SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Table 4: Comparison of structural distributional shifts in terms of OOD robustness and OOD detection. We report the drop in predictive performance of the ERM method measured by Accuracy (left) and the quality of uncertainty estimates of the SE method measured by AUROC (right). The ID to OOD ratio in these experiments is  $70\% : 30\%$  instead of  $50\% : 50\%$  as in the main text.
Popularity	Locality	Density
AmazonComputer	-3.80%	-12.66%	-14.04%
AmazonPhoto	-7.19%	-3.03%	-6.79%
CoauthorCS	-6.51%	-1.89%	-6.64%
CoauthorPhysics	-2.58%	-5.50%	-2.47%
CoraML	-7.56%	-13.63%	-9.49%
CiteSeer	+3.24%	+2.50%	-4.13%
PubMed	+0.46%	-0.67%	-1.76%
OGB-Products	-2.35%	-2.36%	-4.02%
Average	-3.28%	-4.65%	-6.17%
Popularity	Locality	Density
AmazonComputer	87.83	85.77	50.46
AmazonPhoto	91.80	85.78	47.27
CiteSeer	70.56	64.55	58.04
CoauthorCS	83.68	81.13	63.24
CoauthorPhysics	86.70	82.06	39.68
CoraML	82.06	84.63	76.82
PubMed	66.64	63.34	55.79
OGB-Products	82.35	82.44	43.50
Average	81.45	78.71	54.35
Popularity	Locality	Density
ID	OOD	ID	OOD	ID	OOD
ERM	3.0	3.4	2.9	3.1	4.3	4.3
Mixup	1.9	2.6	1.4	2.7	1.4	3.3
EERM	4.9	3.9	5.0	4.0	4.4	4.1
DANN	4.1	4.4	4.3	4.3	3.7	2.6
CORAL	4.1	4.0	4.4	4.9	4.1	3.7
DE	3.0	2.7	3.0	2.0	3.0	3.0
Table 5: Comparison of several graph methods for improving the OOD robustness (left) and detecting the OOD inputs by means of uncertainty estimation (right). For each task, we report the method ranks averaged across different graph datasets (lower is better). The ID to OOD ratio in these experiments is  $70\%$ :  $30\%$  instead of  $50\%$ :  $50\%$  as in the main text.
Popularity	Locality	Density
SE	1.3	2.3	3.7
GPN	3.3	3.9	3.9
NatPN	5.4	4.3	3.7
DE	3.0	1.4	1.9
GPE	3.1	3.9	3.4
NatPE	4.9	5.3	4.4
Now let us discuss the setup based on a more extreme  $90\% : 10\%$  ratio. According to Table 6, the drop in predictive performance under the popularity-based and locality-based shifts reaches more than  $6\%$  instead of  $3\%$  in the previous setup and almost  $11\%$  instead of  $6\%$ , respectively. This was expected and can be explained by the fact that graph models are now tested on the nodes with the most extreme structural properties (i.e., nodes with the lowest PageRank or clustering coefficient), and their inaccurate predictions are no longer compensated by the more accurate ones, which were made on the nodes with far less anomalous properties.
At the same time, the performance drop on the locality-based shift appears to be nearly  $1\%$  on average. This result is quite reasonable since now graph models have access to the whole variety of graph substructures and node features, which allows them to predict equally well for any graph region regardless of its distance to some particular node. The observed effects are interesting and also important to consider when using our approach for evaluating graph models.
Regarding the OOD detection performance, we observe that results on the locality-based shift keep decreasing and reach 73 points, in contrast to 79 points in the previous setup (and this might happen for the same reason as we discussed before). At the same time, the detection metrics on the remaining structural shifts appear to be nearly the same (79 points instead of the previous 81 on the popularity-based shift) or even better on average (63 points instead of 54 on the density-based shift), which is consistent with the changes in predictive performance discussed above.
As for the ranking of different models in Table 7, it remains almost the same, with the most simple methods providing top performance on the majority of prediction tasks.
Table 6: Comparison of structural distributional shifts in terms of OOD robustness and OOD detection. We report the drop in predictive performance of the ERM method measured by Accuracy (left) and the quality of uncertainty estimates of the SE method measured by AUROC (right). The ID to OOD ratio in these experiments is  $90\% : 10\%$  instead of  $50\% : 50\%$  as in the main text.
Popularity	Locality	Density
AmazonComputer	-8.45%	-7.79%	-30.98%
AmazonPhoto	-8.00%	+0.12%	-16.35%
CoauthorCS	-9.79%	-4.42%	-7.76%
CoauthorPhysics	-3.77%	-4.91%	-5.32%
CoraML	-16.10%	+5.52%	-8.03%
CiteSeer	-5.14%	+1.35%	+1.86%
PubMed	+3.33%	+4.70%	-2.63%
OGB-Products	-2.92%	-2.75%	-15.61%
Average	-6.36%	-1.02%	-10.60%
Table 7: Comparison of several graph methods for improving the OOD robustness (left) and detecting the OOD inputs by means of uncertainty estimation (right). For each task, we report the method ranks averaged across different graph datasets (lower is better). The ID to OOD ratio in these experiments is  $90\% : 10\%$  instead of  $50\% : 50\%$  as in the main text.
Popularity	Locality	Density	Popularity	Locality	Density
ID	OOD	ID	OOD	ID	OOD
ERM	3.3	3.7	3.4	3.0	3.3	4.4	SE	1.6	2.6	3.3
Mixup	1.7	2.4	1.7	2.9	1.7	3.0	GPN	3.1	3.0	3.9
EERM	5.0	4.1	5.0	3.7	4.9	3.9	NatPN	5.0	4.4	3.7
DANN	4.3	4.0	4.0	4.3	3.4	3.0	DE	2.6	1.7	1.6
CORAL	3.9	3.6	4.1	5.1	4.3	3.4	GPE	3.6	4.0	4.3
DE	2.9	3.1	2.7	2.0	3.4	3.3	NatPE	5.1	5.3	4.3
C Comparison with the GOOD benchmark from Gui et al. [10]
Our work complements and extends the GOOD benchmark recently proposed by Gui et al. [10]. However, there are several important differences that we discuss in this section.
One of the main properties of the GOOD benchmark is its theoretical distinction between two types of distributional shifts, which are represented through a graphical model. In particular, the authors consider covariate shifts, in which the distribution of features changes while the conditional distribution of targets given features remains the same, and concept shifts, where the opposite situation occurs, i.e., the conditional target distribution changes, while the feature distribution is the same. Although this distinction might be very helpful for understanding the properties of particular GNN models, such exclusively covariate or concept shifts rarely happen in practice where both types of shifts are present at the same time.
To create pure covariate or concept shifts, Gui et al. [10] introduce different subsets of variables that either fully determine the target, create confounding associations with the target, or are completely independent of the target. This has to be properly handled and makes it non-trivial to create distributional shifts on new datasets with this approach. Indeed, the distributional shifts in the GOOD benchmark can be properly implemented only for synthetic graph datasets or via appending synthetic features that either describe various domains as completely independent variables or create the necessary concepts by inducing some spurious correlation with the target. Moreover, the authors claim that, in the case of real-world datasets, one has to perform screening over the available node features to create the required setup of domain or concept shift. This fact implies numerous restrictions on how the data splits can be prepared.
In contrast, our method does not distinguish between covariate and concept shifts and thus can be universally applied to any dataset and does not require any dataset modifications. Importantly, the type of distributional shift and the sizes of all split parts are easily controllable. This flexibility is the main advantage of our approach.
Finally, Gui et al. [10] confirm the importance of using both node features and graph structure. Still, their node-level distributional shifts are mainly based on node features such as the number of words or the year of publication in a citation network, the language of users in a social network, or the name of organizations in a webpage network. As for the graph properties, only node degrees are used in some citation networks. In contrast, we focus on the graph structure and propose diverse structural shifts together with a framework allowing one to easily create splits based on other structural properties.
D Structural properties of the OGB data splits proposed by Hu et al. [12]
In Figures 11-13, we present how the distribution of structural node characteristics, including PageRank, Personalized PageRank, and clustering coefficient, may change across the train and test parts in some OGB datasets, where the distributional shifts are constructed using some domain-specific node feature. This shows that our approach to creating data splits using structural graph properties can be similar to realistic distributional shifts.
(a) PageRank
(b) Personalized PageRank
(c) Clustering coefficient
Figure 11: Structural properties across train and test parts in time-based split of OGB-Arxiv.
(a) PageRank
Figure 12: Structural properties across train and test parts in rank-based split of OGB-Products.
(a) PageRank
Figure 13: Structural properties across train and test parts in species-based split of OGB-Proteins.
E Dataset characteristics
Characteristics of the datasets used in this paper are listed in Table 8.
Table 8: Description of the considered graph datasets for node classification task.
Dataset	# Nodes	# Edges	# Classes	# Features
AmazonComputer	13,381	259,159	10	767
AmazonPhoto	7,484	126,530	8	745
CoauthorCS	18,333	163,788	15	6,805
CoauthorPhysics	34,493	495,924	5	8,415
CoraML	2,995	16,316	7	2,879
CiteSeer	3,327	4,732	6	3,703
PubMed	19,717	44,338	3	8,415
OGB-Products	2,449,029	61,859,140	47	100
F Method configurations
For our main series of experiments with graph models in Sections 5.1 and 5.2, we consider a graph encoder based on three GCN convolutional layers [16]. On top of this graph encoder, we use a single linear layer as the task-specific head, which is used to predict the parameters of categorical distribution in classification tasks, the parameters of Dirichlet distribution for modeling uncertainty, etc. The hidden dimension of our baseline architecture is 256, and the dropout between hidden layers is  $p = 0.2$ . We exploit a standard Adam optimizer [14] with a learning rate of 0.0003 and a weight decay of 0.00001. For an additional series of experiments with an improved GNN architecture, which is discussed in Section 5.3, we reduce the number of graph convolutional layers from 3 to 2, replacing the first one with a pre-processing step based on linear layer, apply the skip-connections between graph convolutional layers, and replace the GCN graph convolution with SAGE [11].
Some of the considered methods for improving the OOD robustness, such as DANN and CORAL, exploit the notion of domains and require the knowledge about which nodes in the training set belong to which domain. Therefore, we need to define domains, which is not straightforward since our node properties are real-valued, not discrete. Thus, we discretize the values into  $k = 10$  non-intersecting domains. After that, we assign a domain index to each node. These indices are treated as labels and used by DANN and CORAL during their training.
G Detailed experimental results
In this section, we provide detailed experimental results.
- Tables 9 — 15 show the predictive performance of various OOD robustness methods on the proposed structural distributional shifts;
- Tables 16 — 22 show the OOD detection performance of various uncertainty estimation methods on the proposed structural distributional shifts.
Table 9: Comparison of several graph methods for improving the OOD robustness in terms of their predictive performance across structural shifts on CoraML dataset.
popularity	locality	density
ID	OOD	ID	OOD	ID	OOD
ERM	85.80 ± 1.10	82.42 ± 0.79	84.47 ± 0.69	72.13 ± 1.74	93.27 ± 0.28	77.33 ± 1.07
Mixup	89.27 ± 0.57	83.65 ± 0.67	86.27 ± 0.57	78.42 ± 0.84	88.20 ± 0.44	78.98 ± 2.10
EERM	88.40 ± 1.22	83.07 ± 0.76	88.80 ± 0.73	69.96 ± 6.62	86.87 ± 0.45	79.07 ± 0.32
DANN	86.00 ± 1.10	79.01 ± 0.77	81.45 ± 0.65	65.03 ± 2.51	91.33 ± 0.76	78.37 ± 2.07
CORAL	85.00 ± 1.11	77.65 ± 0.62	82.33 ± 0.25	66.83 ± 2.24	89.55 ± 0.54	73.34 ± 2.28
DE	87.00 ± 0.00	82.74 ± 0.00	84.67 ± 0.00	75.40 ± 0.00	93.00 ± 0.00	78.90 ± 0.00
ERM + mod	85.33 ± 0.62	82.55 ± 0.56	84.87 ± 0.61	72.59 ± 0.96	94.13 ± 0.69	79.33 ± 0.37
Table 10: Comparison of several graph methods for improving the OOD robustness in terms of their predictive performance across structural shifts on CiteSeer dataset.
popularity	locality	density
ID	OOD	ID	OOD	ID	OOD
ERM	72.43 ± 1.33	72.42 ± 0.37	77.60 ± 0.66	57.03 ± 1.16	73.75 ± 0.96	67.57 ± 0.49
Mixup	72.79 ± 0.74	72.13 ± 0.69	76.82 ± 0.59	56.45 ± 4.53	76.88 ± 1.47	68.59 ± 2.12
EERM	71.47 ± 1.01	70.48 ± 0.66	75.31 ± 0.80	61.95 ± 3.30	76.46 ± 0.92	68.18 ± 0.71
DANN	67.57 ± 0.96	67.84 ± 0.65	71.17 ± 0.33	54.90 ± 2.49	75.07 ± 1.48	61.13 ± 2.82
CORAL	67.87 ± 1.19	67.77 ± 0.38	71.87 ± 0.62	57.16 ± 1.84	73.47 ± 1.22	58.53 ± 2.02
DE	73.27 ± 0.00	72.37 ± 0.00	78.38 ± 0.00	64.71 ± 0.00	74.17 ± 0.00	70.35 ± 0.00
ERM + mod	73.75 ± 0.62	71.73 ± 1.02	76.70 ± 1.14	59.86 ± 2.23	75.80 ± 0.96	68.32 ± 1.01
Table 11: Comparison of several graph methods for improving the OOD robustness in terms of their predictive performance across structural shifts on PubMed dataset.
popularity	locality	density
ID	OOD	ID	OOD	ID	OOD
ERM	86.85 ± 0.12	84.04 ± 0.33	86.75 ± 0.48	81.80 ± 1.07	85.96 ± 0.27	85.29 ± 0.16
Mixup	89.32 ± 0.24	88.21 ± 0.21	87.94 ± 0.37	86.20 ± 0.75	88.72 ± 0.25	88.23 ± 0.16
EERM	86.83 ± 0.12	83.39 ± 0.10	85.67 ± 0.24	84.79 ± 0.18	86.43 ± 0.22	84.10 ± 0.10
DANN	86.26 ± 0.48	84.49 ± 0.18	86.04 ± 0.30	85.41 ± 1.04	87.07 ± 0.24	84.74 ± 0.14
CORAL	86.31 ± 0.48	84.48 ± 0.37	86.00 ± 0.32	85.36 ± 1.63	87.24 ± 0.15	84.86 ± 0.27
DE	87.17 ± 0.00	84.72 ± 0.00	87.07 ± 0.00	82.82 ± 0.00	86.61 ± 0.00	85.97 ± 0.00
ERM + mod	88.83 ± 0.24	87.39 ± 0.27	88.32 ± 0.51	87.58 ± 0.17	88.56 ± 0.43	87.71 ± 0.22
Table 12: Comparison of several graph methods for improving the OOD robustness in terms of their predictive performance across structural shifts on AmazonComputer dataset.
popularity	locality	density
ID	OOD	ID	OOD	ID	OOD
ERM	92.43 ± 0.24	83.18 ± 0.58	93.04 ± 0.38	72.62 ± 0.52	93.10 ± 0.27	85.73 ± 0.60
Mixup	92.27 ± 0.19	66.80 ± 2.84	93.76 ± 0.19	66.94 ± 2.00	92.46 ± 0.44	74.72 ± 1.09
EERM	89.49 ± 1.55	82.08 ± 0.61	92.22 ± 0.37	61.39 ± 0.85	88.09 ± 0.55	79.98 ± 1.28
DANN	91.25 ± 0.37	83.39 ± 1.87	92.30 ± 0.77	68.84 ± 2.53	91.84 ± 0.79	83.89 ± 1.37
CORAL	91.25 ± 0.54	84.77 ± 1.48	92.47 ± 0.18	62.36 ± 1.87	91.40 ± 0.91	85.35 ± 1.58
DE	92.51 ± 0.00	84.17 ± 0.00	93.31 ± 0.00	73.93 ± 0.00	93.24 ± 0.00	86.93 ± 0.00
ERM + mod	90.99 ± 0.16	85.72 ± 0.34	91.74 ± 0.24	63.67 ± 0.35	92.40 ± 0.34	86.17 ± 0.20
Table 13: Comparison of several graph methods for improving the OOD robustness in terms of their predictive performance across structural shifts on AmazonPhoto dataset.
popularity	locality	density
ID	OOD	ID	OOD	ID	OOD
ERM	95.82 ± 0.24	87.63 ± 0.65	93.73 ± 0.29	64.73 ± 3.80	94.64 ± 0.38	91.25 ± 0.16
Mixup	97.99 ± 0.36	87.73 ± 2.39	95.55 ± 0.30	75.36 ± 2.01	95.50 ± 0.41	90.71 ± 2.13
EERM	95.90 ± 0.40	86.84 ± 1.02	91.95 ± 0.21	54.06 ± 0.81	90.46 ± 0.38	88.91 ± 0.31
DANN	95.25 ± 0.26	85.23 ± 0.78	91.81 ± 0.39	48.79 ± 5.04	93.99 ± 0.66	91.87 ± 0.46
CORAL	95.34 ± 0.27	85.91 ± 0.67	91.85 ± 0.51	45.70 ± 3.34	93.64 ± 0.80	91.99 ± 0.43
DE	95.82 ± 0.00	88.82 ± 0.00	93.59 ± 0.00	69.15 ± 0.00	94.38 ± 0.00	92.09 ± 0.00
ERM + mod	97.07 ± 0.35	89.95 ± 0.20	95.01 ± 0.43	57.41 ± 1.79	95.22 ± 0.33	91.97 ± 0.33
Table 14: Comparison of several graph methods for improving the OOD robustness in terms of their predictive performance across structural shifts on CoauthorCS dataset.
popularity	locality	density
ID	OOD	ID	OOD	ID	OOD
ERM	93.60 ± 0.18	90.67 ± 0.20	92.34 ± 0.14	91.22 ± 0.35	94.30 ± 0.17	89.71 ± 0.21
Mixup	94.97 ± 0.33	94.05 ± 0.38	94.05 ± 0.13	91.48 ± 0.16	94.97 ± 0.29	92.11 ± 0.15
EERM	93.82 ± 0.08	91.01 ± 0.15	91.22 ± 0.27	91.95 ± 0.21	93.64 ± 0.23	88.42 ± 0.15
DANN	93.89 ± 0.26	90.59 ± 0.12	91.66 ± 0.37	91.65 ± 1.29	94.60 ± 0.16	89.92 ± 0.29
CORAL	93.80 ± 0.14	90.78 ± 0.13	91.69 ± 0.23	91.37 ± 0.48	94.42 ± 0.07	89.72 ± 0.22
DE	93.68 ± 0.00	90.93 ± 0.00	92.64 ± 0.00	91.76 ± 0.00	94.55 ± 0.00	90.05 ± 0.00
ERM + mod	95.35 ± 0.15	95.00 ± 0.06	94.77 ± 0.09	94.06 ± 0.11	96.67 ± 0.11	93.64 ± 0.16
Table 15: Comparison of several graph methods for improving the OOD robustness in terms of their predictive performance across structural shifts on CoauthorPhysics dataset.
popularity	locality	density
ID	OOD	ID	OOD	ID	OOD
ERM	93.60 ± 0.18	90.67 ± 0.20	92.34 ± 0.14	91.22 ± 0.35	94.30 ± 0.17	89.71 ± 0.21
Mixup	96.96 ± 0.11	94.05 ± 0.74	97.27 ± 0.11	93.39 ± 0.95	96.80 ± 0.08	84.57 ± 0.85
EERM	95.93 ± 0.10	93.37 ± 0.08	94.04 ± 0.16	92.32 ± 0.11	95.26 ± 0.03	93.98 ± 0.06
DANN	96.56 ± 0.14	93.73 ± 0.25	96.64 ± 0.23	91.35 ± 1.12	96.08 ± 0.17	95.05 ± 0.11
CORAL	96.51 ± 0.10	93.58 ± 0.29	96.93 ± 0.08	86.21 ± 0.86	95.97 ± 0.09	95.00 ± 0.17
DE	93.68 ± 0.00	90.93 ± 0.00	92.64 ± 0.00	91.76 ± 0.00	94.55 ± 0.00	90.05 ± 0.00
ERM + mod	95.35 ± 0.15	95.00 ± 0.06	94.77 ± 0.09	94.06 ± 0.11	96.67 ± 0.11	93.64 ± 0.16
Table 16: Comparison of several graph methods for uncertainty estimation in terms of their OOD detection performance across structural shifts on CoraML dataset.
Popularity	Locality	Density
SE	75.67 ± 0.85	87.13 ± 0.74	81.55 ± 0.45
NatPN	47.31 ± 9.44	81.66 ± 3.05	69.05 ± 3.05
GPN	66.95 ± 1.15	73.09 ± 2.21	70.95 ± 1.58
DE	70.55 ± 0.00	93.32 ± 0.00	84.46 ± 0.00
NatPE	37.59 ± 0.00	78.48 ± 0.00	65.83 ± 0.00
GPE	65.67 ± 0.00	74.23 ± 0.00	71.07 ± 0.00
SE + mod	55.66 ± 0.25	72.97 ± 0.71	68.27 ± 0.58
Table 17: Comparison of several graph methods for uncertainty estimation in terms of their OOD detection performance across structural shifts on CiteSeer dataset.
Popularity	Locality	Density
SE	68.01 ± 1.23	89.89 ± 0.56	66.90 ± 0.41
NatPN	31.18 ± 1.54	91.96 ± 3.14	59.77 ± 1.54
GPN	61.99 ± 1.74	69.53 ± 5.10	57.41 ± 1.72
DE	56.22 ± 0.00	98.18 ± 0.00	70.48 ± 0.00
NatPE	28.07 ± 0.00	89.02 ± 0.00	58.10 ± 0.00
GPE	63.01 ± 0.00	73.89 ± 0.00	57.94 ± 0.00
SE + mod	54.36 ± 0.56	78.74 ± 0.55	61.03 ± 0.73
Table 18: Comparison of several graph methods for uncertainty estimation in terms of their OOD detection performance across structural shifts on PubMed dataset.
Popularity	Locality	Density
SE	68.60 ± 0.34	66.34 ± 1.03	58.60 ± 0.39
NatPN	49.61 ± 3.17	58.91 ± 2.89	51.00 ± 1.45
GPN	72.30 ± 0.29	69.63 ± 4.84	62.04 ± 0.68
DE	74.31 ± 0.00	72.19 ± 0.00	63.04 ± 0.00
NatPE	46.94 ± 0.00	56.41 ± 0.00	49.54 ± 0.00
GPE	72.62 ± 0.00	66.07 ± 0.00	62.58 ± 0.00
SE + mod	53.68 ± 0.12	54.86 ± 0.72	52.25 ± 0.13
Table 19: Comparison of several graph methods for uncertainty estimation in terms of their OOD detection performance across structural shifts on AmazonComputer dataset.
Popularity	Locality	Density
SE	88.52 ± 0.37	86.48 ± 0.80	44.24 ± 0.57
NatPN	53.83 ± 15.71	59.92 ± 6.59	44.42 ± 3.44
GPN	81.23 ± 1.44	82.84 ± 2.56	43.92 ± 2.56
DE	82.53 ± 0.00	83.54 ± 0.00	50.88 ± 0.00
NatPE	41.33 ± 0.00	51.93 ± 0.00	42.09 ± 0.00
GPE	81.35 ± 0.00	81.97 ± 0.00	43.81 ± 0.00
SE + mod	60.54 ± 0.41	74.09 ± 0.64	59.27 ± 0.44
Table 20: Comparison of several graph methods for uncertainty estimation in terms of their OOD detection performance across structural shifts on AmazonPhoto dataset.
Popularity	Locality	Density
SE	92.05 ± 0.27	93.29 ± 0.60	41.08 ± 1.89
NatPN	79.72 ± 7.29	65.66 ± 11.98	54.88 ± 17.40
GPN	80.85 ± 2.64	90.90 ± 2.82	49.97 ± 2.24
DE	90.72 ± 0.00	96.66 ± 0.00	51.33 ± 0.00
NatPE	80.43 ± 0.00	59.74 ± 0.00	64.69 ± 0.00
GPE	82.95 ± 0.00	91.61 ± 0.00	51.74 ± 0.00
SE + mod	66.93 ± 0.47	83.09 ± 1.57	51.36 ± 0.94
Table 21: Comparison of several graph methods for uncertainty estimation in terms of their OOD detection performance across structural shifts on CoauthorCS dataset.
Popularity	Locality	Density
SE	83.25 ± 0.40	85.74 ± 1.06	50.91 ± 0.26
NatPN	59.99 ± 5.57	64.53 ± 12.82	58.47 ± 8.22
GPN	63.43 ± 0.46	64.84 ± 1.57	56.81 ± 0.54
DE	82.50 ± 0.00	87.54 ± 0.00	54.47 ± 0.00
NatPE	61.00 ± 0.00	60.55 ± 0.00	62.63 ± 0.00
GPE	62.12 ± 0.00	63.76 ± 0.00	57.15 ± 0.00
SE + mod	58.05 ± 0.71	64.43 ± 1.40	56.99 ± 0.26
Table 22: Comparison of several graph methods for uncertainty estimation in terms of their OOD detection performance across structural shifts on CoauthorPhysics dataset.
Popularity	Locality	Density
SE	86.60 ± 0.29	87.73 ± 0.51	37.50 ± 0.32
NatPN	56.74 ± 3.68	59.54 ± 23.75	58.81 ± 5.96
GPN	67.48 ± 0.45	74.05 ± 2.67	50.99 ± 0.53
DE	85.94 ± 0.00	91.93 ± 0.00	36.15 ± 0.00
NatPE	56.11 ± 0.00	46.14 ± 0.00	56.94 ± 0.00
GPE	66.29 ± 0.00	72.16 ± 0.00	51.89 ± 0.00
SE + mod	64.84 ± 0.73	88.02 ± 0.31	44.13 ± 0.31
H Visualization of distributional shifts in graph domain
In Figures 14-20, we provide the visualizations of different structural shifts in graph domain for all the considered graph datasets: ID is blue, OOD is red. Some graphs have multiple connected components — in that case, we keep only the largest one.
Figure 14: Visualization of structural shifts in graph domain for CoraML dataset.
(a) Popularity
(a) Popularity
Figure 16: Visualization of structural shifts in graph domain for PubMed dataset.
Figure 15: Visualization of structural shifts in graph domain for CiteSeer dataset.
(c) Density
(b) Locality
Figure 17: Visualization of structural shifts in graph domain for AmazonPhoto dataset.
(b) Locality
Figure 18: Visualization of structural shifts in graph domain for AmazonComputer dataset.
(a) Popularity
Figure 19: Visualization of structural shifts in graph domain for CoauthorCS dataset.
(a) Popularity
Figure 20: Visualization of structural shifts in graph domain for CoauthorPhysics dataset.
I Visualization of distributional shifts in node feature space
In Figures 21-27, we provide the visualizations of different structural shifts in node feature space for all the considered graph datasets: ID is blue, OOD is red. The pictures are obtained by reducing the original node feature space into the 2D space of t-SNE representations.
Figure 21: Visualization of structural shifts in node feature space for CoraML dataset.
(a) Popularity
(a) Popularity
Figure 23: Visualization of structural shifts in node feature space for PubMed dataset.
Figure 22: Visualization of structural shifts in node feature space for CiteSeer dataset.
(c) Density
(b) Locality
Figure 24: Visualization of structural shifts in node feature space for AmazonPhoto dataset.
(a) Popularity
Figure 25: Visualization of structural shifts in node feature space for AmazonComputer dataset.
(a) Popularity
Figure 26: Visualization of structural shifts in node feature space for CoauthorCS dataset.
(a) Popularity
Figure 27: Visualization of structural shifts in node feature space for CoauthorPhysics dataset.