Wu et al. (023b). In contrast, we decompose EBM learning into representation learning and energy learning. delivering better detection capahility without OOD exposure (see Sec. 4.2). Additionally, benefits from powerful energy construction and effective graph encoder, we do not require energy propagation, thus keeping high performance across homophilic and heterophilic graphs.
that are common in many real-world applications.
CONCLUSION
We introduce novel approach, DeGEM, for graph OOD node detection, overcoming the heterophily issue and the computational challenges associated with MCMC sampling in large graphs. By decoupling the learning process into a GNN-based graph encoder and an energy head, we managed to leverage the GCL algorithm and classification loss to learn robust node representations and perform efficient MCMC sampling in the latent space, circumventing the need to directly sample the adjacency matrix. The design of DeGEM, featuring Multi- Graph encoder and a Recurrent Update mechanism, facilitates the incorporation of topological information into node representations, which is crucial for OOD detection in graph-structured data. Extensive experimental evaluations have validated the effectivvness of DeGEM, which not only exhibits superior performance on both homophilic and heterophilic graphs compared baselines with/without OOD exposure, but also outperforms methods trained with OOD exposure in a label-insufficient scenario.
10

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ACKNOWLEDGMENTS
Xiaochun Cao's work is partially supported by the Shenzhen Science and Technology Program (No. KQTD20221 101093559018) and by the National Natural Science Foundation of China (No. 6241 12410540034). Pengwen Dai's work is supported by the National Natural Science Foundation of China (No. 62302532). Jing Tang's work is partially supported by National Key R&D Program of China under Grant No. 2023YFF07251 100 and No. 2024YFA 1012011, by the National Natural Science Foundation of China (NSFC) under Grant No. 62402410 and No. U22B2060, by Guang- dong Provincial Project (No. 20X30N by Guangdong Basic and Applied Basic Research Foundation under Grant No. 2023A1515110131, by Guangzhou Municipal Science and Technology Bureau under Grant No. 2023A0310667 and No. 2024A04J4454, by Guangzhou Municipal Education Bureau (No. 2024312263), and by Guangzhou Municipality Big Data Intelligence Key Lab (No. 2023003J)10)), Guangzhou Industrial Information and Intelligent Key Laboratory Project (No. 2024A0310628) and Guangzhou Municipal Key Laboratory of Financial Technology Cutting-Edge Research (No. 2024A03J0630).
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Qitian Wu, Yiting Chen, Chenxiao Yang, and Junchi Yan. Energy- out- out-of--dstribbe di-tiiiibbt detection for graph neural networks. In The Eleventh International Conference on Learning Representations, 2023a. URL TtLhttps:: //pprrevieeeeeeeeeeeemeem net wtw.:oodddmooo2222m
Qitian Wu, Yiting Chen, Chenxiao Yang, and Junchi Yan. Energy-based out-of-distribution detection for graph neural networks. International Conference on Learning Representations (ICLR), 2023b.
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Jianwen Xie, Yang Lu, Ruiqi Gao, and Ying Nian Wu. Cooperative learning of energy-based model and latent variable model via meme teaching. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.
Jianwen Xie, Zilong Zheng, and Ping Li. Learning energy-based model with variational auto-encoder as amortized sampler. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pp. 10441-10451, 2021.
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Lingxiao Zhao and Leman Akoglu. On using classification datasets to evaluate graph-level outlier detection: Peculiar observations and new insights, 2021.
Xujiang Zhao, Feng Chen, Shu Hu, and Jin-Hee Cho. Uncertainty aware semi-supervised learning on graph data, Advances Neural Information Processing Systems, 13:12227-12836 2020a.
Xujiang Zhao, Feng Chen, Shu Hu, and Jin-Hee Cho. Uncertainty aware semi-supervised learning on graph data. Advances in Neural Information Processing Systems, 33: 1287--1286, 2020b.
Jiong Zhu, Yujun Yan, Lingxiao Zhao, Mark Heimann, Leman Akoglu, and Danai Koutra. Beyond homophily in graph neural networks: current limitations and effective designs. In Proceedings of the 34th International Conference Neural Information Processing Systems, pp. 7793-7804, 2020a.
Jiong Zhu, Ryan A Rossi, Anup Rao, Tung Mai, Nedim Lipka, Nesreen K Ahmed, and Danai Koutra. Graph neural networks with heterophily. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pp. 11168-11176 2021.
Yanqiao Zhu, Yichen Xu, Feng Yu, Qiang Liu, Shu Wu, and Liang Wang. Deep graph contrastive representation learning. arXiv preprint 2rXiv:2006. 04131, 2020b.
Ey Zisselman and Aviv Tamar. Deep residual flow for out of distribution detection. Proceedings of the IEEE/CVF Conference Computer Vision and Patern Recognition, pp. 13994 14003, 2020.
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A POTENTIAL BROADER IMPACT
This paper presents work whose goal to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here.
B LIMITATIONS
DeGEM has achieved excellent performance, but the results are nearly maxed out on some datasets. need better and more challenging benchmarks evaluate performance. Additionally, DeGEM shares the same drawback EBMs trained with MLE: training requires MCMC sampling. However, still shows superior performance. Besides, the cost of MCMC sampling can be reduced by cooperative learning (Xie et al. 2018 2021
Luo et al. 2024b).
C ADDITIONAL RELATED WORKS
Graph Contrastive Learning, Contrastive learning (CL) stands as a widely applied self-supervised learning method, aiming derive informative sample representation solely from feature information. The main idea of CL is to align the representations of similar samples in close proximity while driving apart the representations of dissimilar samples. Witnessing the remarkable advancement of Graph Neural Networks (GNNs), substantial amount of recent research has focused on Graph Contrastive Learning (GCL) (Veličković et al.) 2018 Peng et al. 2020 Hassani & Khasahmadi
2020: Zhu al., 2020b Mo et al.) 2022). DGI (Velicković et al. 2018) learns by maximizing mutual information between node representations and corresponding high-level summaries of graphs. GRACE (Zhu et al., 2020b) maximizes the agreement of corresponding node representations two augmented views for graph. SUGRI. (Mo et al. 2022) explores the complementary information from structural and neighbor information
maximize inter-class variation and minimize intra- class variation through triplet losses and an upper bound loss, while removing the need for data augmentation and discriminators. Our work builds upon the foundation of GCL, where we employ DGI as part of learning the graph encoder that extract graph topology information. We also establish the relationship between DGI and EBM: DGI can be understood as an EBM trained by noise contrastive estimation. To further enhance the capabilities of the proposed method, we design recurrent update to let the energy head and DGI promote each other in training (see Sec. 3.3 and experiments show that this to be greatly beneficial for OOD detection (see Sec.
Node Anomaly Detection (NAD). NAD is a binary classification task, that directly categorizes nodes into two different categories: normal and anomalous. Some works (Dong et al., 2025a [0024] 2025b Zhao & Akoglu 2021; Gong et al.) 2023) have been proposed to handle this problem. In contrast, node OOD detection requires balancing the ability to classify in nn-distribution nodes and detect out-of-distribution nodes.
D DERIVIATIONS
D.1 TRAINING EBMS
Given Boltzmann distribution po(x) expl -E((x))
its negative log- likeelood (NLL) Lo is:
Zo
Lo -Eeepp((e)
log log)o(2))
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To minimize Lo, first need to calculate the gradient of log po(x) w.r.t. 0:
exp (=Eo(x))
Vo log (o)) Vo log
Zo
V Eg (a)
Vo log Zo
(o
Vo log Σ exp (-Fo(z))
1
-VoFo())
Vo (x
Ex expl (- (x())
-Ve + Σ exp (E)(())
car exp (-Li(xx)) VeEe(x)
VeEs(x) Σ (oE
(o) Ex)pp(() (obo(Tt
So, we have
VeLo V (aaxps(a log po(c)
Eappi(a))
[o Be)pa(#) [o
Therefore, the EBM loss can be reformulated as:
N
1 M
Lg Empps(()
EØ(x)]
Eg (a))
Eo(E)
Ex(
Eo
Σ M
Eo(ã,)
N
j=1
Additionally, minimizing Lo is equivalent to minimizing the KL divergence KL (pa (z))
Vo Σ pa( log Pa(æ:)
VOKT. VOKT. (pa(x)))
(x)
Vo log Vo Σ pa(x)) log (x
V "gEm=pa( logpa(z))
V Vo Voln~na(a)) logpo(z))
0 + VeLo = VeLo.
D.2 REGULARIZZTIONN FOR EBM Loss
Additionally, follow (Du & Mordatch 2019) to adopt the L2 regularization for energy magnitudes of both true data and sampled data when computing Leem during training, as otherwise while the difference between true data and sampled data preserved, the actual values would fluctuate to numerically unstable values. For an energy function E.(.), true data æ; ~ pa(x), and sampled data 2; ~ (
N Eg (xx)
M M Σ J=1
N +
M
1 M 1=1
Lobm
Ea(i,) tc
Ex(x,)²
1=1
where e is a coefficient, we set here.
D.3 SAMPLING FROM EBM
Given Boltzmann distribution (a = exp(-Fale Ea(ar) Zn
the gradient of logPP(()) w.r.t a is:
cxp
(-oo(x))
(ogpo(x) = V. log
7o
Va log Zo
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Algorithm 1 Training algorithm of DeGEM.
Require: graph G = (X,A), A), learning rate 7, classification loss weight &, MCMC step size A,
MCMC noise variance o², MCMC steps K, MCMC samples M, epoch number E.
Ensure: optimized models {ga. Dig. fw. la).
1: Initialize weights {ao, 40. wo, Φo};
2: Initialize MCMC replay buffer s 2;
3: for < 1 to E do
4: Get node embeddings (h:)
(oi,
5:
6: # MCMC sampling
7:
for sample node = to M do
h ~ with 95% probability and U otherwise;
8:
9:
for sample step k 1 to do
10:
h (k) h (k-1) AV (h) ( + c( end for
c(0) ~
NN(0,09);
11:
hj sq 1:
12:
13:
BU {hi}:
end for
15: Sampled embeddings (h, 1
17:
# compute losses
Compute Letms Las, and Las via recurrent update (Sec. 3.4
19:
20:
# update models
21:
ae Qe-1 1) (La {Les);
22: we t ch-11 1 Laii
23:
W. w 1 Vehmii
24: de f Φc-1 (E
25: end for
26: â + ag, +b + DE. as + WE. ô - ΦF.
so we can first initialize a sample using uniform distribution (0) U, and utilize multi steps stochastic gradient make the sample follow distribution po(x:):
(k) = (k-1) + AV log po (x (k- (k-1))) 3))+++ + (k)
= (k-1) A En (x (1))
where (k) ~N(00, o?). After K steps, (K) is the output of K-step MCMC sampling.
E IMPLEMENTATION DETAILS
E. TRAINING ALGORITHM
The detailed training algorithm is shown in Algorithm 1 We first obtain node embeddings (Line 4), and then sample nodes by a K-step MCMC sampling (Line 7-15). Then the recurrent update conducted for computing losses Lehms Let. and Las (Line 18). Finally, update the parameters (Line 21-24).
Furthermore, follow (Du & Mordatch 2019) to sample relay buffer. The initialization of MCMC chain plays crucial role in mixing time, but langevin dynamics does not place restrictions on sample initialization given sufficient sampling steps. Thus which we preserve previously generated samples and use either these samples or uniform noise for initialization.
sample replay buffer B in
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Table 6: Search space of hyper-parameters:
Search Space
learning rate r
{0.05, 0.01, 0.001, 0,0005, 0.0001]
weight decay
{0.0, 0.01, 0.001, 0.0005, 0.0001)
dropout
{0.0, 0.1, 0.2, 0.3, 0.4, 0.5)
(0, 0.1, 0.3, 0.5, 0.7, 0.9, 1)
(0, 0.1, 0.3, 0.5, 0.7, 0.9, 1]
{0, 0.3, 0.5, 0.7,0 0.9, 1}
(I. 10)
o²
(0.01, 0.005)
{
(0.01. 0.05, 0.1, 0.1.0.
E.2 DATASET AND SPLITS
For Twitch that has several subgraphs, we use DE as ID dataset and the other five subgraphs as OOD datasets. The subgraph ENGE used as OOD exposure dataset for training the OOD Expo baselines. These subgraphs are different in size, edge densities, and degree distribution, thus can be regarded as samples from different distribution (Wu et al. 202T).
For Arxiv that 2015 used as ID data, those after 2017 are used OOD data, and those between are used OOD exposure data for training the OOD Expo baselines.
single graph, we divide the nodes into three parts: the papers published before
For Cora, Amazon, Chameleon, Actor, and Cornell that have clear domain information, we synthesize OOD data in three ways (Wu et al. 2023a). Structure manipulation (S): use the original graph as ID data and adopt stochastic block model randomly generate OOD graph. Feature interpolation (F): use random interpolation to create node features for OOD data and the original graph as ID data. Label leave-out (L): use nodes with partial classes as ID and leave out others for OOD.
We split the ID dataset as 10%//0%/80% (train/valid/test),, and use all the nodes in OOD dataset for evaluation. Additionally, an extra OOD dataset is also used in training for OOD exposure (OE, Energy-FT, and GN+S)
E.3 SEARCH SPACE FOR
The detailed search space of to conduct random search for byper-parameters.
shown in Tab. We use Optuna (Akiba al. 2019
F COMPLEXITY ANALYSIS
Compared to other baselines, our additional computational complexity comes from the introduced GCL. and MCMC sampling.
Suppose that the GCN is used as the graph encoder in other baselines, with m the edge numbers, n the node numbers, do the initial node dimension, d1 the latent dimension of node representation, and the number of layers (propagation numbers). So other baselines have a complexity of (Lmddgdi)
In terms of our methods, the GCL processes both positive nodes and negative nodes, but the prop- agation for positive nodes conducted in pre-process, with only the transformation remaining in training iterations. So the complexity for GCL is OLLndod; + Lmddod)) se (LLmdod;) ). since normally m >> n, where n is the number of nodes. The MCMC sampling has a complexity of O ((nd,dd rdd)) MLP), and K is the number of steps of MCMC sampling.
(nd, d2 K), where d2 is the hidden dimension of Energy Function (2-layer
This actually suggests that method has scalability comparable to GNNs, since the seemingly computationally intensive MCMC steps do not suffer from edges, with complexity only scaling linearly with the number of nodes.
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Table 7: The average time consumption (s) of one-run DeGEM training, including the total time, the time consumption of GCL algorithm and MCMC sampling.
Cora
Amazon
Twitch
Arxiv
Chamelcon
Actor
Cornell
Avg
Total
15.26
17.71
GCL
0.71
0.90
0.67
10.08
0.58
1.02
0.18
2.02
6%
MCMC
5.15
5.45
4.18
15.12
4.53
6.82
2.61
6.27
19%
Table 8: The average training time consumption (s) (for one-run) comparison between baselines and
DeGEM.
Cora
Amazon
Twitch
Arxiv
Chamelcon
Actor
Cornell
Avg
17.98
24.45
104.16
6.35
3.16
23.70
ODIN
6.43
31.65
42.72
225.78
10.23
10.68
5.38
47.55
Mahalanobis
28.77
96.66
187.42
4956.43
36.32
71.06
20.40
771.01
Energy
21.40
26.36
117.02
7.14
7.78
4.10
26.94
GKDE
5.21
10.58
14.15
45.06
5.37
7.72
4.48
13.22
GPN
15.28
27.88
23.55
0OM
14.39
22.51
11.43
N/A
OODGAT
5.18
23.13
33.80
144.25
7.78
8.70
4.56
32.49
GNNSafe
5.59
22.28
26.42
136.55
7,55
8.12
3.83
30.05
OE
5.72
24.40
27.39
129.01
7.63
8.38
4.27
29.54
Energy-FT
28.37
131.99
9.30
5.14
30.66
GNNSafe++
8.23
25.68
31.14
138.09
9.66
10.43
7.13
32,91
DeGEM
11.64
15.26
15.54
149.17
10.40
17.71
5.52
32.18
Table 9: Average OOD detection performance measured by AUC1 on synthetic-Cora datasets with various homophily level.
0.4
GNNSafe DeGEM
69.02
70.65
72.73
73.72
79.08
79.12
99.66
96.56
98.39
99.11
99.86
99.96
G TIME CONSUMPTION
The detailed time consumption of DeGEM training is shown in Tab. In the GCL algorithm, pesssage-passing for positive samples is handled during preprocessing, and the augmentation step only involves shuffling the rows of the node feature matrix. This makes the entire GCL learning process highly efficient. Furthermore, MCMC sampling is conducted in the latent space, which is low-dimensional and topelogy-free, resulting in an acceptable time consumption. Overall, GCL learning and MCMC sampling account for only 6% and 19% of the total training time, respectively.
Tab. compares the training time consumption (s) of baseline models and our proposed DeGEM. Thanks to preprocessing node features and MCMC sampling in the latent space, DeGEM requires only slightly more time during the training phase compared to other baselines. Specifically, training DeGEM (32.18s) is just 7% slower than GNNSafe (30.66s), while delivering sigmificant performance improvements, particularly heterophilic graphs.
H EVALUATION ON SYNTHETIC GRAPHS
We conduct additional experiments on varying the homophilic ratio using synthetic graphs ZZuu al. (222I). We report the AUROC results synthetic-Cora in Tab. and we use the Feature OOD setting.
As the homophilic ratio decreases from 1.0 to 0.0, the performance of GNNSafe decreases signifi- cantly from 79.12 to 69.02. In contrast, DeGEM consistently maintains a high performance across homophilic ratios, achieving around 99 in most cases. This indicates that GNNSafe struggles to han- dle heterophily, whereas our proposed method demonstrates strong performance on both homophilic and heterophilic graphs.
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Table 10: Detailed OOD detection performance measured by AUC1 AUPRI FPR95| / Acct on homophilic datasets with three OOD types (Structure manipulation, Feature interpolation, Label leave-out). The cells filled in gray are methods tailored for graph- inputs.
Methodd
Eapa
OOD
AUCT
AUIFET F1999)
Stuucuree
Acdi
AUCC
Fealure FFR95]
Acc1
AUIC
Label
F1RO5]
Acc
ADCC
Ang FTRO5]
Acr1
MSP
No
75.05
52.13
85.23
77.30
55.87
75.26
05.29
31.30
91.10
73.24
43.41
85.29
54.01
68.57
66.64
$0.96
ODIN
No
30:57
19.26
07.00
24.50
21.19
12.02
09.
77.50
20. 20.20
15.11
59.90
87.56
26.02
17:14
99.12
79.92
Mahulanobis
No
41.03
23.12
99.41
71.90
63.93
41.65
74.30
67.30
39:83
67.53
98.92
57.47
54.45
11
78.34
Energy
79.45
54.23
79:10
76:30
72.30
$2.96
OKDE
64.49
36.57
50.75
44.20
94.59
75.85
GPS
No
87.31
12.01
81.00
95 05
71.76
45.61
75.
34
85.71
51.00
01.12
57
73.00
50.55
$3.96
DODGAT
No
53.75
30.55
96.38
34.90
57.03
37.64
98.80
14.50
95.33
69.58
12.68
89.56
68.78
52.27
71.63
46.32
GNNNatc
No
67.96
79.20
76.48
75.30
02.15
66-46
49.41
75.40
02. 56
61.55
34.45
55.92
00.84
87.44
53.66
19.87
OE
59.05
78.10
GNNSuEe
32.90
DeURoN (Ounst
No
09.85
99.81
0.33
88.30
99.34
49.50
0..66
80.30
97.58
93.63
12.37
93.04
99.13
47.64
4.43
87.18
Pboto
97.02
04.13
90.15
96.20
96.13
IDIN
No
1.50
26.70
105.00
01.42
5.31
76.93
19974
42.91
10:00 16
24.55
00.00
06.18
6.15
26.06
99.55
94.67
Mahilanokis
No
62.40
70.20
99:75
03.44
72.47
66.92
$3.03
02.58
00.90
47.36
79.12
95.34
65.22
62.49
67.62
94.14
Frergy
No
09:94
09:93
0..07
03:06
08.30 00.51
06.
3.83
03:76
07.13
06.11
0.555
05:88
08:30 03.53
97:47
6.06
99.83
ResitualFoow
No
82.58
85.75
94.51
03.55
75.45
50:42
59.00 05
03.55
78.31
59.08
98.20
93.55
79.75
83.08
97.39
93.55
GKDE
02.85
04.80
89:50
73.80
85:55
74.32
73:30
OODGAAT
25.42
70.93
74.80
81.72
GNNNSate
66.69
09.26
11100
41.
98. 67
581
946
98.14
OF
You
09.84
09.82
0.09
02.55
58 60
96.3
3.46
01.84
06.3 58
05:30
16.63
03.96
98:5
97.34
6.73
93.90
Eneegy FT
Yos
05.93
01.93
0.10
03.83
98. 96
2.72
90.26
97.23
6.50
87.44
59.033
19.00
3105
GINNSalew
Yos
99.60
09.77
0.08
93.04
09.57
09:42
0.87
92.76
07.57
97.38
5:36
05.52
08.91
08.02
1.84
93.77
DeGEMM
100
100,00
99.11
99.73
0.74
94.76
Twitch
AUCT
AUPRRT
AUCT
AUPRC
AUCT
AUPR
ADCC
AUPRI
Aug.
Asc1
MSP
No
66.08
16.35
191.5
42.0
53.65
99.85
57.41
71:31
99.75
56.93
67.14
99.70
65.89
ODIN
No
43.97
59.89
97.44
51.894
55.13
96.15
40.53
63.70
96.99
48.45
59.24
96.88
62.15
Mahulanohie
No
06.73
58.91
03.90
49.00
53.26
05. 73
56.34
97.83
44.94
56.15
98.87
62:95
Energy
No
58.42
08.55
90..5
72.90
75.26
$1..0
09.90
79.26
$6.11
67.07
20.36
65.99
ReiiuuFFux
74.31
77.84
06.9
71.13
66.37
$3.99
93.07
83.35
$9.10
60.29
DODGLAT
No
77.35
64.45
$3.64
17.22 77
10.05
13
63.20
76,
11.51
60.24
GNNNSule
No
51.00
$7.41
80.79
79.03
$1.54
68.51
$2.93
96.45
57.00
71.00
75.13
68.79
66.18
TIR
Yoe
64.32
77.11
48.67
$2.99
67.00
02.81
89 m4
K1
82. 14
23.51
18.41
66.71
Energy FT
Yes
03.55
96.62
$2.85
70.89
75.99
$7.47
92.71
15.09
10.58
57 38
10.43
46.97
63.82
CINNSSacow
Vox
07.75
9.78
02:11
54.57
99.87
09.94
34.76
65.1
DetSEN
07.41
Arslc
AUCT
AU/RR7
2015
19995
AUCC
AUIPR
2019
11R91
AUCC
A1IP%
2020
119911
AUCC
ALIPR
Ang
119991
Acct
63.52
75.56
$3.87
UDIN
64.01
55.17
97.85
42.63
98.24
38.96
76.45
99:37
41.47
98:50
49.29
Malulamobis
No
57.08
65.09
93.69
56.76
$7.85
94.01
56.92
65.05
95.01
56.92
09.63
14.24
$1.50
Energy
No
6401
72.30
18.00
65.99
67:80
70.37
91.35
15.35
06 M6
77.33
17:09
53.93
RosiuniFlo
No
62.38
71,02
91.09
63.82
67.23
91.26
62.21
89.22
92.74
62.91
25.83
91.70
$4.94
GKDI
66:45
76:45
36.19
71.79
16.00
$6.90
89:01
$6.50
31:15
00M
COM
OOM
000
0000
COM
DODGAAT
15.93
GNNNSate
75.46
09.20
79002
71.43
$6.36
DE
Yes
70.51
77.35
11.45
11.75
13.47
$0.19
75.12
91.24
77.06
72.46
$1.31
79.56
$3.17
Eurzy-FT
You
75.93
83.99
68.81
20.13
90.19
68.71
52.05
05:44
$6.11
$0.04
R8.54
62.91
39
GNNbsatere
Yes
75.76
82.15
75.54
76.95
78
3390
56.18
78.96
88.79
71.09
08:05
DeGEM (Ouns)
$1.30
$1.20
$6.00
90.30
5601
$4.44
58.20
I DETAILED EVALUATION RESULTS
The detailed results of OOD detection on homophilic and heterophilic graphs are shown Tabs.
and [] respectively.
The detailed results of OOD detection on varying the available label ratio are shown in Tab. 12
The full results of the ablation study regarding the different components are shown in Tab.
The detailed results of OOD detection performance of different GCL algorithms are shown in Tab.
20

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Published as a conference paper at ICLR 2025
Table 11: Detailed OOD detection performance measured by AUC AUPR FPR95| Acct heterophilic datasets with three OOD types (Structure manipulation. Feature interpolation, Label leave-out). The cells filled in gray are methods tailored for graph-format inputs.
Mathods
OOD
Structure
EPRO5L
FFR05.
AccT
Churndun
MSP
No
09.25
05.35
1,41
33.41
20.52
21.15
86.39
32.42
53.32
11.78
93.67
63.89
74.51
$0.95
59.52
36.57
Mahulanabiic
35.95
91.91
05.25
100.00
$6.15
59.92
75.25
70.18
38.09
73.20
79405
38,80
ResiduzFlos
No
48.34
42.46
$9.39
34.50
53.82
53.61
99.91
34.50
56.16
$4.94
90.34
34,50
52.74
50.34
99.55
$4.50
GKD6
No
06.05
96.65
15:05
30.12
07:35
70:37
91.25
31.55
00.35
77.43
73
00.00
74.56
81.
61.30
35.15
GPS
No
$2.50
87.44
100.00
20.41
64.59
53.79
$8.76
30.00
72.58
82.83
77.85
34.71
73.52
76.02
88.57
28.70
DODGAT
58.05
100.00
95.83
52.18
73.72
43,43
48.00
52.94
93.64
38.94
UE
Yee
09:00
$2.70
35.49
75.
M4.02
70.83
00.22
11.02
$3.31
36.87
Essry-FF
Via
98.95
99.15
0,61
38.01
65.
64.56
$4.51
38.50
75.53
84.45
59
39.30
79.80
82.72
54.00
38.74
Y09
36.23
59.25
100000
35.59
57.80
59.51
93.94
37:41
50.65
73.35
$7.00
02.35
48.23
03.37
93.67
58.75
DeGEM (Oung)
0.00
1.10
89:68
94.62
30:04
64.46
96:46
96.04
69.07
MSP
No
71.38
73.30
92.35 99
21.64
59.11
$3.50
92.14
24.36
56.12
13
59:31
37.37
61.37
70.35
91.19
27.25
ODIN
No
42.32
43.97
$9.65
23.30
52.57
40.79
93.55
23.50
65.10
00.12
55.58
35.79
53.35
50.02
92.53
27.35
29.96
35.00
31.95
08.15
71.79
26.64
ResiduaFlow
$0.68
50.15
97.55
27.63
GKDE
No
71.21
69.11
05.99
15.55
$9.00
$0.60
90.67
19.15
$3.92
$4.44
03.50
33.00
61.00
58
92:19
26.35
GPS
No
78.55
73.13
$2.50
15.07
02.13
56.27
93. 42
20.31
25.04
01.13
25.12
38.05
71,92
75.15
23.22
25.65
DODGAT
No
$1.25
$6.46
94.83
23.65
$2.00
48.76
05.3 20
25.77
65.30
63.65
$4.84
36.30
56.21
51.5 61.31
01.56
25.57
GNNSaate
No
51.76
43.55
59.05
26.30
50.06
06.38
95.02
25.30
53.50
$3.00
57.81
27.95
34.08
57.5
94.16
30.14
FT
55.75
29.13
UNNNuleer
Ye9
58,89
63:44
98.89
25.15
02.64
62.03
93.00
25.31
52.36
54.47
02
37.23
57.55
94.10
29.22
DeGEM (Ours)
No
99.76
99.65
0.34
31.97
99.98
99.98
0.05
33.87
100,00
100.00
0.08
36.02
99.93
99.85
0.31
33.95
MSP
No
81.50
84.58
02.55
41.50
71.89
61.46
54.15
41.54
68.50
65.15
$0.42
63.81
111 11.93
77.05
86.56
49.61
ODIN
No
67.59
57.50
99.91
35.75
95.76
57.82
75.96
38.30
42.16
66.71
89.47
63.81
94.76
70.65
85.11
46.89
Mahalansibie
No
60.54
69.08
02.00
35.75
70006
00.00
02.90
43.54
81.73
03:73
07.59
62.35
104.
77
94.30
47.21
Enczyy
$6.54
$5.61
$3.06
63:70
83590
63.35
OKDE
199.97
GPN
09.65
00.86
03.54
12.01
81.95
97.79
00.53
63.81
165.90
$3.16
40.07
DODGAT
No
67.16
66.13
97.81
42.18
69.42
61.27
99.45
42.56
68.52
$2.45
$1.55
64.75
95.30
09.95
92.55
49.95
FNNNSale
No
74.65
87.84
04.64
25.17
76.32
81.44
$8.90
41.50
68:17
90:37
68.47
63.81
115.10
$2:15
81.46
41.49
55.74
57.14
72.47
DeURMM (Ouns)
No
97.97
96.11
5.46
45.30
100,00
10010
0.00
65.31
98.90
96.74
65.79
77.14
333.73
47
2575
63.55
21

---

Published as a conference paper at ICLR 2025
Table 12: Detailed OOD detection performance on varying the available label ratio, measured by AUCT AUPR FPR954 / Acct.
0OD
Structure
Feature AUPRT
Label
Expo
ABC
AUPPR
FPROS]
Asz?
ATCC
FPROS]
Axcf
AECC
AUPRY
FPROS)
AsoT
AUCC
AUPRT
FFR03]
Auc
Label Raic 1088
UE
Yes
51.47
30.21
93.09
42.50
62.60
42.52
89.11
39.30
50.01
67.94
$3.87
74.37
65.36
47.01
83,09
$5.39
Energy
Yes
37.80
92.84
51.50
05 89
47.46
96.00
$3.99
57.0%
73.90
00.31
08.39
72.00
53.00
93.
38.60
81.72
76.58
Label Rate
DE
Ves
67.64
45.88
$6.95
68.30
$2.14
67.48
66.06
71.60
96.12
68.05
59.74
87.56
79.63
50.45
70.92
75.85
Energy VT
Ves
67:51
42.20
79.99
70100
77.15
59.17
71.10
70.00
$7.89
08:25
45.54
89.24
77.58
50.56
65.56
76.65
GNNSafeee
Ven
52.21
67.91
76.19
70.60
$7.76
75.59
60.9
71.90
19. 41
73.81
17.02
13.51
85.47
72.40
57.3 11.74
77.04
DeGEM
No
98.45
95.88
6.76
79.90
98.72
97.08
4.95
78.40
95.40
$2.90
14.91
91.14
97.53
91.95
8.87
$3.15
79.20
91.13
40.10
$2.95
GNNSufeee
76.50
94.51
32.90
92.45
90.51
$1.30
DxGEM
99.93
99.81
0.38
84.20
99.98
99.93
0.15
$3.20
97.38
93.61
12:27
93.04
99.16
97.78
4.25
86.81
Twitch
Expo
ES
EPROS]
ACCC
AUPRI
FPROS
AECC
AUPRR
FFROS)
AUCC
FFR0S
Acct
Label
UE
67.85
90.12
61.77
92.78
72.32
62:55
Eacry-FT FT
Ves
85.09
90.80
73,86
$5.98
89.06
76.28
67.36
91 166
62.46
86.14
90..4
70,62
63.54
UNNSafee+
Ves
75.41
82.92
$4.90
76.55
62.00
XT.79
51.24
501
M1:35
77.75
$5.01
14.30
63104
DeGEM
No
93.63
97.09
73.97
96.21
97.93
17.69
94.15
97.39
62.55
94.67
97:47
$1.41
60.29
Label
GNNSufese
40.98
99.36
99.65
64.04
DrGEM
93.06
47.10
65.77
96.50
98.85
16.58
90. 45
96.35
66.15
94.31
97.21
80.51
61
Label Rate 10855
OE
Yea
64.52
77.11
88:47
$2.99
62.00
02.81
50
$1.45
$4.01
62.38
73.55
85.43
64.71
Energy- FT
Ves
93.55
95.62
42.85
70.89
73.99
87.47
97.71
98.69
10.58
87.38
90.43
46.97
63.82
9.70
99.94
24.76
6883
DeGEM
43.90
94.75
97.57
29:11
64.51
00D
Stradiure
Fosture
Label
Ang
Expo
AUC
AUPK!
[9R95]
Acct
AUCT
AUPKI
19R95]
Acct
AUCT
AUPRT
BPR95L
Aoc7
AUCT
AUPRI
Acc1
Label Rate 13%
GNNSafcc+
100.00
DeGEM
69.29
Label Rate 5OFE
Ul:
Ves
98.14
05.33
53.67
33.52
67.75
65. 30
$8.23
32.03
57.
77.05
54:47
35.57
70.32
79.70
67.46
02.15
Eacryy-T FT
Via
76.85
83.30
100.00
29.55
55.29
53.24
05.30
31.54
73.62
$3.80
70.39
39.36
63.60
74.14
85.59
64.43
UNNSStee+
31.95
54.80
100100
36.36
55.79
51.30
94.30
34.12
50.47
72.90
53.99
38.45
65:07
91.75
64:04
DeGEM
0.89
53.48
1.21
95.82
13,10
61.64
83.76
35.49
51.23
58.95
99.15
0.61
65.19
94.51
38.89
$4.45
54,00
63.82
GNNSafee+
36.21
100.00
36.59
93.94
50.65
73.35
M3.67
6583
DeGEM
No
99.99
99.98
0.80
57.82
99.70
99.51
1.10
$7.93
$9.65
94.62
30,04
64.46
96.46
98.04
13.38
64.51
Coruill
Expo
AUPRI
EFRUS]
Acet
AUPRI
FFROS
Acet
AEC1
AUPRI
FFR05
Acct
Label Rate
UI:
78.38
21.47
95.05
47.36
61:03
94.54
29
58.5
06 90
99.74
65.71
58.00
94.7%
02:55
Energy FT
Ves
65.18
76.90
100.00
43.54
59.50
61.45
100.00
43.54
$3.41
74.09
97.37
64.76
60.36
70.78
99.12
63.54
LNNSSifre
Ven
72.43
801
99:45
1401
75.64
$7.04
95.
43.54
52:3 99
75.90
$6.34
61.81
09.50
70.51
94.14
6314
DeGEM
No
90.62
09.50
2.19
14.97
100,00
100000
EUBS
17.01
S3JBS
97.35
31.58
63.51
97.57
98.95
11.25
60.29
57
GNNSuferr
$4.38
91.50
95.63
77.94
85.28
64.14
DrGEM
No
97.84
92.02
3.28
44.23
99.85
99.85
1.9
58.78
91.23
96.73
44.34
73.33
96.33
96.20
16.30
61.44
Tabel Rato: 100155
OE
Yes
92.05
79.76
74.32
26.53
77.29
62.74
55.74
37.41
55.79
74.93
92.11
57.14
71.71
72.47
74/05
64:71
Energy
Ves
30.42
95.71
92.90
43.54
51.19
90.55
71.58
39.05
[5:81
$3.53
39.47
02.56
75.51
33.93
84.6
63.83
76
84
79,
81.65
65.83
URGEM
0U00
96.74
2575
22

---

Published as a conference paper at ICLR 2025
Table 13: Detailed ablation study results measured by AUCT AUPRT FPR95, Acct. 'Eprop' stands for energy propagation. 'MLE-Energy" indicates whether the energy head is trained via maximum likelihood estimation, otherwise, the node energy is obtained by classification logits Classify-Energy 'GCL' indicates whether graph contrastive learning algorithm is employed when training the graph encoder.
Chando
AUBOOCT
Struttrre
IPRSS
Acci
AttSOCT
.1191
FTSM.
Acci
AUBOCT
Lake
[TRSS)
AUBOC
Arg
14:3
66.33
1316
05.32
75.00
13.50
11.5
01.33
12.43
21.5
0515
01.36
22.30
07.03
05
90
6.36
16.30
1546
02.55
14.34
27.00
12:50
7:41
4.13
0145
13:71
53.15
58.36
47:02
71.34
111
13.0
28.18
00.05
27.50
19:58
02.55
13.39
23.32
08.36
08:0
mm
0:44
81
63.84
18.75
00:43
04.55
811
15.00
04.07
14
3635
12.00
13.34
08
21
9
11:33
$3.00
19.33
171
22.45
0632
1433
98.68
08.27
11.05
26.5
Tymp
Hemp
G0L
MII
186
A1991
1981))
-1991
19955)
Domp
56.43
66.50
16.00
72:31
73.25
$2.01
0033
15.36
06:11
51.00
14.34
$5.55
65.50
23.10
07.05
76.05
13:00
73
11.30
10.31
3.54
12.31
03.12
14.33
01-3
03.34
13.50
08.00
13.30
16:10
0113
14.
$3.31
21.41
04.03
06.3
DIGMM
10:11
$7.00
3.51
$1.01
15.65
17.53
00.20
(henskon
Eprp
MILE Freng
G1L
MII
Ell
Label
Are
30
25:38
13000
25:30
31:40
18.33
7877
02:18
78.33
10:09
13.0
88:00
13.04
16:28
04.03
18.15
28:35
13.00
18.21
$3.55
07:35
56.65
01:09
00
55.0
03.18
71.34
09:11
48.5
0.60
03.34
1.4
25.13
02.00
04.30
11.41
15352
93.33
52.41
03:25
11.27
55
26-11
08.3
09.23
15.44
3.70
14.04
08.0
94.90
01.97
02.08
03:23
03.02
04.3
BNGEM
96.99
19.30
1.60
2032
19.35
59.55
1.13
$3.00
18.00
04.02
33.66
$0.55
96.66
18.04
13:35
64.51
Gomil
Tynp
Chull-
MLE.
Touere
Label
Ang
13.00
13.5
35.11
13.55
18.57
$1.7
3.44
53.55
04.3
19:11
62.54
68.90
03.3
13:54
02:77
08:11
0110
08
m1
51.0
11:1
19
08.14
14:11
16.25
01.00
07.00
00.35
10.00
10.30
$2.00
1530
0.31
5038
23.25
07.0
Table 14: The detailed OOD detection performance of different GCL algorithms, measured by AUC1 / AUPR FPR95
AUC1
ALPPR
AUC:
AUPR.
FP9951
GEN
70.85
44.12
W7.67
GRACE
42.24
90.69
89.71
7N.84
91.75
$3.01
SUICRI
45.75
54.52
57.75
$2.41
5K.78
39.53
Twitch
FR
ALPR.
AUC1
AUPR?
EPR95
E2995)
AUPR1
EPR951
GUN
64.14
94.67
GRACE
67.66
73.59
83.15
63.54
64.71
93.67
66.15
72.59
86.65
63.78
89. 48
SUCERI
$4.23
61.52
70,90
75.30
Chamelonn
Structure
Feature
Label
GCN
73.99
83.52
GRACE
00.64
56.38
SLIGRI
62.71
$3.00
54.32
Structure AUPPR:
Feature
Label
GCN
77.25
83.09
100.00
73.54
76.47
86.84
$1.58
93.45
GRACE
74.86
86.44
65.45
79.77
SUGRL
55.31
96.17
63.19
68.51
86.34
64.69
01.42
60.23
23


Reviews:
[{"review_id": "p3o0jbKRSM", "reviewer": "Reviewer_6WFP", "summary": "This paper studies out-of-distribution detection on graph data. The authors argue that existing models for this problem derives node-wise energy function that ignores the inter-dependence among node instances for uncertainty modeling and relies on homophily assumption for energy propagation. To address these limitations, the authors propose a new energy-based model as well as a new training scheme that can address graph heterophily and enables effective training of the energy model. Experiments on benchmark datasets show the consistent improvements of the proposed model over state-of-the-arts.", "questions": "1. Is there any intuition why the proposed model can address graph heterophily?\n\n2. What is the computational complexity of the training algorithm and how does it compare with the other models?\n\n3. What is the time/memory costs of the model and how does it compare with the other models?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well motivated and studies an important problem in graph learning\n\n2. The proposed model seems reasonable and sound\n\n3. The experiment results are promising and solid", "weaknesses": "1. The argument that existing works on out-of-distribution detection resort to energy function independent for each node is arguably not true. There is a recent paper [1] that considers Dirichlet energy and extends the node-wise energy-based modeling for accommodating the neighborhood information in the graph.\n\n2. The proposed model is computationally expensive, and more justification on the necessity of the proposed components that complex the model is needed. Particularly, compared to GNNSafe and other peer models, the additional computational cost of the proposed method seems considerably large, that questions the practical efficacy and scalability of the model in large datasets.\n\n[1] Graph Out-of-Distribution Detection Goes Neighborhood Shaping, ICML 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LcYosq2XGR", "reviewer": "Reviewer_ta9f", "summary": "This paper, titled “Decoupled Graph Energy-based Model for Node Out-of-Distribution Detection on Heterophilic Graphs,” presents a new model, DeGEM, aimed at detecting Out-of-Distribution (OOD) data in nodes on heterophilic graphs. By decoupling the graph encoder from the energy head, DeGEM addresses challenges in traditional methods, such as reliance on homophily assumptions and the complexity of MCMC sampling in graph structures. Experimental results show that, even without exposure to OOD data, DeGEM outperforms state-of-the-art models on both homophilic and heterophilic graphs.", "questions": "1.Although the decoupled design reduces dependency on graph structure and improves computational efficiency, could the introduction of multiple components (such as GCL, conditional energy, and recurrent updates) make the model structure overly complex? Could future versions simplify key components to reduce implementation difficulty?\n\n2.While the paper includes some ablation studies, why were independent tests not conducted for specific modules like the Multi-Hop encoder, Energy Readout, and Conditional Energy? Could future research add such independent testing to more comprehensively assess the contributions of each component?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.The DeGEM model enhances OOD detection performance on graphs by decoupling the graph encoder and energy head, which helps avoid the performance degradation seen on heterophilic graphs due to homophily assumptions. This design demonstrates strong generalizability.\n\n2.By moving MCMC sampling to the latent space, DeGEM reduces computational complexity, avoiding direct sampling on graph structures and achieving good scalability.\n\n3.The paper conducts extensive experiments on both homophilic and heterophilic graphs, showing performance improvements across different graph types, which illustrates the model’s practical potential.\n\n4.DeGEM enables effective OOD detection without the need for OOD data during training, making it highly valuable for real-world applications where unsupervised adaptability is crucial.", "weaknesses": "1.Although the decoupled design reduces dependency on graph structure and improves computational efficiency, the introduction of multiple components (such as GCL, conditional energy, and recurrent updates) adds complexity to the model structure. I suggest that key components be simplified in future versions.\n\n2.While the paper includes some ablation studies, it does not discuss independent contributions from specific components such as the Multi-Hop encoder, Energy Readout, and Conditional Energy modules. Further independent testing of these components is recommended for future research.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "bsUn8zYrm3", "reviewer": "Reviewer_JnX7", "summary": "The paper proposes a Decoupled Graph Energy-based Model (DeGEM) for detecting out-of-distribution (OOD) nodes on graphs, specifically addressing challenges on heterophilic graphs where existing models struggle. Extensive experiments validate DeGEM's superior performance, highlighting its robustness and scalability.", "questions": "It appears that the uncertainty estimation methods proposed in previous GNN literature [1, 2] could also be applied for OOD node detection. Would it be possible for the authors to consider including these methods as baselines?\n\n[1] Huang, Kexin, et al. \"Uncertainty quantification over graph with conformalized graph neural networks.\" Advances in Neural Information Processing Systems 36.\n\n[2] Hart, Russell, et al. \"Improvements on Uncertainty Quantification for Node Classification via Distance Based Regularization.\" Advances in Neural Information Processing Systems 36.", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is easy to follow.\n\n2. OOD node detection is an important topic in the graph machine learning literature.\n\n3. The model design is technically novel.\n\n4. The performance of the proposed model surpasses that of the existing baseline methods.", "weaknesses": "1. More detailed explanations are needed regarding the state-of-the-art performance on homophilic graph datasets. The paper introduces specific components designed to address the heterophily phenomenon in graph datasets, which lead to their effectiveness on heterophilic datasets. However, it remains unclear why these components are also beneficial for homophilic data. Providing a rationale or theoretical basis for this performance would strengthen the paper's quality.\n\n2. Given the paper’s focus on heterophilic graphs, it would be helpful to include experiments that analyze how varying levels of heterophily affect model performance. This could be similar to Figure 2 in [1], which explores performance across different heterophily levels, offering valuable insights into the model's adaptability.\n\n[1] Zhu, Jiong, et al. \"Graph neural networks with heterophily.\" Proceedings of the AAAI conference on artificial intelligence. Vol. 35. No. 12. 2021.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1gcTkS9nkA", "reviewer": "Reviewer_fY4f", "summary": "This work proposes a method named DeGEM for graph OOD node detection, which can overcome the heterophily issue and the computational challenges.", "questions": "Please see the weaknesses listed above.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- This work seems to be theoretically solid and technically sound.\n- Extensive experiments show that the proposed method has a promising performance.\n- The authors provided the source code in Supplementary Material, facilitating good reproducibility of this work.", "weaknesses": "- In Abstract, the authors argue that GNNSafe has significant performance degradation on heterophilic graphs. However, there is a recent work [1] that has similar consideration.\n- In Introduction, the authors only analyze the limitation of one recent graph OOD method named GNNSafe. However, as discussed in Section 5, there are many existing graph OOD methods. What are the fundamental research challenges that cannot be addressed by these existing methods?\n- There are some minors. For example, the text font in tables is too small. It would be better to provide all the source scripts as well as the used datasets and give the anonymous repo link in the text.\n\n\nRefs:\n\n[1] Ren, Lingfei, Ruimin Hu, Zheng Wang, Yilin Xiao, Dengshi Li, Junhang Wu, Jinzhang Hu, Yilong Zang, and Zijun Huang. \"Heterophilic Graph Invariant Learning for Out-of-Distribution of Fraud Detection.\" In ACM Multimedia 2024. 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)