Robust Counterfactual Explanations on Graph Neural Networks

Abstract: Massive deployment of Graph Neural Networks (GNNs) in high-stake applications generates a strong demand for explanations that are robust to noise and align well with human intuition. Most existing methods generate explanations by identifying a subgraph of an input graph that has a strong correlation with the prediction. These explanations are not robust to noise because independently optimizing the correlation for a single input can easily overfit noise. Moreover, they are not counterfactual because removing an identified subgraph from an input graph does not necessarily change the prediction result. In this paper, we propose a novel method to generate robust counterfactual explanations on GNNs by explicitly modelling the common decision logic of GNNs on similar input graphs. Our explanations are naturally robust to noise because they are produced from the common decision boundaries of a GNN that govern the predictions of many similar input graphs. The explanations are also counterfactual because removing the set of edges identified by an explanation from the input graph changes the prediction significantly. Exhaustive experiments on many public datasets demonstrate the superior performance of our method.


Reviews and author responses:

 We thank the reviewer for the reply. Below is our response.
  
**Q9: Re Q7.4: In my opinion, the most severe problem with this paper is the relatively handwavy discussion around the concept of robustness. I would have expected a clear definition of robustness and a clear definition of how a method can be defined robust to noise. I have also observed that another reviewer had the same issue with the definition of robustness.**  

**A9:** As systematically studied in [Ghorbani et al. AAAI’19], the robustness of an explanation method refers to the extent that explanations are stable against small perturbations applied to the input data. Ghorbani et al. measure the robustness of explanation methods by how much explanations change when a fixed amount of perturbation is applied to input data.
Following the idea by Ghorbani et al., the robustness of an explanation method is a “continuous variable” instead of a “binary one”. This means, instead of claiming whether a single explanation method is robust or not, it is more meaningful to discuss whether one explanation method is more robust than another explanation method.  
As discussed in Lines 349-350, we compare the robustness of a series of explanation methods by quantifying how much an explanation changes after perturbing the input graph by noise. When input graphs are perturbed by the same amount of noise, we can compare the robustness of different explanation methods by comparing the extents of changes of the produced explanations. This aligns well with the concept of explanation robustness studied by Ghorbani et al.  
*Ghorbani, A., Abid, A., & Zou, J. (2019). Interpretation of Neural Networks Is Fragile. Proceedings of the AAAI Conference on Artificial Intelligence, 33(01), 3681-3688. https://doi.org/10.1609/aaai.v33i01.33013681*


**Q10: Re Minimality and Sparsity: Your answer does not really clarify my question because it is still not true that you explicitly minimize for the amount of edges to be modified.**
  
 **A10:** As stated in Lines 288-292, we applied sparsity constraints in a way similar to [41] to select only a small number of edges in the input graph G as part of our explanation. The constraints $\mathcal{R}\_{sparse}$ and $\mathcal{R}\_{discrete}$ are explicitly written as parts of the loss function $\mathcal{L}(\theta)$ in Equation (8). Thus, we are explicitly minimizing $\mathcal{R}\_{sparse}$ and $\mathcal{R}\_{discrete}$ when minimizing the loss function $\mathcal{L}(\theta)$. Since minimizing $\mathcal{R}\_{sparse}$ and $\mathcal{R}\_{discrete}$ reduce the number of selected edges, we are explicitly reducing the amount of edges to be modified.

**Q11: Re LDBs and GNNs: After re-reading your paper and the reply I am still not sure I understand what are the relationships between LDBs and GNNs. If there are not, what is the goal of LDBs? What are the possible alternatives? Why LDBs are the most suitable tool for this step in the algorithm?**
  
  **A11:** As stated in Lines 149-158, for GNNs adopting piecewise linear activation functions, the relationship between the LDBs and a trained GNN is that the LDBs are the decision hyperplanes of the GNN. That is, the LDBs carry the real decision logic of a GNN. 
The goal of using LDBs is to produce the explanations that truthfully reflect the real decision logic of a GNN. If we aim to interpret the decision logic of a GNN, then we have to interpret the LDBs of the GNN because the LDBs are exactly the decision hyperplanes of the GNN [4]. 

**Q12: Re Q5.1: If I understand correctly your method also optimizes for each single instance (in the same way as PGExplainer also does) so I do not really see how your method "fundamentally differs" from the others?**  
  
**A12:** As explained in Lines 153-168, our explanations are produced from the decision regions formed by the LDBs of a GNN. For each decision region, the corresponding LDBs form the boundaries of the decision region, and they are shared by all the input graphs covered by the decision region. These LDBs explicitly characterize the common decision logic of the GNN on all the input graphs covered by the decision region. Therefore, the common decision logic extracted by our method is not optimized for a single input graph; instead, it is optimized to explicitly capture the common decision logic of as many input graphs as possible.  
Unlike PGExplainer, when optimizing explanation for a single input graph, our loss terms in Equations (9) and (10) explicitly use the LDBs (i.e., $\mathcal{B}\_i$) of the corresponding decision region. Since the LDBs carry the common decision logic shared by all the input graphs covered by the decision region, the explanation of an input graph is not optimized independently from the other input graphs. 
In summary, our method explicitly leverages the common decision logic of the GNN to produce explanations. This is substantially different from PGExplainer, and it is also why we can observe the significant difference between the performance of PGExplainer and RCExplainer in the reported experimental results.

**Q13: Re Q5.2: The discussion on L_some and L_opp is not clear at all, in my opinion. I still believe that the two loss functions are very similar as L_some tries to minimize the difference between predictions and L_opp tries to change prediction.**  
  
**A13:** $\mathcal{L}\_{same}$ and $\mathcal{L}\_{opp}$ implement two different requirements on explanations. The requirements are described in Lines 247-250, and we discuss them as follows.  
**Requirement 1** (implemented by $\mathcal{L}\_{same}$): The prediction of the GNN on the subgraph $G\_S$ of $G$ should be consistent with the prediction on $G$. Since $G\_S$ is induced by the set of edges $S$ that are selected as our explanation, if the prediction on $G\_S$ is different from the prediction on $G$, then the explanation will not be trustworthy. According to [21, 41], requirement 1 is reasonable for a trustworthy explanation.  
**Requirement 2** (implemented by $\mathcal{L}\_{opp}$): The prediction of the GNN on the subgraph $G\_{E \setminus S}$ should be substantially significantly from the prediction on $G$. Recall that $G\_{E \setminus S}$ is the remainder subgraph after removing $S$ from $G$. If the prediction of $G\_{E \setminus S}$ remains the same as the prediction of $G$, then the original prediction is not dominated by $S$, which implies $S$ is not a trustworthy explanation. Therefore, requirement 2 is also a reasonable requirement for a trustworthy explanation.  
Therefore, $\mathcal{L}\_{same}$ and $\mathcal{L}\_{opp}$ implement two different requirements of a trustworthy explanation. Optimizing one requirement does not necessarily optimize the other one. It is important to optimize $\mathcal{L}\_{same}$ and $\mathcal{L}\_{opp}$ simultaneously.




 I am still not convinced by some of your replies. I will comment them all here in this box instead of dividing it into three parts.

Re Minimality and Sparsity: Your answer does not really clarify my question because it is still not true that you explicitly minimize for the amount of edges to be modified.

Re LDBs and GNNs: After re-reading your paper and the reply I am still not sure I understand what are the relationships between LDBs and GNNs. If there are not, what is the goal of LDBs? What are the possible alternatives? Why LDBs are the most suitable tool for this step in the algorithm?

Re Q5.1: If I understand correctly your method also optimizes for each single instance (in the same way as PGExplainer also does) so I do not really see how your method "fundamentally differs" from the others?

Re Q5.2: The discussion on L_some and L_opp is not clear at all, in my opinion. I still believe that the two loss functions are very similar as L_some tries to minimize the difference between predictions and L_opp tries to change prediction.

Re Q7.4: In my opinion, the most severe problem with this paper is the relatively handwavy discussion around the concept of robustness. I would have expected a clear definition of robustness and a clear definition of how a method can be defined robust to noise. I have also observed that another reviewer had the same issue with the definition of robustness.

 **Q7.1: Lines 7-8: there is no experimental support for this claim.**

**A7.1**: "aligning well with human intuition" means a counterfactual explanation is easy to be comprehended by a human. It has been demonstrated by psychologists that counterfactuals elicit causal reasoning in humans [1, 2, 3] and are thus more intuitive. There have also been user studies to confirm that users prefer counterfactual explanations over case-based reasoning [4, 5]. 

Our method is designed to produce counterfactual explanations; thus the produced explanations are expected to be easy to comprehend. We have also shown some qualitative results in Table 4 of Section H of the appendix. As illustrated in Section H, the results in Table 4 shows that the proposed RCExplainer can accurately identify the target motifs that aligns well with human understanding. We will describe this more precisely and also add more qualitative results to the revised paper.


[1] Ruth Byrne. 2008. The Rational Imagination: How People Create Alternatives to Reality. The Behavioral and brain sciences 30 (12 2008), 439–53; discussion 453. https://doi.org/10.1017/S0140525X07002579

[2] Ruth M. J. Byrne. 2019. Counterfactuals in Explainable Artificial Intelligence (XAI): Evidence from Human Reasoning. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI-19. International Joint Conferences on Artificial Intelligence Organization, California, USA, 6276– 6282. https://doi.org/10.24963/ijcai.2019/876

[3] D. Kahneman and D. Miller. 1986. Norm Theory: Comparing Reality to Its Alternatives. Psychological Review 93 (1986), 136–153

[4] Reuben Binns, Max Van Kleek, Michael Veale, Ulrik Lyngs, Jun Zhao, and Nigel Shadbolt. 2018. ’It’s Reducing a Human Being to a Percentage’: Perceptions of Justice in Algorithmic Decisions. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems (CHI ’18). Association for Computing Machinery, New York, NY, USA, 1–14. https://doi.org/10.1145/3173574.3173951

[5] Jonathan Dodge, Q. Vera Liao, Yunfeng Zhang, Rachel K. E. Bellamy, and Casey Dugan. 2019. Explaining Models: An Empirical Study of How Explanations Impact Fairness Judgment. In Proceedings of the 24th International Conference on Intelligent User Interfaces (IUI ’19). Association for Computing Machinery, New York, NY, USA, 275–285. https://doi.org/10.1145/3301275.3302310


**Q7.2: Lines 14-15: Authors do not do any human experiments, so they cannot claim that their method aligns well with human intuition.**

**A7.2**: please refer to the answer to Q7.1.


**Q7.3: Line 114: the sentence “directly optimize for explainability of an explanation” means nothing. How do you define “explainability”?**

**A7.3**: The word “explainability” is not well defined, and we will rectify this wording in the final paper. What we intend to express is as follows. In our method, our loss function consists of a term ($\mathcal{L}\_{same}$) that enforces the resulting explanation to justify the original prediction while other loss term ($\mathcal{L}\_{opp}$) preserves the counterfactual nature of the explanation. Both characteristics aim to make the explanation more human-interpretable. 


**Q7.4: Lines 126-131: the definition is not a proper definition. What is “slight,” how do you quantify a “slight change”? How do you define “stable”?**

**A7.4**: The "slight changes" refers to minimizing the changes made on an input graph G. This is achieved by the sparsity constraints of $\mathcal{R}\_{sparse}$ and $\mathcal{R}\_{discrete}$ introduced in lines 288-291 of the paper. “Stable” means the explanations should not change when slight changes to input graph are applied. This is demonstrated by the experiment results in Figure 2. We will clarify this in our revised paper.


**Q7.5: Line 230: I’ve found no support for this claim at all anywhere.**

**A7.5**: We have explained and justified this claim in lines 226-230. Since GNNs make decisions by aggregating messages passed on edges, including edges in explanation would also provide insights about interactions between nodes that are important for the prediction. Therefore, highlighting edges identify important nodes along with the crucial specific interactions between such nodes.



**Q7.6: Line 302-303: have you tried to reuse the network without retraining? Does it work as claimed in the paper?**

**A7.6**: In a general setting, since producing explanations is an unsupervised task, the same data is used for training (if applicable) and evaluation of the explanation methods. 
However, to be useful in an inductive setting, the explanation model needs to generalize well on the new unseen input graphs that were not used in the training. 

We have thoroughly examined our explainer by training it only on a part of the whole data and then evaluating the performance on the remaining data. The test data was not used in any phase of the training of explanation network. We used 80% data for training and 20% data for testing. We found that the explainer’s performance on the test data remains consistent with the performance on training data. The results of RCExplainer on test split were consistent with the results reported in the experiment section of the paper. Therefore, like PGExplainer [21], it is perfectly suitable to be used in inductive setting on new input graphs. We will add these details to the paper. 



**Q8: The claim that the method is robust to noise is weakly supported by the experiments, in my opinion. For instance, in Mutagenicity, PGExplainer is close (although the error bars are separate) to the other two methods. Also, the robustness feature is poorly motivated in the paper. Why is it essential to be robust? If you don’t mention proper motivations, the problem definition feels rather artificial.**

**A8**: As shown in Figure 2, our method does slightly better than PGExplainer on Mutagenicity but is significantly better than PGExplainer on other 2 datasets. For instance, our method shows up to 35% and 17% better robustness than PGExplainer on BA-2Motifs and NCI1 respectively. PGExplainer does better on Mutagenicity dataset than the other datasets because the samples of Mutagenicity  dataset share a lot of common structures such as carbon rings, which makes it easier for the neural network trained by PGExplainer to identify these structures in the presence of noise.  However, for BA-2Motifs and NCI1, this is harder as the graphs in these datasets share very few structures. Therefore, the performance of PGExplainer drops significantly. We discussed this reasoning in lines 369-376.

We motivate the robustness of GNN explanations as follows.

The robustness of DNN explanations has been studied in computer vision domain, and the key idea aligns quite well with our work. According to Ghorbani et al. (Interpretation of Neural Networks is Fragile, AAAI2019, https://arxiv.org/pdf/1710.10547.pdf), Ghorhani et al. claimed that many interpretations on neural networks are fragile because they can “generate adversarial perturbations that produce perceptively indistinguishable inputs that are assigned the same predicted label, yet have very different interpretations.” Here, the concept of “fragile” is the opposite of “robustness”. An interpretation is said to be fragile if ”systematic perturbations can lead to dramatically different interpretations without changing the label”. Otherwise, the interpretation is said to be robust. 

In order to show that our robustness definition in the graph domain aligns well with the robustness concept proposed by Ghorhani et al. AAAI 2019, we elaborate the key idea of robust explanations in graph domain as follows. Given an input graph that is perturbed/modified by random noise, if the GNN's prediction on the perturbed input graph does not change, then the corresponding explanation should not change dramatically. Following this intuition, an explanation is said to be "robust", if it stays stable when an input graph is perturbed by random noise, but the prediction made by the GNN on the perturbed input graph stays unchanged. Otherwise, the explanation is not robust, which corresponds well with the concept of “fragile” proposed by Ghorbani et al. Robust explanations are not sensitive to the artifacts of the input data which makes them more trustworthy than not-robust explanations. 

 We thank the reviewer for valuable comments. Below is our response.

**Q1: First and foremost, in all of the counterfactual explanation techniques, there is usually a component of “minimality” that is missing from this work. In general, one seeks to make a few modifications as possible to the input sample. This aspect is totally absent in this research and, in my opinion, is of fundamental importance in this area. In fact, even in the experimental section, one can see that authors always measure fidelity as a function of sparsity but, typically, one should seek the smallest modification possible and then measure fidelity, sparsity, and size of the recommendation.**

**A1**: As illustrated in Lines 288-292, we applied sparsity constraints in a similar way as [41] to select only a small number of edges in the input graph G as part of our explanation. So, yes, we do have the component of “minimality” in our formulation.

It is true that we measure fidelity at different levels of sparsity in our experiments. This is necessary due to the following reasons. 

First, we need to comprehensively compare the explanation quality of all baseline methods at different levels of sparsity. As shown in Figure 1, the sparsity of explanations largely affects the fidelity performance. Thus, one cannot tell the clear advantage of one method A over another method B, if A outperforms B at one sparsity level, but performs worse than B at another sparsity level. Our evaluation method is also consistent with the evaluation used by [43] where fidelity vs sparsity plots are used for comparison of SubgraphX with the other baselines. As shown in Figure 1, the proposed RCExplainer consistently outperforms the other baseline methods at every level of sparsity. Thus, we can comprehensively observe the advantage of RCExplainer over the baseline methods.

Second, we need to fairly compare with some baseline methods that do not precisely control the sparsity of explanations. Since sparsity largely affects fidelity, it is unfair to compare the fidelity of two explanations A and B with different levels of sparsity. As discussed in lines 698-703 of the paper and mentioned in [43], the baseline methods SubgraphX [43] and PGM-Explainer [34] cannot precisely control the sparsity of explanations. To fairly compare with these methods, we have to compare the fidelity performance of explanations on similar levels of sparsity, which requires to compare the fidelity performance at different sparsity levels. Furthermore, for methods that could control the sparsity of explanations, the sparsity could be optimized to the extreme (e.g., selecting only one edge as explanation). However, for methods that could not control the sparsity of explanations, the sparsity of explanation is determined by data. These methods usually select more than one edge as explanation. If we ask all methods to seek the smallest modification possible (i.e., the largest sparsity), then we are going to compare the fidelity of explanations at different sparsity levels, for example, a method A may only select 1 edge as the explanation, but another method B may select 10 edges as the explanation. A comparison like this is unfair to explanations with higher sparsity level. 



**Q2: The first half of the intuition in 4.1 makes sense, but this formulation also means that you have to use piecewise linear activation functions for this method to work. You want to extract a decision region with 2 properties:
Should be induced by a subset of LDBs in H
Should cover many instances in the training set, and all covered graphs should be predicted as the same classification**

**A2**: It is true that the proposed RCExplainer has to work on GNNs with piecewise linear activation functions. Considering that a large proportion of existing GNN models are adopting piecewise linear activation functions such as ReLU, the proposed RCExplainer should be general enough to apply to a large family of GNNs.  How to model the decision regions of GNNs that do not use piecewise linear activation functions is a very interesting research topic, and we plan to investigate this direction in the near future. There are some studies [1] that use piecewise linear activations to approximate other activations. Therefore, our current work focused on piecewise linear activations may be used to extend to other activations in future.

[1] Hu X, Liu W, Bian J, Pei J. Measuring model complexity of neural networks with curve activation functions. InProceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining 2020 Aug 23 (pp. 1521-1531). 



**Q3: Also, it is not really clear how LDBs are related to GNNs (not really considering that you base your formulation only on GCNs) as it could be used, in principle, in all deep learning mechanisms, right? In fact, I would like the authors to comment on how the method described in 4.2 is related to GNN.**

**A3**: Yes, in principle, the concepts of LDB and decision region are applicable to general deep neural networks. Therefore, the decision region extraction method proposed in Section 4.2 potentially may be extended to other deep learning related tasks. 

Extracting decision region is an algorithmic component of the proposed RCExplainer. The extracted decision regions are intermediate results of RCExplainer, they cannot be directly used as explanations. To generate high-quality explanations on GNNs, we carefully design the explanation algorithm in Section 4.3. Both the decision region and the explanation algorithm are important for RCExplainer to work well on GNNs. 

While extracted decision regions may be a general vehicle to deliver explanations in other tasks, to make decision regions work in other tasks, careful task-related adaptation and extension need to be explored systematically, which are beyond the scope of this paper.



**Q4: I find the end of 4.1 and 4.2 a bit hard to follow, but we see here that it involves sampling from H, which does not seem very robust because then the partitioning of O^d is dependent on which LDBs were sampled from H. The authors even say that r(P,c) induced by LDBs in the \tilde{H} might have instances that aren’t predicted as the correct class. Do the explanations change depending on the sampling of H? If so, this is not robust at all.**

**A4**: Yes, the explanations may change depending on the samples of $\mathcal{H}$. As shown in Figure 7 of the Appendix, the number of sampled LDBs affects the fidelity performance of RCExplainer at all levels of sparsity. However, this does not mean our explanations are not robust. As discussed in lines 771-777 and shown in Figure 7 in the Appendix, the Fidelity performance of RCExplainer converges fast and becomes stable at all levels of sparsity when the number of sampled LDBs increases from 50 to 70. This means that the explanations produced by RCExplainer are robust when enough LDBs are sampled. We have conducted more experiments to test a larger range of the number of sampled LDBs on the other data sets. The results consistently show the fast convergence of the fidelity performance when large enough number of LDBs are sampled. 

We have also conducted qualitative studies to verify this robustness across multiple instances of RCExplainers trained with different sampled LDBs. We are also conducting the robustness experiment under this setting where we compare explanations of two different instances of RCExplainer on same input graphs. We use AUC for this comparison which is computed in similar fashion as described in lines 353-355. Our preliminary results on Mutagenicity and BA-2Motifs show that AUC score exceeds 0.95 on both of the datasets after both the models are trained for 200 epochs. As training progresses further, we see that AUC increases further and approaches 1. This is solid proof of our model's robustness. We will add these results to the paper once the experiments are complete on all of the datasets.

In lines 203-204, we mentioned that “the convex polytope $r(\mathcal{P},c)$ induced by subset of boundaries in $\tilde{\mathcal{H}}$ may contain instances that are not predicted as class $c$.”. Here, an instance not predicted by the GNN as class $c$ does not mean the instance is not predicted as the correct class. It only means the instance is predicted by GNN to be a different class from class $c$. 

According to [4] and illustrated in lines 149-158, every convex polytope $r(\mathcal{P},c)$ induced by the full set of LDBs $\mathcal{H}$ is guaranteed to contain input graphs that are predicted by the GNN as class $c$. However, since $\tilde{\mathcal{H}}$ is a subset of $\mathcal{H}$, we may miss some LDBs in $\mathcal{H}$. Therefore, the $r(\mathcal{P},c)$ induced by $\tilde{\mathcal{H}}$ may miss some LDBs. Thus, the $r(\mathcal{P},c)$ induced by $\tilde{\mathcal{H}}$  may cover a larger area in the feature space $\mathcal{O}^d$, which means it may include some input graphs that are not predicted by the GNN as class $c$. However, this does not affect the robustness of RCExplainer, because most of the input graphs contained in $r(\mathcal{P},c)$ will still be predicted by GNN as class $c$. Thus, $r(\mathcal{P},c)$ is still carrying the GNN’s common decision logic, which is the key to robust explanation.  This is supported by the results in Figure 7 of Appendix, where we show after sampling large enough number of decision boundaries, the performance of model maximizes and saturates. This clearly indicates that such instances even if they might exist do not affect the performance of the model.

**We answer the remaining questions in next comments.**

 **Q5.1: They say existing methods are not robust to noise bc they optimize for individual inputs. They cite PGExplainer here as one of the methods. Still, unless I did not understand correctly, it does not optimize for individual inputs and instead learns a general explanation model applied to individual inputs.**

**A5.1**: If we decompose the loss function used to train PGExplainer, we can see that the basic loss unit of this loss function is to optimize the mutual information between the explanation subgraph and the prediction of the GNN with respect to a single input graph.  The final loss function is simply an expectation of the basic loss unit over all input graphs. Therefore, at the core, PGExplainer is optimizing the mutual information for every single input graph. This is substantially different from our method, because RCExplainer exploits the common decision logic for many similar input graphs and then uses this as a context for training to generate robust explanations. As a result, we can see from Figure 2 that the robustness performance (i.e., measured by AUC) of RCExplainer consistently outperforms that of PGExplainer.


**Q5.2: The comparison to CF-GNNExplainer is not fair, in my opinion. The method they propose is similar to CF-GNNExplainer in the way they build the CF explanation, but it has not been properly acknowledged in the paper. Even the loss functions are very similar, but this is not properly acknowledged, as well. Both methods learn a sparsifier that is then thresholded to find the edges to be removed. Also, I do not quite understand why authors refer to CF-GNNExplorer as an adversarial attack method. From my understanding of the paper, the main goal of CF-GNNExplainer is to build CF explanations and not adversarial samples.**

**A5.2**: We compare CF-GNNExplainer and the proposed RCExplainer because they both focus on producing counterfactual explanations. Our experiments adopt standard evaluation metrics widely used to evaluate the quality of explanations on GNNs [28, 43]. All comparisons are conducted on standard public datasets that are widely adopted to compare the performance of GNN explanations [21, 41, 43]. Therefore, our comparison with CF-GNNExplainer is fair.

Our method is substantially different from CF-GNNExplainer. First, our method relies on decision regions extracted from the decision logic of GNN to make our counterfactual explanations robust. Second, our loss function is also developed based on the extracted decision regions, which is a novel mechanism different from CF-GNNExplainer. Last, when producing explanations, we optimize our loss function collectively on the complete dataset while CF-GNNExplainer optimizes on each sample independently. All of these make our formulation substantially different from the one proposed in CF-GNNExplainer.

As for the sparsity constraint, we clearly stated in line 288 that we follow the idea of [41] to enforce sparsity and discreteness constraints.


To further demonstrate the significant difference between the proposed RCExplainer and CF-GNNExplainer, we have conducted extensive experiments to comprehensively compare the performance of the two methods.

Please refer to the following pdf file for the results (we will add these results in our revised paper) :  

https://drive.google.com/file/d/10C0wXG9Xsqljzef39ZwTgvgi3ea__-MF/view?usp=sharing  

As shown in the experimental results provided above, the AUC performance and Fidelity performance of RCExplainer are consistently better than CF-GNNExplainer on all data sets. These results concretely demonstrate the substantial difference between the two methods.

CF-GNNExplainer produces explanations by only focusing on changing the model’s prediction on the masked input graph. This is similar to the process of generating adversarial attacks. For an explanation generated by CF-GNNExplainer, removing the subgraph induced by the explanation may change the prediction of GNN; however, the GNN’s prediction on the induced explanation subgraph can be different from the GNN’s original prediction on the input graph. This means the explanation generated by CF-GNNExplainer may not justify the original model’s prediction very well, thus it may not be a good explanation to GNN’s predictions. In our method, our loss function consists of a term ($\mathcal{L}\_{same}$) that enforces the resulting explanation to justify the original prediction. The other loss term ($\mathcal{L}\_{opp}$) preserves the counterfactual nature of the explanation. Therefore, explanations generated by out method are likely to justify GNN’s original predictions better. 

We agree that straight-forwardly regarding CF-GNNExplainer as an adversarial attack method is inappropriate. We will revise it and describe above points more clearly in the revised paper.

**Q6: The baselines mentioned in the experiments are not CF explanations at all. This makes the experiment not convincing at all. Also, the methods are for explaining node classifiers and not graph classification models. I would appreciate some thoughts on this point. The only proper baseline method is CF-GNNExplainer, but it is reported in the Appendix, and I would not like to comment on the results there.**

**A6**: We compare our method with the baseline methods because all the methods have the same goal to generate explanations on GNNs. From this perspective, all the methods are highly related and are comparable.
The evaluation metrics used in our paper are general metrics widely used to evaluate the quality of explanations on GNNs. We compare our method with the rest of methods on three aspects: fidelity, robustness, and efficiency. These three aspects reflect general properties desired in all kinds of explanations. They are not specific to counterfactual explanations. Therefore, the evaluation metrics are valid, appropriate, and fair with respect to all compared methods, which makes our experimental results convincing.

We elaborate the evaluation metrics as follows.

Fidelity is an important evaluation metric to verify if the generated explanations are faithful to the underlying GNN model. Fidelity was proposed by [28] which does not focus on counterfactual explanations. Fidelity vs sparsity plots are also used by [43] for comparing SubgraphX, GNNExplainer and PGExplainer. None of these methods focus on counterfactual explanations. 

Similarly, robustness is also desired in all explanation methods as robust explanations tend to not overfit on artifacts in the input graph. If the explanations change dramatically when an input graph is slightly perturbed such that GNN’s prediction doesn’t change, this indicates that the explanation method overfits on some artifacts in input and is not faithful to the behavior of underlying GNN model.  Therefore, robustness is essential for building user’s confidence in the explanation methods.

Efficiency is equally valid for all methods and has been used in [21,43] for such a comparison.

All methods that we use for comparison in the experiments are applicable to graph classification tasks. GNNExplainer, PGExplainer and SubgraphX report results on at least 2 graph classification datasets and PGM-Explainer reports results on 1 graph classification dataset. For graph classification results of GNNExplainer, please refer Figure 4 of [41]. For graph classification results of PGExplainer, please refer Table 2 of [21], and for Subgraphx please refer Figures 3, 4 and 5 of [43]. Graph classification results of PGM-Explainer can be found at Table 1 and Figure 5 of [34]. We use the original implementations of these methods for producing the results. More details are in lines 661-676 of Appendix E. CF-GNNExplainer is the only method that has no default support for graph classification and therefore does not report any results on graph classification datasets (see Conclusion of [20]). Hence, we did not add CF-GNNExplainer as a baseline to the experiments on graph classification datasets. Therefore, our experiments are fair to these baselines.

Since CF-GNNExplainer was originally designed for node classification, graph convolutions were done only on the computation graph of the target node. To extend this to graph classification, graph convolutions need to be done on the entire graph and final embeddings of all nodes should be pooled to predict the class. With these modifications, we have been able to extend CF-GNNExplainer to graph classification and have conducted thorough experiments on CF-GNNExplainer. Below are the detailed results.

https://drive.google.com/file/d/10C0wXG9Xsqljzef39ZwTgvgi3ea__-MF/view?usp=sharing  

Here is the summary of the results:

**Fidelity**:
We find that though CF-GNNExplainer demonstrates better fidelity than rest of the baselines. This is expected as it is designed to generate counterfactual explanations which are expected to demonstrate higher fidelity. However, it performs significantly worse than our method. This is perhaps because it optimizes each sample independently and does not explicitly use common decision logic of GNN that is shared by group of input graphs.

**Robustness**:
We also see that its robustness is substantially worse than our method.  This demonstrates how optimizing each sample independently makes it sensitive to artifacts in input graph, and its explanations can change dramatically after applying slight perturbations to the input graph.

**Efficiency**:
Lastly, its inference runtime is significantly worse than our method and PGExplainer. This is because, like GNNExplainer, it needs to be optimized separately on each input graph and cannot be used in inductive setting.

We will add these results to our paper and make our evaluation code public. 

**We answer the remaining questions in the next comment**.

 We thank the reviewer for valuable comments. Below is our response.

**Q1: the paper mentioned that the explanation align well with human intuition by removing the set of edges. It is less convincing to claim the human intuition here without user studies.**

**A1**: "aligning well with human intuition" means a counterfactual explanation is easy to be comprehended by a human. It has been demonstrated by psychologists that counterfactuals elicit causal reasoning in humans [1, 2, 3] and are thus intuitive. There have also been user studies to confirm that users prefer counterfactual explanations over case-based reasoning [4, 5]. 

Our method is designed to produce counterfactual explanations; thus the produced explanations are expected to be easy to comprehend. We also showed some qualitative results in Table 4 of Appendix: the proposed RCExplainer can accurately identify the target motifs that align well with human understanding. We will describe this more precisely and add more qualitative results to the revised paper.
 


[1] Ruth Byrne. 2008. The Rational Imagination: How People Create Alternatives to Reality. The Behavioral and brain sciences 30 (12 2008), 439–53; discussion 453. https://doi.org/10.1017/S0140525X07002579

[2] Ruth M. J. Byrne. 2019. Counterfactuals in Explainable Artificial Intelligence (XAI): Evidence from Human Reasoning. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI-19. International Joint Conferences on Artificial Intelligence Organization, California, USA, 6276– 6282. https://doi.org/10.24963/ijcai.2019/876

[3] D. Kahneman and D. Miller. 1986. Norm Theory: Comparing Reality to Its Alternatives. Psychological Review 93 (1986), 136–153

[4] Reuben Binns, Max Van Kleek, Michael Veale, Ulrik Lyngs, Jun Zhao, and Nigel Shadbolt. 2018. ’It’s Reducing a Human Being to a Percentage’: Perceptions of Justice in Algorithmic Decisions. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems (CHI ’18). Association for Computing Machinery, New York, NY, USA, 1–14. https://doi.org/10.1145/3173574.3173951

[5] Jonathan Dodge, Q. Vera Liao, Yunfeng Zhang, Rachel K. E. Bellamy, and Casey Dugan. 2019. Explaining Models: An Empirical Study of How Explanations Impact Fairness Judgment. In Proceedings of the 24th International Conference on Intelligent User Interfaces (IUI ’19). Association for Computing Machinery, New York, NY, USA, 275–285. https://doi.org/10.1145/3301275.3302310

**Q2: the extracting decision optimization algorithm has been leveraged by SCSC. I'm curious how to make the overall efficiency much better? More complexity analysis would be more trustworthy.**

**A2**: One technical contribution we made in this paper is to prove that the optimization problem in Equation (4) is a SCSC problem, which is NP-hard according to [5]. The SCSC problem has been well investigated in literature. Since solving the SCSC problem is not the major focus of this paper, we adopt a classical greedy solution proposed by [36]. [36] also establishes in-depth theoretical analysis on the complexity as well as the solution quality of the greedy algorithm. We will clarify this in the revised paper.

We would also like to highlight that LDB extraction is part of training and does not affect inference time of our explanation model.  Moreover, we find empirically that LDB extraction phase takes no more than 2 minutes for each of the three datasets that we use in the experiments. Our training time on each dataset is at par with the training time of PGExplainer. Here is the training time comparison:

||Mutagenicity|BA_2Motifs|NCI1|
|--|----------|----------|----|
|PGExplainer| 30.0s ± 3.5|5.5s ± 1.3|30.7s ± 4.5|
|RCExplainer (Ours)|33.2s ± 2.4|5.8s ± 1.4|29.2s ± 3.6|

The mentioned training time (in seconds) is per epoch and we use 500 epochs in training of both the methods. We will add the training times to the paper. More training details are mentioned in lines 676-688 of Appendix E.


**Q3: In order to better evaluation the explainability, how to quantify the significance of G_{E/S} changes? In [1], it use 'faithfulness' to justify the explainability. I hope similar notion could be adopted as explainability evaluation.**

**A3**: The ‘fidelity’ [28,43] metric used in our paper is a standard evaluation metric to measure the faithfulness of the explanations on graph predictions. Fidelity is defined as the drop in the confidence of a model’s original predictions after masking the explanation in the input graph. This can be written as $\phi(G\_{E}) – \phi(G\_{E/S})$. As we can see that this metric clearly emphasizes the significance of generated explanation in the original prediction. Smaller the value of prediction confidence $\phi(G\_{E/S})$  on subgraph $G\_{E/S}$, larger is the fidelity, which further indicates higher faithfulness. 

The definition of fidelity also aligns well with the definition of “comprehensiveness” which is a faithfulness metric defined in [1]. The comprehensiveness is defined as change in the prediction score after removing the rational from the input. Therefore, comprehensiveness emphasizes the role of identified rational in the prediction and ensures all the input features important for the prediction are selected as part of the rational. In other words, the rational is comprehensive.

[1] ERASER : A Benchmark to Evaluate Rationalized NLP Models
 

 We thank the reviewer for valuable comments. Below is our response.

**Q1: Definition of counterfactual explanation: According to the previous, the definition of counterfactual explanation is “if X had been different, Y would not have occurred”, rather than “if X had not occurred, Y would not have occurred” as claimed by the authors. That means, the authors only replace the subgraph (X) with the empty set, rather than the reference subgraph (that could be the empty or other subgraphs). Moreover, recent studies find that simply replacing the features of interest (X) with the empty could cause the distribution shift from the complete graphs. Please explain or clarify these points.
Explaining image classiﬁers by counterfactual generation. In ICLR, 2019.**

**A1**: There are several versions of counter-factual explanations in literature. We are using the definition of counter-facutal explanation given in the 4th paragraph of Section 6.1 of the book "Interpretable Machine Learning: A Guide for Making Black Box Models Explainable" written by Christoph Molnar (https://christophm.github.io/interpretable-ml-book/). In this book, a counterfactual explanation is defined as: "A counterfactual explanation of a prediction describes the smallest change to the feature values that changes the prediction to a predefined output." As introduced in Lines 309-311, the proposed RCExplainer uses a set of edges of the input graph as explanations. Since an edge of the input graph only have two status: exist and not-exist, changing the status of an existing edge is equivalent to removing it. Therefore, if we treat X as a set of edges that explains the prediction of a GNN, then following the definition proposed by Christoph Molnar, changing the status of X means removing the edges contained in X from the input graph. 

When generating X, we strictly follow the definition proposed by Christoph Molnar to require the set of edges to be the minimum. This is achieved by applying the sparsity constraints illustrated in Lines 288-292. We will clarify this more clearly in the revised paper.

The studies in the ICLR 2019 paper is very interesting. The authors agree that masking interpreted image area by black/empty pixels may cause distribution shift of images. However, there has been no similar study on graphs yet. The latest counterfactual explanation method on graph neural networks proposed in [20] generate explanations by removing edges of input graphs. Therefore, the proposed RCExplainer follows this routine to generate explanations as a set of edges of the input graph. Recall that RCExplainer applies a sparsity constraint in lines 288-292 to reduce the number of edges forming an explanation, and removing a small number of edges from an input graph is less likely to cause a large distribution drift.



**Q2: Definition of noise: I cannot find a clear definition of noise in this paper. I think it’s necessary to clarify it when you are discussing robustness. For example, what does 30% noise mean? Adding random noise whose magnitude is 30% of the original node feature or randomly adding or deleting 30% edges or do them both?**

**A2**: We define x% of noise as follows:
We randomly perturb features of x% nodes to flip their labels, and also simultaneously replace x% of edges randomly in the graph. This replacement of edges is done by randomly removing x% of the existing edges and adding same number of new edges randomly in the graph.  This is briefly mentioned in lines 359-360.

Please note that adding noise to the input graph can change the GNN’s prediction. If the prediction changes, the explanation cannot be expected to remain consistent for the perturbed input graph. Therefore while computing robustness, only those perturbations are considered valid that do not change the GNN’s original prediction. The perturbations that flip the GNN’s prediction are ignored and not considered for the evaluation. 

We will explain this more clearly in our revised paper.



**Q3: Robustness in GNN explanation field: It will be much better if the authors can explain more why the robustness defined in this paper is important. In CV domain, after adding noise to the panda picture, the NN cannot classify it correctly anymore while human being can’t see any difference between the original pic and the noisy one. However, taking mutagenicity for example, adding or deleting 30% of edges will make the original molecule a totally different one even though the GNN doesn’t change the prediction (noxious or not). In this case, it’s reasonable for an explainer to identify a different subgraph. So, I don’t think evaluating the robustness based on the similarity between S’ and S makes perfect sense to me.**

**A3**: The robustness of DNN explanations in CV domain has a similar meaning as the robustness of explanations in graph domain. According to Ghorbani et al. (Interpretation of Neural Networks is Fragile, AAAI2019, https://arxiv.org/pdf/1710.10547.pdf), Ghorhani et al. claimed that many interpretations on neural networks are fragile because they can “generate adversarial perturbations that produce perceptively indistinguishable inputs that are assigned the same predicted label, yet have very different interpretations.” Here, the concept of “fragile” is the opposite of “robustness”. An interpretation is said to be fragile if “systematic perturbations can lead to dramatically different interpretations without changing the label”. Otherwise, the interpretation is said to be robust. It is worth mentioning that Ghorbani et al. are using the intersections and rank order correlation between interpretations on images to evaluate the fragile level of interpretations (in Figure 3 of the paper). This aligns quite well with evaluating the robustness of explanations by the similarity of explanations S and S’.

To show that our robustness definition in the graph domain aligns well with the robustness concept proposed by Ghorhani et al. AAAI 2019, we elaborate the key idea of robust explanations in graph domain as follows. Given an input graph that is perturbed/modified by random noise, if the GNN's prediction on the perturbed input graph does not change, then the corresponding explanation should not change dramatically. Following this intuition, an explanation is said to be "robust", if it stays stable when an input graph is perturbed by random noise, but the prediction made by the GNN on the perturbed input graph stays unchanged. Otherwise, the explanation is not robust, which corresponds well with the concept of “fragile” proposed by Ghorbani et al. Robust explanations are not sensitive to the artifacts of the input data which makes them more trustworthy than not-robust explanations. 


Now we discuss the example of mutagenicity raised by the reviewer. If deleting 30% of edges of an original molecule but the GNN still predicts the same label, then it is highly possible that that graph structure that dominates the GNN’s prediction is still dominating the prediction. Since an explanation should focus on the dominating graph structures, it should not change dramatically. 

In conclusion, it is meaningful to evaluate the robustness of explanations by the similarity between S’ and S.


**Q4: The ablation study of the counterfactual term. The effectiveness of the counterfactual part needs to be demonstrated by setting λ=1or deleting the counterfactual term L_opp in eq(8).**

**A4**: We present such an ablation study on BA-2Motifs in lines 778-783 of Appendix G. As shown in Figure 8, weighing the counterfactual term $\mathcal{L}\_{opp}$ less in the net loss by setting $\lambda$ close to 1 causes the fidelity to drop significantly across all levels of sparsity.  
We see similar trend for the other datasets and the results can be seen at the below link where we show the results of ablation study for $\lambda$ on multiple datasets .

https://drive.google.com/file/d/1iQ_OYK6FHZ7tBYlC39_3-nIT_Ud47U_s/view?usp=sharing

We also see that when $\lambda=0.2$ and both of the loss terms: $\mathcal{L}\_{same}$ and $\mathcal{L}\_{opp}$  are active,  the common decision logic of GNN is effectively leveraged by RCExplainer to produce superior robustness performance.

We will include these results in the revised paper.


The paper proposed a method for robust counterfactual explanation on GNN which leverages an extraction decision boundaries shared by multiple samples. It is an interesting framework and the experimental results justified the proposed method.  the topic of robust counterfactual explanations in GNN is certainly of high interest for the community and it has been, yet, poorly explored.
As such the work as very high potential impact. The proposed approach tackles specifically the issue of providing subgraph explanation with robustness. The proposed method is neat and the paper is easy to follow. However, I have a few concerns.

1, the paper mentioned that the explanation align well with human intuition by removing the set of edges. It is less convincing to claim the human intuition here without user studies.

2, the extracting decision optimization algorithm has been leveraged by SCSC. I'm curious how to make the overall efficiency much better? More complexity analysis would be more trustworthy.

3, In order to better evaluation the explainability, how to quantify the significance of G_{E/S} changes? In [1], it use 'faithfulness' to justify the explainability. I hope similar notion could be adopted as explainability evaluation.

[1] ERASER : A Benchmark to Evaluate Rationalized NLP Models

 n/a

The paper presents an approach to GNN prediction explanations based on providing modifications to the input samples to change the prediction outcomes. The idea is that those modifications are the ones that can make a human understand the reason for a prediction. For example, a post is considered misinformation due to its connections to other misinformation posts. Those connections, i.e., edges, constitute the explanation for marking a post as misinformation.
The idea presented in the paper consists of distilling the model learned by the GNN into a set of polytopes that bound the decision regions. This aims to make the explanation model more robust to “noise” (intended as random connections in the input graphs).
The loss function presented in the paper is then used to modify the input graph to move the sample away from the original region to a different region.
The experimental results are presented in graph classification, but the method can be extended to node classification.  The paper, in my opinion, has some important and strong aspects that make it interesting for the community.
In particular, I believe that the following are its main strengths:
* The method is general and can be used for the three tasks usually studied in graph learning
* Once the boundaries are found, the method consists of moving a sample from one boundary to another to change the prediction outcome. This makes the method efficient at explanation time
* The experiments are conducted on datasets that are standard for this task.

The paper, and the research presented in the paper, suffer from the following limitations:
* First and foremost, in all of the counterfactual explanation techniques, there is usually a component of “minimality” that is missing from this work. In general, one seeks to make a few modifications as possible to the input sample. This aspect is totally absent in this research and, in my opinion, is of fundamental importance in this area. In fact, even in the experimental section, one can see that authors always measure fidelity as a function of sparsity but, typically, one should seek the smallest modification possible and then measure fidelity, sparsity, and size of the recommendation.
* The first half of the intuition in 4.1 makes sense, but this formulation also means that you have to use piecewise linear activation functions for this method to work. You want to extract a decision region with 2 properties:
   * Should be induced by a subset of LDBs in H
   * Should cover many instances in the training set, and all covered graphs should be predicted as the same classification
* Also, it is not really clear how LDBs are related to GNNs (not really considering that you base your formulation only on GCNs) as it could be used, in principle, in all deep learning mechanisms, right? In fact, I would like the authors to comment on how the method described in 4.2 is related to GNN.
* I find the end of 4.1 and 4.2 a bit hard to follow, but we see here that it involves sampling from H, which does not seem very robust because then the partitioning of O^d is dependent on which LDBs were sampled from H. The authors even say that r(P,c) induced by LDBs in the \tilde{H} might have instances that aren’t predicted as the correct class. Do the explanations change depending on the sampling of H? If so, this is not robust at all.
* Comparison to existing methods is weak:
   * They say existing methods are not robust to noise bc they optimize for individual inputs. They cite PGExplainer here as one of the methods. Still, unless I did not understand correctly, it does not optimize for individual inputs and instead learns a general explanation model applied to individual inputs.
   * The comparison to CF-GNNExplainer is not fair, in my opinion. The method they propose is similar to CF-GNNExplainer in the way they build the CF explanation, but it has not been properly acknowledged in the paper. Even the loss functions are very similar, but this is not properly acknowledged, as well. Both methods learn a sparsifier that is then thresholded to find the edges to be removed.
      * Also, I do not quite understand why authors refer to CF-GNNExplorer as an adversarial attack method. From my understanding of the paper, the main goal of CF-GNNExplainer is to build CF explanations and not adversarial samples.
* The baselines mentioned in the experiments are not CF explanations at all. This makes the experiment not convincing at all. Also, the methods are for explaining node classifiers and not graph classification models. I would appreciate some thoughts on this point.
   * The only proper baseline method is CF-GNNExplainer, but it is reported in the Appendix, and I would not like to comment on the results there.
* There are some unjustified claims in the paper:
   * Lines 7-8: there is no experimental support for this claim.
   * Lines 14-15: Authors do not do any human experiments, so they cannot claim that their method aligns well with human intuition.
   * Line 114: the sentence “directly optimize for explainability of an explanation” means nothing. How do you define “explainability”?
   * Lines 126-131: the definition is not a proper definition. What is “slight,” how do you quantify a “slight change”? How do you define “stable”?
   * Line 230: I’ve found no support for this claim at all anywhere.
   * Line 302-303: have you tried to reuse the network without retraining? Does it work as claimed in the paper?
* The claim that the method is robust to noise is weakly supported by the experiments, in my opinion. For instance, in Mutagenicity, PGExplainer is close (although the error bars are separate) to the other two methods.
   * Also, the robustness feature is poorly motivated in the paper. Why is it essential to be robust? If you don’t mention proper motivations, the problem definition feels rather artificial.

Minor remarks:
* Requirements require → Can you change this? It sounds a little bit odd :)
* The sentence at lines 143-144 is broken; I cannot parse it
* Line 147. Does FC actually mean FC + nonlinearity?
* Can you improve the definition of what the set P is? It is not really clear. Is it a set of convex polytopes, or is it a set of LBDs?
* Please define the positive and negative side at Line 273.
* What is the * operator in Eq. (9)?
* In references [1], [15], and [46] the year is missing
 Limitations are not properly addressed in Section 6.

This paper focuses on the counterfactual explanations of GNNs, and proposes a method to identify a subgraph of an input graph whose absence could cause the change of prediction. In particular, the proposed method starts from extracting the decision regions over similar graph instances: the subsets of linear decision boundaries (LDBs), which is later used to formulate the counterfactual loss function. Then a NN is used to predict the probability of each edge. Furthermore, a counterfactual term is included in the loss function to ensure the counterfactual property of explanations. Experiments are done on several datasets to illustrate the proposed model.  Originality: Though some concepts like LDB and counterfactual have been used in other domains, to my knowledge, the framework proposed by this paper is original in the field of GNN explanation. However, the definition of counterfactual explanation seems broken. Please refer to the tech comments below.

Quality: This paper is well-written and the authors make their idea and the workflow clear. However, the reviewer thinks more experiments are needed to demonstrate the effectiveness of the proposed method. In addition, the definition of noise in this paper is not very clear, without which robustness can’t be clarified as well. Details will be described in following tech comments.

Clarity: the structure of this paper is well organized. It starts with the intuition and then get into the tech details. Each module(e.g. LDBs, counterfactual subgraph) is introduced following the pipeline of the proposed method, which makes the paper structure quite clear. 

Significance: The experiments are conducted on three small datasets and consider several relevant baselines for GNN explanation. The results are significantly better than the baseline approaches on criteria fidelity and AUC save for the AUC performance on Mutagenicity. 

Tech Comments: 

1. Definition of counterfactual explanation: According to the previous, the definition of counterfactual explanation is “if X had been different, Y would not have occurred”, rather than “if X had not occurred, Y would not have occurred” as claimed by the authors. That means, the authors only replace the subgraph (X) with the empty set, rather than the reference subgraph (that could be the empty or other subgraphs). Moreover, recent studies find that simply replacing the features of interest (X) with the empty could cause the distribution shift from the complete graphs. Please explain or clarify these points.
    - Explaining image classiﬁers by counterfactual generation. In ICLR, 2019.

2. Definition of noise: I cannot find a clear definition of noise in this paper. I think it’s necessary to clarify it when you are discussing robustness. For example, what does 30% noise mean? Adding random noise whose magnitude is 30% of the original node feature or randomly adding or deleting 30% edges or do them both?

3. Robustness in GNN explanation field: It will be much better if the authors can explain more why the robustness defined in this paper is important. In CV domain, after adding noise to the panda picture, the NN cannot classify it correctly anymore while human being can’t see any difference between the original pic and the noisy one. However, taking mutagenicity for example, adding or deleting 30% of edges will make the original molecule a totally different one even though the GNN doesn’t change the prediction(noxious or not). In these case, it’s reasonable for an explainer to identify a different subgraph. So, I don’t think evaluating the robustness based on the similarity between S’ and S makes perfect sense to me.

4. The ablation study of the counterfactual term. The effectiveness of the counterfactual part needs to be demonstrated by setting λ=1or deleting the counterfactual term L_opp in eq(8). Please refer to the main review.

Meta-review: This paper presents a new method to produce robust counterfactual explanations for the predictions of GNNs. However, reviewers raised several concerns about the method, the experiments, and the writing. Therefore, the paper cannot be accepted in the current form.

Decision: accept