On Locality in Graph Learning via Graph Neural Network

Abstract: Theoretical understanding on the learning process of graph neural network (GNN) has been lacking. The common practice in GNN training is to adapt strategies from other machine learning families, despite the striking differences between learning non-graph and graph-structured data. This results in unstable learning performance (e.g., accuracy) for GNN. In this paper, we study how the training set in the input graph effects the performance of GNN.  Combining the topology awareness of GNN and the dependence (topology) of data samples, we formally derive a structural relation between the performance of GNN and the coverage of the training set in the graph. More specifically, the distance of the training set to the rest of the vertexes in the graph is negatively correlated to the learning outcome of GNN. We further validate our theory on different graph data sets with extensive experiments. Using the derived result as a guidance, we also investigate the initial data labelling problem in active learning of GNN, and show that locality-aware data labelling substantially outperforms the prevailing random sampling approach. 

Reviews and author responses:

 Thanks for providing a more clear revision regarding reviewer's comment. As stated by other reviewers as well, we found the results of this paper are based on some very strict assumptions, e.g. homophily and $\epsilon=0$. The author modify the theorem 1 and 2 by replace $p$ with distortion rate $\alpha$. However, I also find the introduction of one-to-one mapping g is not clear and only used in proof as index. The existence of such mapping should be discussed. Also a PAC-Learning error tolerance $\epsilon$, instead of 0, is more welcomed to see. I understand the experiment part is difficult to improve due to the limited time in rebuttal. I recommend author conduct experiments on large-scale benchmark as current three small graph and small training size is not convincing to a broader audience. Therefore, I hold my score but would like to see this work get improved. 

 The reviewer appreciates the authors' response. However, the major concerns of the reviewer were not well-addressed.

1. The theorem is still presented based on the existence of some constants $\delta$ and $\delta'$. The dependence of these values on the training set is still not presented, but ''hidden'' in the proof. In this way, the conclusion about the better selection of training sets based on the theorems is questionable. Since such a conclusion is made assuming $\delta$ and $\delta'$ will not change with the selection of the training set, which the reviewer finds invalid. Also, the introduction of one-to-one mapping g seems to be irrelevant as it is stated in the theorem. The effects need to be clarified.

2. The reviewer again tried the best to read the corresponding part in appendix B to understand theorem 1 and justify the conclusion and the theory but found the proof not clearly written and there seem to be some mistakes that prevent the reviewer to judge the correctness of the theorem, e.g.,
i. Cross-referencing errors: In appendix b the proof of 11 13 15 was referring to equations 22-28 that appear later;
ii. before (14), it is mentioned there exist constant $\alpha>1$, but $\alpha$ is graph distortion, and it is not guaranteed to be greater than 1.

Therefore, the reviewer decided to keep the rating unchanged.


 Thank you for your response. Unfortunately, I am not willing to increase my scoring of this paper. However, I would encourage the author's to continue to thoroughly pursue this line of work, as it is of interest to the community.

 Again, we want to thank all the reviewers for their comments and suggestions, which have helped improve the paper. 
We have tried to address all the comments and suggestions in the revision. The following is a summary of changes we have made to the revision:
1)	We have modified and revised the theory, hoping that the current version should be more explicit and easier to process. 
  * Remove the probabilistic distance preserving definition and relate the distance preserving property with distortion of the transformation (Page 3 Definition 1). 
  * Divide the original theory into two parts: i) prediction power of GNN on different test groups when trained on a fixed training set (Page 4 Theorem 1), and ii) prediction performance (test accuracy) of GNN on different pairs of training/test sets (Page 5 Theorem 2).
  * Added additional experiment (Page 7 Fig. 6) to validate the modified result

2)	We added references to some concurrent studies that have obtained similar observations and that have proved similar results (Page 2 Sec. 2).

3)	We added more comparison methods in the application sections (Page 9 Table 3) to further showcase the effectiveness of our devised strategy in initial data labelling. 


4)	We have addressed the presentation issues raised by some of the reviewers: 
  * re-arrange the related work section 
  * remove distracting discussion in the theoretical section


 Thank you so much for the reply and clarification.

>Both work are relevant, [1] is relevant from theoretical perspective. [2] can be applied to the same setting as yours and they do not generate training distributions (the claim of locality is very similar with your definition) but generalize across different training and testing distribution. In your case, the locality can be seen as different from testing distribution (IID in most case). And what I'm curious is you are trying to do active learning (find nodes to annotate while a lot of real-world applications may not have this luxury as they have a biased annotation already ) based on your proposal. However, [2] seems to be more general that help training with any given localized training data.

Even though current existing applications of GNN might come with (bias) labelled data, annotating the data is an important fundamental problem for the following reasons: 1) if we can label data well in the first place, it will substantially facilitate GNN learning; 2) there can be new graph systems with new data, and 3) one may wish to leverage the existing data set to solve a new problem (with new labels). We will further clarify this and discuss about [2] in the revision. 

> I suppose random sample with small sample size cannot guarantee IID and why does it need to be IID? I think the localized training is naturally non-IID otherwise as you suggest the nodes will probably span around the graph and distance become smaller.

We do not assume that the training set/test set/data population are IID (we are a bit confused what you may refer to as IID in our paper and will be much appreciated if you can clarify a bit more). For our experiment, we follow the standard training pipeline of GNN to avoid bias and show that the distance among these random partitions of labelled data is related to the performance (test error) of GNN. This is the same experimental setting as in [1]. The only difference is in the size of the training set. In our case, we want to vary the distance among nodes and hence choose a smaller size of the training set.


 >However, we failed to see how [2] is directly related to our work. Based on our understanding, [2] studies the problem that there is a mismatch between the distributions for generating training set and test set, which is quite different from what we focus on.

Both work are relevant, [1] is relevant from theoretical perspective. [2] can be applied to the same setting as yours and they do not generate training distributions (the claim of locality is very similar with your definition) but generalize across different training and testing distribution. In your case, the locality can be seen as different from testing distribution (IID in most case). And what I'm curious is you are trying to do active learning (find nodes to annotate while a lot of real-world applications may not have this luxury as they have a biased annotation already ) based on your proposal. However, [2] seems to be more general that help training with any given localized training data.

> Even though the size of the training set is small, we do not think the trend/relation we showed in the experiments is due to the variance of small training set size. 

I suppose random sample with small sample size cannot guarantee IID and why does it need to be IID? I think the localized training is naturally non-IID otherwise as you suggest the nodes will probably span around the graph and distance become smaller.


 Dear Reviewer:

Thank you for the detailed review and comments. Please find our response to your concerns/questions below:

> Technical novelty and significance: This paper misunderstands generalization, as it proposes a node sampling scheme for the training set based on a fixed test set of nodes...

Our study aims to show that there exists a structural relation between the distance of training/test set and the performance of GNN. In our experiments, we followed the standard GNN training pipeline and computed shortest path distances between vertexes in the training set and test set with label information of the test set only after training is completed. To apply our results, we considered the initial data labelling problem where there is no label/test set. Essentially, we did not use any label information of the test set in selecting the train set in our experiments. We are sorry if our writing has confused you, but it would be great if you could elaborate on where and how the misunderstanding could have been generated.

> Correctness: The theoretical results are undermined by the fact that Assumption 3 is unreasonable...

Assumption 3 states that if a vertex in the test set is closed to a training vertex of the same class in the graph, this vertex would be mapped into a simple neighbourhood of the training vertex in the embedding space (where loss values of abstracted points display monotonic behaviour with respect to the distance to the centre of the neighbourhood). The objective of this study is to see if we could use network principles/structure and structural behaviour of GNN to derive useful information/insight in GNN learning for node classification. Our theory is closely tied to network homophily; if a data set/the problem does not display network homophily property, then our theory indeed does not hold. However, because network homophily is a common network principle that governs many real-world networks, we believe our result and observation provide useful insight/guidance in many situations such as the one we presented in Section 4 (page 8). In the example you gave, if the problem is cast into a node classification problem, the two users are not considered close in our framework as they have different labels (different labels/preferences on horror movies). 

The reason we use a probabilistic definition of distance preservation but proved for the case that ‘’distance-preservation property of p = 1’’ is as follows: The main objective of our theoretical analysis is to show that there exists a theoretically plausible structural relation between the distance of train/test set and the performance of GNN; then we can use this understanding into designing algorithms in a number of applications such as initial data labelling. We could relax p to be another value, use expectation in our theorem/derivation and obtain similar results. We felt doing the latter does not add too much additional value to our main objective (to show a theoretically plausible structural relation between the distance of train/test set and the performance of GNN). In addition, in our experiments, we first validated whether GNNs indeed preserve distance: as shown in Fig. 4 (Page 7) and Appendix E.2, GNNs do have a strong distance preserving property. 

> Empirical novelty and significance: The experiments are restricted to the Cora dataset, and the improvements reported in Tables 1 and 2 are marginal. The authors should run experiments on other node classification datasets.

As stated at the beginning of the experiment section, we only included figures from Cora dataset due to the page limit. We have experiments on other node classification data sets included in the Appendix. The reason why the improvement might appear marginal is that we used small training sets (to create variance on average distance) and divided all 100 partitions into three regions. The improvement in the tables is the average improvement of the partitions in each region. 

> Related work is typically presented in Section 2...

We will definitely try to address these presentation issues in the revision.



 Dear Reviewer:

Thank you for the detailed review and comments. We are glad that you enjoyed the paper. Please find our response to your concerns/questions below: 

> Definition 1: It is unclear at what point in this statement the proper context is in place to make a probabilistic claim about some map. Is there some probability measure on the metric space according to which the points were sampled? Please explain why the probability p is needed in this definition.

> Theorem 1: My confusion here ties back to my question about Definition 1. Why are we dealing with probabilities p, particularly when we put p=1 here? And if M is a GNN with distance preservation, do we mean as a map from the set of nodes with the shortest path distance to the embedding space? Your choice to use d for many different metric spaces has made this confusing. Please clarify the reasoning behind this. It is also strange that you used the double bars to refer to what I think is the cardinality of the set of nodes in B≤||V||/2: did you mean to do this?

The reason we include probability p in the distance preservation property is that GNN has a strong distance preserving property but this does not necessarily hold for all pairs of vertexes. There might exist some vertexes whose embeddings violate this property, while the structural relation we proposed is still exhibited, as illustrated in our experiments. In our analysis, we proved the case when GNN has a perfect distance preserving property, in order to obtain a clean and clear result.  It is indeed possible to relax our theorem/analysis to allow p to be less than 1. We just did not find this to add too much additional value to the message we try to deliver. When we say a GNN map is distance preserving, it means that the relative order of elements in the two metric spaces is preserved. Sorry for the confusion caused by the notation and thank you for catching the typo (|V|). We will have them addressed in the revision.

Assumption 3: I appreciated the empirical validation of this assumption, and I do think it would be too hard to show this holds purely theoretically. However, I do not find the justification of "the assumption seems to hold empirically" to be a useful explanation of this assumption. Please provide some insight into why you think a well-trained graph neural network would satisfy this property. In particular, it is currently unclear what kind of assumptions on the data are necessary, or at least suggestive, for this to be true.
We briefly discussed this on page 6 under ‘’Network Homophily’’. Many real-world networks display the network homophily property, and this principle is also behind the design of many GNNs. If a dataset or the problem does not display network homophily, then our theory probably does not hold. After you point this out, we realize that we indeed did not discuss much on the data set. Thank you for the comment. We will address this in the revision and bring this discussion earlier in the presentation.

> Section 4: I found this application of your results quite pleasant. This section does a good job demonstrating the broader scope of your results, and how they can be used to motivate graph sampling strategies. I found the experimental comparisons lacking, though. For instance, there are many better sampling methods one could imagine that would probably fare better in finding a good set of nodes. One good one would be selecting most important nodes according to their eigenvector centrality, or betweenness centrality. These approaches reflect the role of a node in the global graph better, and would make for a stronger comparison with the proposed program. Additionally, spectral methods from the graph signal processing literature ought to be included, such as [1].

In our study, we mainly considered spatial GNN, and thank you for your suggestions. We will definitely include more comparisons of different sampling strategies in the revision.


 Dear Reviewer:

Thank you for the detailed review and comments. Please find our response to your concerns/questions below:

> Missing some important references...

Thank you so much for these latest references! We have looked into both [1] and [2] and found that the setting/problem studied in [1] is indeed closely related to the problem we studied here. However, we believe our focuses are different. Their focus is on the data population distribution, but our focus is on what the graph structure can tell us about the performance of GNN training.
As echoed in [1], there is currently very limited study on the generalization of GNNs on node-level tasks.  We believe our study has been complementary to [1] in the case of geodesic distance. We will include a more in-depth discussion about [1] in the related work section in the revision.

However, we failed to see how [2] is directly related to our work. Based on our understanding, [2] studies the problem that there is a mismatch between the distributions for generating training set and test set, which is quite different from what we focus on. 

>I.I.D. condition cannot hold in the analysis. Most of the plot and experiments are conducted via multiple random sampling with small training size (35 on Cora, which is 5 per class)...

>The experiment section is weak, the extreme small number training seeds leads to bad generalization already. Based on my understanding, the main outcome of the theorem 1 is to study the correlations in Figure 4,5,6, which is quite similar with [1,2]...

Theorem 1 states that the average distance must improve (decrease) for a certain threshold before performance improvement can be guaranteed.  The goal of the experiments is to validate some of the assumptions and that there indeed exists a relation between the distance of train/test set and GNN performance. In our experiment, we followed the standard GNN training pipeline and randomly selected vertexes for the training set and showed that the average distance of the (random) training set does correlate with the performance (test error). We agree that a small sample size brings more variance. The reasons we chose a small training set are as follows:

1) For our experiments, we need the shortest path distance between all pairs of vertexes in the graph, and it is not very feasible to compute this with a large graph data set (complexity too high). 

2) Because of 1), we choose the three standard and relative small datasets for our experiment.
Because of the power-law property of real-world networks and the size of the data set, if we chose a large training set, this would result in very little change on the average distance.

3) Even though the size of the training set is small, we do not think the trend/relation we showed in the experiments is due to the variance of small training set size. In Section 4 (Page 8), we conducted experiments on bias sampling for the active learning process and showed that our metric also outperformed the commonly used one. We will add more experiments on larger datasets in the revision.

> The writing is hard to follow and lack of theoretical justifications In section 3.2, the technical contributions are scattered. Mentioning Loss Landscape and Network homophily makes me distracted from the main focus of the paper...

We are sorry that our discussions on network homophily and loss landscape have caused distractions. The purpose of these discussions is to provide a more intuitive graphical illustration of the idea behind our theory and to connect it with existing network theory. We will definitely try to address this in the revision. 

We did not avoid discussion on node features. The key motivation/belief/focus we have for this study is that graph structure provides useful information on the learning process of GNN. As we discussed in the network homophily session, the reason for this could be the underlying organizing principle of the real-world network, where closed vertexes tend to have similar behaviour (feature). Thanks for the comments. We will try to make this more explicit and mentioned it earlier in the revision. 

As for ϵ=0, we discussed it right under Assumption 2. We believe that it is theoretically feasible to have zero loss on the training data (one could have a neural network layer with enough neurons to memorize each training sample, assuming there are no identically featured subgraphs). Having this assumption simplifies the analysis. In practice, we expect the GNN to have a small training loss; as demonstrated in our experiment, that is sufficient to exhibit the proposed structural relation.

 Dear Reviewer:

Thank you for the detailed review and comments. Please find our response to your concerns/questions below:

> The major result Theorem 1 is proved under assumptions 1-3 and can only be applied to GNNs ``distance-preservation property of p = 1 as given in Definition 1.'' However, the reviewer found these assumptions too strong to be applied to GNNs. For example, assumption 2 does not hold for any training dataset that cannot be perfectly separated for a classification task, which is the typical case...
 
The reason we use a probabilistic definition of distance preservation but proved for the case that ‘’distance-preservation property of p = 1’’ is as follows: The main objective of our theoretical analysis is to show that there exists a theoretically plausible structural relation between the distance of train/test set and the performance of GNN; then we can use this understanding into designing algorithms in a number of applications such as initial data labelling. We could relax ``p’’ to be another value, use expectation in our theorem/derivation and obtain similar results. We felt doing the latter does not add too much additional value to our main objective (to show a theoretically plausible structural relation between the distance of train/test set and the performance of GNN). In addition, in our experiments, we first validated whether GNNs indeed preserve distance: as shown in Fig. 4 (Page 7) and Appendix E.2, GNNs do have a strong distance preserving property. 
 
As for Assumption 2, we believe it is theoretically feasible to have zero loss on the training data (one could have a neural network layer with enough neurons to memorize each training sample, assuming there are no identically featured subgraphs). Having this assumption simplifies the analysis. In practice, we expect the GNN to have a small training loss; as demonstrated in our experiment, that is sufficient to exhibit the proposed structural relation.

> Both Proposition 1 and Theorem 1 depend on several constants:r_v, B, and delta. However, these constants are actually related to the graph, and such relationships are intertwined in the proof. For example, in Appendix B, it is claimed delta=S-S' and delta>(\beta-1)S, where S and S' are some quantities related to graph and testing data. Hence, it seems to the reviewer delta is actually not a constant. Also, these "constants" seem to be intertwined, hence it is not clear whether they actually "exist". Please clarify all these constants and how they are related to the data and each other for the reviewer to better evaluate the generality and usefulness of the theory.

We completely agree that these constants are related to the graph and the GNN. For example, r_v is the width of the neighbourhood in the embedding space that displays simple monotonic behaviour. We do not think it feasible to theoretically show how this width is related to the graph and the GNN, according to our current understanding. We compensated this with a rather comprehensive experiment, as illustrated in Fig. 5. 

You are right that delta depends on the choice/size of the train set. The reason we need the threshold (B) on the size of the training set is that we only guarantee the structure of the loss function in the embedding space in a local region. We want to use this predictable local region to cover the test set in the embedding space so that we can derive a structural result on them. Therefore, we use the size of the training set to achieve this.  We aimed to state/prove that given Train Set 1, Train Set 2, and Test Set, there exists a finite constant delta if the average distance/size of Train Set 1 (to the Test Set) and average/size distance of Train Set 2  (to the Test Set) satisfy the premise, and then we can guarantee an improvement. Sorry for the confusion. We will clarify it further in the revision. 

> The analysis only considers the case where the testing sets are already revealed in the training stage (transductive setting), it would be great if the authors could provide some insight on the inductive setting as well.

To derive a relation between the training set and the performance of GNN, we need a way to ``measure/relate’’ the training set to the generalization/test error. One can achieve this by making assumptions on how the training set is related to the underlying data population; or in our case here, using the information on a given test set. In Section 4 (Page 8), we applied our result to active learning of GNN for the initial labelling problem: in that setting, no label information of any vertex is given; training the GNN on a vertex set with a good coverage allows the GNN to generalize better comparing to other commonly used strategies. This implies that coverage of the training set is a good indicator of the generalization performance of GNN.  




This paper aims at providing a theoretical understanding of GNNs. Specifically, in terms of how the training set in theinput graph effects the performance of GNN on the testing set. 
 This paper studies the problem of understanding GNNs, which is of great interest and importance. Following are some questions that need to be clarified and addressed:

1. The major result Theorem 1 is proved under assumptions 1-3 and can only be applied to GNNs ``distance-preservation property of p = 1 as given in Definition 1.'' However, the reviewer found these assumptions too strong to be applied to GNNs.
For example, assumption 2 does not hold for any training dataset that cannot be perfectly separated for a classification task, which is the typical case. Also, it is not clear why GNNs would satisfy distance preservation with prob 1, since GNN is a highly nonlinear mapping. To improve the usefulness of the analysis, it should be provided with a certain probability value, instead of only p=1.

2. Both Proposition 1 and Theorem 1 depend on several constants:r_v, B, and delta.  However, these constants are actually related to the graph, and such relationships are intertwined in the proof. For example, in Appendix B, it is claimed delta=S-S' and delta>(\beta-1)S, where S and S' are some quantities related to graph and testing data.  Hence, it seems to the reviewer delta is actually not a constant. Also, these "constants" seem to be intertwined, hence it is not clear whether they actually "exist". Please clarify all these constants and how they are related to the data and each other for the reviewer to better evaluate the generality and usefulness of the theory.

3. The analysis only considers the case where the testing sets are already revealed in the training stage (transductive setting), it would be great if the authors could provide some insight on the inductive setting as well.

 While the paper considers an important theoretical problem, the reviewer finds the assumptions not justified and the constants claimed to exist not well-defined. The reviewer already went through the appendix to find the definition of the constants, but failed to understand why they are constants, and why they exist. Hence, the reviewer considers the submission not good enough at this stage.

This paper draws connection between performance of GNN and training set coverage in the graph. Specifically, it proposes theoretical study towards structural relation between them in terms of graph distance and empirical classification loss. A set of experiments are designed to validate the theory and assumption on three graph dataset. Further, the proposed idea is used in active learning as guidance. ## Strength
1. **The illustration of the paper is clear, especially Fig.2 and 3.**
2. **The experiments on active learning with the proposed theory is interesting and promising.**

## Weaknesses
1. **Missing some important references.** The author claims there is no existing study on dependency of the data samples and performance of GNN models. However, there are couple of work [1,2] study the generalization from non-I.I.D training data. See next point for more details.
1. **I.I.D. condition can not hold in the analysis.**
Most of the plot and experiments are conducted via multiple random sampling with small training size (35 on Cora, which is 5 per class). However, as theorem 1 states, the average distance can not guarantee the vertexes closer to $D$ than $D'$. Moreover, as we all know, smaller sample size brings more variance to the uniform sampling. Hence, I believe I.I.D training data is not robust during author's presentation, which makes this work more comparable with [1,2] in a generalized non I.I.D analytical setting.
1. **The writing is hard to follow and lack of theoretical justifications**
In section 3.2, the technical contributions are scattered. Mentioning Loss Landscape and Network homophily makes me distracted from the main focus of the paper. Instead, I expect to see more discussion on node features, since the $M$ in theorem lack discussion of node features $X$, which I deem essential in analysis of any neural networks. Both [1] and [2] specifically include $X$ in discussion. In other words, this paper use assumption to avoid discussion on node features makes me a bit uncomfortable. I don't think you can let $\epsilon=0$ from universal approximation of GNNs.
1. **The experiment section is weak, the extreme small number training seeds leads to bad generalization already.**
Based on my understanding, the main outcome of the theorem 1 is to study the correlations in Figure 4,5,6, which is quite similar with [1,2]. Specifically, you can see correlations from [1,2] is much more clear, partially because they use the embedding distance as the main measurement. However, the usage of graph distance here, again, do not consider the node features. Also, the performance of Table 1 and 2 can not align with existing semi-supervised GNN because it only uses 5 per class training seeds while 20 per class is more common. Fewer training seeds make the model worse generalization on testing and not convincing. I suggest you can use more biased samples and larger sample size (as of PPR-sampler in [2]). I also think experiments on three small benchmark is not sufficient for the significance of the study. 

[1] Ma, Jiaqi, Junwei Deng, and Qiaozhu Mei. "Subgroup Generalization and Fairness of Graph Neural Networks." NeurIPS 2021.

[2] Zhu, Qi, et al. "Shift-Robust GNNs: Overcoming the Limitations of Localized Graph Training Data." NeurIPS 2021. Overall, I think the problem this paper studied is interesting and exciting. However, there are already bunch of studies on this "localized training data". So I expect more new results and observations. This paper is not directly improving the generalization performance under arbitrary samples while more useful in active learning when you have the luxury to select training samples. I would like to see a more comprehensive version in the future considering more justifications on current assumptions and experiments. Therefore, I lean towards reject this time.

This paper considers the problem of training set selection for graph neural network training. It shows that the generalization from the training to the test set in a well-trained graph neural network is closely tied to the shortest path distance in the graph, which motivates training strategies that "cover" the graph in this sense. The authors apply these ideas in a subset selection program, and verify their results with numerical experiments. I enjoyed this paper quite a bit. The presentation was (for the most part) clear and simple, and experiments were carefully chosen. Some technical details weren't satisfactory, and some comparisons were lacking, as I explain below.

Definition 1: It is unclear at what point in this statement the proper context is in place to make a probabilistic claim about some map. Is there some probability measure on the metric space according to which the points were sampled? Please explain why the probability $p$ is needed in this definition.

Proposition 1: I don't have a critique of the technicality of this result, but it would be helpful to the reader to clarify that this does not relate to distances between nodes in the graph at this point. Perhaps it would be useful to add a line emphasizing that claim is only made for abstract points in the embedding space, and not for embedded nodes in that space.

Assumption 3: I appreciated the empirical validation of this assumption, and I do think it would be too hard to show this holds purely theoretically. However, I do not find the justification of "the assumption seems to hold empirically" to be a useful explanation of this assumption. Please provide some insight into *why* you think a well-trained graph neural network would satisfy this property. In particular, it is currently unclear what kind of assumptions on the data are necessary, or at least suggestive, for this to be true.

Theorem 1: My confusion here ties back to my question about Definition 1. Why are we dealing with probabilities $p$, particularly when we put $p=1$ here? And if $M$ is a GNN with distance preservation, do we mean as a map from the set of nodes with the shortest path distance to the embedding space? Your choice to use $d$ for many different metric spaces has made this confusing. Please clarify the reasoning behind this. It is also strange that you used the double bars to refer to what I think is the cardinality of the set of nodes in $B \leq ||V||/2$: did you mean to do this?

Experiments: I found the experiments in this paper quite convincing. Although I had a complaint about it earlier, Figure 4 is illustrative in showing the relationship between graph distances and embedding distances in Assumption 3. I also found Figure 5 convincing, as it clearly showed that those nodes located far away from the training set had high loss, due to unpredictability of the GNN acting on those regions of the graph. It would have been nice to see comparisons to some more GNN architectures, such as ChebNets, for instance.

Section 4: I found this application of your results quite pleasant. This section does a good job demonstrating the broader scope of your results, and how they can be used to motivate graph sampling strategies. I found the experimental comparisons lacking, though. For instance, there are many better sampling methods one could imagine that would probably fare better in finding a good set of nodes. One good one would be selecting most important nodes according to their eigenvector centrality, or betweenness centrality. These approaches reflect the role of a node in the global graph better, and would make for a stronger comparison with the proposed program. Additionally, spectral methods from the graph signal processing literature ought to be included, such as [1].

[1] Anis, Aamir, Akshay Gadde, and Antonio Ortega. "Efficient sampling set selection for bandlimited graph signals using graph spectral proxies." IEEE Transactions on Signal Processing 64.14 (2016): 3775-3789. I found this paper to be well-written, focused in its approach, and convincing in its argument. I do have some issues that I hope the authors will address, so I will give this paper a weak accept for now.

This paper derives theoretical results showing that, in node classification problems, GNN performance is negatively correlated to the distance of the nodes in the training set to the other nodes of the graph. Numerical results are presented to validate these findings. Technical novelty and significance:
This paper misunderstands generalization, as it proposes a node sampling scheme for the training set based on a fixed test set of nodes. The test set is a construct, hence, the accuracy results obtained on the test set should be seen as a proxy to measure generalization. In practice, we want to have good classification accuracy for all nodes, so picking training nodes according to their distance to a given test set skews the GNN to only perform well on this specific choice of test nodes. The authors should rethink their sampling approach. It might also be a good idea to refer to the literature of graph signal sampling, e.g. (Anis, 2016).

Correctness:
The theoretical results are undermined by the fact that Assumption 3 is unreasonable. This assumption is problem-dependent and unrealistic in general. It is very strong to assume that just because two nodes are close on the graph they will have the same embedding. For instance, in movie recommendation on a user similarity network, which can be modeled as a node classification problem, you may have two users (nodes) that are similar because they tend to like the same movies of the comedy genre, but that will disagree in their ratings of the horror movie under consideration because they have a different taste for horror movies. This type of relationship will typically not be learned through locality, but through permutation equivariance which allows exploiting symmetries between this node neighborhood and other node neighborhoods with the same preference makeup (i.e., neighbors who like the same comedy movies, but dislike the same horror movies) where ratings are available. The assumption that the GNN model has the distance preservation property in Theorem 1 is also too strong and hard to verify in practice.

Empirical novelty and significance:
The experiments are restricted to the Cora dataset, and the improvements reported in Tables 1 and 2 are marginal. The authors should run experiments on other node classification datasets.

Other comments:
- Related work is typically presented in Section 2. The authors should cite papers discussing other mathematical properties of GNNs such as permutation invariance and stability.
- The notation in 2.1 is confusing. X and Y are used to denote vertices, representations and sets.
 This paper misunderstands generalization, as it proposes a node sampling scheme for the training set based on a fixed test set of nodes. The test set is a construct, hence, the accuracy results obtained on the test set should be seen as a proxy to measure generalization. In practice, we want to have good classification accuracy for all nodes, so picking training nodes according to their distance to a given test set skews the GNN to only perform well on this specific choice of test nodes. The authors should rethink their sampling approach. 

Meta-review: This paper analyses generalization ability of graph neural network from the aspect of the distance between the test data point and the training data point, where the labels of a part of the vertexes are observed as the training data and a test data point is selected from the remaining vertexes. The theoretical result indicates that if the training data "cover" the whole vertexes of the graph, then the test accuracy will be better. This theoretical finding is supported by some numerical experiments. 

The problem this paper considers is interesting and would be worth investigation. However, the theoretical results presented in the paper are based on quite strong assumptions, and the statement of the theorem is not well exposed.  
- First, the paper assumes that a distortion map is obtained by training and the training procedure can produce zero training errors. Although these assumptions are far from obvious in practice, the paper lacks justification of these assumption. Hence, these assumptions seem to be made only for the sake of proof.  
- Second, the constants appeared in the theorems are not correctly specified. How different constants are correlated is not properly exposed.  

As for the experiments, they are not so strong: only Cora is experimented and training data size is small.  
For these reasons, this paper is not sufficiently matured to appear in ICLR.

Decision: Reject