From GNNs to Trees: Multi-Granular Interpretability for Graph Neural Networks

[Q2] Ablation Study for Perturbation Module.

Thanks for the valuable comment. We have additionally designed two model variants for the Perturbation Module to conduct ablation experiments. Experiments were conducted across various datasets, with results for classification accuracy and interpretability accuracy presented in Figure 16 (page 22). Moreover, in the initial manuscript, we have also provided a similar experiment (Figure 7, Page 10 and Figure 15, Page 22). The experimental results indicate that TIF outperforms the other two variants in both classification and interpretability tasks. This suggests that TIF’s perturbation structure effectively captures and learns information that benefits both classification tasks and interoperability. These additional experiments have been updated in Figure 16 (page 22) of the revised manuscript.

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Ref:

[1] Yuwen Wang, Shunyu Liu, Tongya Zheng, Kaixuan Chen, and Mingli Song. Unveiling Global Interactive Patterns across Graphs: Towards Interpretable Graph Neural Networks. In SIGKDD (ACM Knowledge Discovery and Data Mining), pages 3277–3288, 2024.

We are glad that the reviewer appreciates our work as a useful contribution to the community. We have carefully revised the manuscript according to your constructive suggestions. Below we address the main points raised in the review.

[W1] & [Q1] (1) Definition of Interpretation in This Work, (2) the Reason Why the Coarsened Graph Can Serve as an Interpretation, (3) an Example of the TIF Model.

Thanks for the valuable comment. (1) In our work, the interpretability is defined from a multi-granularity perspective, including tree nodes with coarsened graphs of varying granularities, and the selected root-to-leaf path.

(2) The coarsened graphs generated by our model can illustrate and reveal the structural features of the corresponding original graph data at a given granularity [1] (as shown in Figures 4-5, 8-11 of the coarsened graph). This provides structural explanations and learning guidance specific to the chosen granularity.

(3) We now additionally provide an example in Appendix B.1 (Page 16,18, Figure 8), which includes the input graph, the root-to-leaf path, the coarsened graphs of each layer, and the final prediction. Moreover, in the initial manuscript, we have also provided a similar example. Figure 4 (Page 10) demonstrates that the coarsened graphs deliver effective interpretability.

We observe that TIF effectively captures both local substructures in finer explanations and global graph patterns in coarser explanations, ensuring that key features at different granularities are preserved. These clarifications and additional experiments have been updated in Appendix B.1 (Page 16,18, Figure 8) of the revised manuscript.

[W2] (1) The Experimental Metric That Reflects Sparsity, (2) Original Graph Is Meaningless to Practical Applications.

Sorry for the confusion. (1) In our ablation studies of the initial manuscript (Page 9-10, 20-22), we have introduced the compression ratio as a sparsity metric to reflect the sparsity of the generated explanations. (2) The original graph is not included in the scope of our explanations. Instead, our generated explanations are based on coarsened graphs of different granularities (Page 2, Figure 1 and line 72).

[W3] Definition of (1) Path Importance and (2) Ratio q.

(1) Sorry for the misunderstanding. Path Importance is defined as a measure to determine the significance of each path by analyzing its utilization frequency across a sample set and quantifying it using normalized entropy. This metric helps identify potential issues of imbalance, where certain paths are either excessively or insufficiently utilized. In the initial manuscript, we have provided the definition of Path Importance in Section 4.1.3 METRICS on Page 7, line 374.

(2) Thanks for the valuable comment. The ratio q is defined as the measure of compression applied to the graph structure and node features between adjacent layers of the model during pooling. These clarifications have been updated on Page 9, line 482 of the revised manuscript.

[W4] Detailed Explanation of Figures 4 and 5.

Thanks for the valuable comment. (1) The standard for explaining quality is the ability to accurately capture the important features and structural information at each granularity level. Different colors represent structural information learned or captured from the previous level of granularity. (2) For example, in Figure 4, the model selects a finer graph with three-ring structures and one chain structure, which corresponds to the structure of the original graph. Then, a coarser graph with three branches is selected, which also corresponds to the previously chosen finer graph's structure. (3) The comparison in Figure 5 mainly serves to demonstrate that TIF can provide multi-granularity explanations, whereas models like GIP cannot. These models can only recognize single structural information, such as rings or non-rings. Different colors represent the structural information learned or captured from the previous level of granularity. Therefore, models like GIP only provide a template based on the entire graph, so the generated explanation for GIP is depicted in gray. These additional clarifications have been updated on Page 19, Appendix B.3.

[W5] Notations in Figure 7.

Sorry for the confusion. We have completed the missing symbol definitions and added a description of a specific model variant in Figure 7, ensuring that Figure 7 aligns with the corresponding discussion. These clarifications have been updated in Figure 7 (Page 10) of the revised manuscript.

Dear reviewer,

We are glad that the reviewer appreciates our attempt, and sincerely thank the reviewer for the constructive comments. As suggested, we have additionally added some clarifications about our framework and included further experiments, such as an example of the TIF model and an ablation study for the perturbation module. Please let us know if you have other questions or comments.

Since half of the allocated time for reviewer-author discussion has already elapsed, we sincerely look forward to your reevaluation of our work and would very appreciate it if you could raise your score to boost our chance of more exposure to the community. Thank you very much!

Best regards,

The authors of TIF

[Q1] Unclear Representation in Fig 1.

Thanks for the valuable comment. We have added the correspondence between the different granularity levels of proteins and the "finer graph," "moderate graph," and "coarser graph" in Figure 1. These interpretability terms of multi-granularity are inspired by the distinct granularity between enzyme and non-enzyme, ranging from functional groups via amino acids to protein molecular level [14-15]. In this context, the "finer graph" refers to the explanation at the functional group level. The "moderate graph" represents the explanation at the amino acid level. The "coarser graph" represents the explanation at the protein molecular level. We previously mentioned this example in the abstract and introduction of the initial manuscript. These clarifications have been updated in Figure 1 (Page 2) of the revised manuscript.

[Q2] Interpretation of Eq (5).

Sorry for the confusion.

(1) $\hat{A} = A + I$ is the adjacency matrix of the original graph with self-loops. $A^{(l+1)}$ represents the coarser graph's adjacency matrix at layer $l+1$.

(2) In Eq. (5), the adjacency matrix $\hat{{A}} = {A} + {I}$ corresponds to the first layer of the graph, where it has the same dimensions $n \times n$ as the original adjacency matrix ${A}$. As the graph undergoes coarsening in subsequent layers, the number of nodes decreases, and the dimensions of the adjacency matrix are reduced accordingly.

(3) Eq. (5) represents a link prediction loss, which is used to ensure that the connectivity of the graph is maintained during the graph aggregation and compression process. The purpose of this loss function is to minimize the difference between the original adjacency matrix and the aggregated adjacency matrix, thus ensuring that the structure and connectivity of the graph are preserved as much as possible during aggregation.

[Q3] Theoretical Justification of Explanation Accuracy.

Thanks. In our initial manuscript, we followed this used in existing works [1-9], which is a widely recognized practice within the community and has been broadly adopted. To clarify, the explanation accuracy metric includes a prediction consistency formula [13]:

$\text{ACC}_{\text{exp}} = \frac{|f(G) = f(G_s)|}{|T|}$

Here, $f(G)$ and $f(G_s)$ represent the predictions on the original graph $G$ and explanation subgraph $G_s$, respectively. The metric evaluates whether the predictions match, relying on the softmax output to determine the predicted class. Since softmax provides the probability distribution, the highest-probability class is often interpreted as the model's most confident prediction. Thus, existing works [1-9] consistently treat the GNN’s softmax output as a confidence measure for experiments involving explanation accuracy.

[Q4] Graph Kernels' Capability.

Sorry for the confusion. Graph kernels are primarily used in the Consistency metric [9]. The random walk graph kernel measures the similarity between two graphs by counting the number of shared paths, leveraging the path-sharing hypothesis to effectively capture both local and global structural features of graphs. Its theoretical foundation includes the construction of direct product graphs and the mapping properties of kernel methods, making it suitable for assessing the similarity between graphs and patterns in global interaction modeling [4, 9, 16]. The use of graph kernels and the Consistency metric in interpretability research is widely recognized and accepted.

[Q5] Explanation Experiments on the Mutag Dataset.

Thanks. In the original manuscript, we have conducted explanation experiments on the MUTAG dataset. On this dataset, we performed a series of interpretability experiments, including Explanation Accuracy, Consistency, Path Consistency, and Path Importance. The results are presented in Table 2, Figure 3 (b), and Figure 3 (c). According to the experimental results, compared to all other models listed in the experiment, our model achieves the best performance on the MUTAG dataset in terms of classification accuracy, path consistency, and path importance. Additionally, it achieves the second-best F1 score and explanation accuracy. This demonstrates that our model performs exceptionally well in both classification and explanation tasks on the MUTAG dataset.

[W2] Effectiveness of Evaluation Metrics.

Thanks. In our initial manuscript, we have provided all widely recognized and commonly used evaluation metrics in existing interpretability research, including explanation accuracy, consistency, and visual comparisons. These evaluation metrics strictly adhere to the standards established in existing works [1-9]. Additionally, considering the tree structure of our model, we have designed several tailored evaluation metrics, such as Path Consistency and Path Importance.

[W3] & [Q6] Lack of Visual Comparisons Experiments.

Thanks. (1) In the initial manuscript, we have already provided visualization comparison experiments between TIF and GIP (Page 9-10, Figure 4).

(2) As suggested, we have additionally included visualization comparison experiments between TIF and the PortGNN model, as well as the GAST model. The results show that compared to GIP, GSAT, and ProtGNN, TIF's capability to span from fine-grained local interactions to coarse-grained global structures provides a more transparent and interpretable decision-making process, elucidating how various levels of graph information contribute to final model predictions. These additional results have been updated in Figure 11 (Page 19) of the revised manuscript.

[W4] Running Efficiency.

Thanks for the valuable comment. As suggested, we have additionally analyzed the theoretical complexity and compared the time consumption of our proposed framework with several interpretable baselines.

(1) We calculated the theoretical complexity of the proposed TIF framework. For time complexity, updating node embeddings requires $O(N \cdot d^2)$ for dense graphs and $O(E \cdot d)$ for sparse graphs, while clustering and perturbation modules add $O(M \cdot N \cdot d \cdot c)$. Overall, the time complexity is $O(N \cdot d^2 + M \cdot N \cdot d \cdot c)$ (dense graphs) or $O(E \cdot d + M \cdot N \cdot d \cdot c)$ (sparse graphs). For space complexity, dense graphs require $O(N^2)$ for adjacency storage, while sparse graphs need $O(E)$. Adding node embeddings, perturbations, and routing parameters results in $O(N^2 + N \cdot d \cdot (1 + M \cdot c))$ for dense graphs or $O(E + N \cdot d \cdot (1 + M \cdot c))$ for sparse graphs. These clarifications have been updated in Appendix D.2 (Pages 22-23) of the revised manuscript.

(2) We present the time required to complete the training of each interpretable model. The results of the time consumption in Table 13 show that our method is only slightly less efficient than KerGNN and GIP. Given that our model outperforms KerGNN and GIP in terms of prediction and explanation performance on the vast majority of datasets, we believe that this slight additional time cost is justified. These additional results have been updated in Appendix D.2 (Pages 22-23, Table 13) of the revised manuscript, as follows:

Table 13 (Page 23): Time consumption of different methods. The table shows the time required (in seconds) to finish training for each interpretable model on various datasets. “*” indicates the method requires additional pre-training process which takes nearly 72 hours.

[W5] Theoretical Basis of Proposed Algorithms.

Sorry for the confusion. Existing interpretability methods primarily stem from two aspects: human-cognitive logic interpretability [1-9, 12] and mathematical interpretability [10-11].

In this work, we primarily follow the paradigm of human-cognitive logic interpretability. We observe that existing methods often adopt an inflexible approach that captures graph connectivity at a single specific level, whereas real-world graph tasks typically involve relationships at multiple granularities (e.g., relevant interactions in proteins range from functional groups to amino acids and even to protein domains [14-15]). To address this limitation, we propose TIF, which has been extensively validated through a series of quantitative and qualitative experiments designed in strict adherence to the standards established in existing works [1-9], demonstrating the effectiveness of our approach.

We greatly appreciate the reviewer for the insightful comments, which helped improve the quality of the paper significantly. We have carefully revised the manuscript according to your valuable suggestions. Below we address the main points raised in the review.

[W1] Various Subscripts Making It Difficult to Follow.

Sorry for the confusion. We have reorganized the formula expressions throughout the paper. Subscripts are consistently used to represent the internal logic of matrices and classification tasks, while superscripts are used to indicate the logic related to the tree model structure. Moreover, we provide a series of symbol tables to further elaborate on the specific meanings of each subscript. These clarifications have been updated in Table 5-11 (Page 16-17) of the revised manuscript, as follows:

Table 5: Node-related Symbols.

Table 6: Feature and Weight-related Symbols.

Table 7: Graph Structure-related Symbols.

Table 8: Clustering-related Symbols.

Table 9: Loss and Regularization-related Symbols.

Table 10: Path and Routing-related Symbols.

Table 11: Parameters and Hyperparameters.

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Ref:

[1] Zhitao Ying, Dylan Bourgeois, Jiaxuan You, Marinka Zitnik, and Jure Leskovec. GNNExplainer: Generating explanations for graph neural networks. In NeurIPS (Advances in Neural Information Processing Systems), 2019.

[2] Hao Yuan, Haiyang Yu, Jie Wang, Kang Li, and Shuiwang Ji. On explainability of graph neural networks via subgraph explorations. In ICML (International Conference on Machine Learning), 2021.

[3] Hao Yuan, Jiliang Tang, Xia Hu, and Shuiwang Ji. XGNN: Towards model-level explanations of graph neural networks. In SIGKDD (ACM Knowledge Discovery and Data Mining), 2020.

[4] Aosong Feng, Chenyu You, Shiqiang Wang, and Leandros Tassiulas. KerGNNs: Interpretable graph neural networks with graph kernels. In Proceedings of the AAAI Conference on Artificial Intelligence, 2022.

[5] Jun Yin, Chaozhuo Li, Hao Yan, Jianxun Lian, and Senzhang Wang. Train Once and Explain Everywhere: Pre-training Interpretable Graph Neural Networks. In NeurIPS (Advances in Neural Information Processing Systems), 2023.

[6] Junchi Yu, Tingyang Xu, Yu Rong, Yatao Bian, Junzhou Huang, and Ran He. Graph Information Bottleneck for Subgraph Recognition. In ICLR (International Conference on Learning Representations), 2020.

[7] Siqi Miao, Mia Liu, and Pan Li. Interpretable and generalizable graph learning via stochastic attention mechanism. In ICML (International Conference on Machine Learning), 2022.

[8] Yongduo Sui, Xiang Wang, Jiancan Wu, Min Lin, Xiangnan He, and Tat-Seng Chua. Causal attention for interpretable and generalizable graph classification. In SIGKDD (ACM Knowledge Discovery and Data Mining), 2022.

[9] Yuwen Wang, Shunyu Liu, Tongya Zheng, Kaixuan Chen, and Mingli Song. Unveiling Global Interactive Patterns across Graphs: Towards Interpretable Graph Neural Networks. In SIGKDD (ACM Knowledge Discovery and Data Mining), pages 3277–3288, 2024.

[10] Sahil Verma, Varich Boonsanong, Minh Hoang, Keegan Hines, John Dickerson, and Chirag Shah. Counterfactual explanations and algorithmic recourses for machine learning: A review. In ACM Computing Surveys, volume 56, number 12, pages 1–42, 2024. ACM New York, NY.

[11] Rui Wang, Xiaoqian Wang, and David I. Inouye. Shapley Explanation Networks. In International Conference on Learning Representations, 2021.

[12] Hao Yuan, Haiyang Yu, Jie Wang, Kang Li, and Shuiwang Ji. On explainability of graph neural networks via subgraph explorations. In ICML (International Conference on Machine Learning), 2021.

[13] Yiqiao Li, Jianlong Zhou, Sunny Verma, and Fang Chen. A survey of explainable graph neural networks: Taxonomy and evaluation metrics. arXiv preprint arXiv:2207.12599, 2022.

[14] Hao Hu, Wei Hu, An-Di Guo, Linhui Zhai, Song Ma, Hui-Jun Nie, Bin-Shan Zhou, Tianxian Liu, Xinglong Jia, Xing Liu, et al. Spatiotemporal and direct capturing global substrates of lysine-modifying enzymes in living cells. Nature Communications, 15(1):1465, 2024.

[15] Yansheng Zhai, Xinyu Zhang, Zijing Chen, Dingyuan Yan, Lin Zhu, Zhe Zhang, Xianghe Wang, Kailu Tian, Yan Huang, Xi Yang, et al. Global profiling of functional histidines in live cells using small-molecule photosensitizer and chemical probe relay labelling. Nature Chemistry, pages 1–12, 2024.

[16] SVN Vishwanathan, Karsten M. Borgwardt, Nicol N. Schraudolph, and others. Fast computation of graph kernels. In NIPS (Advances in Neural Information Processing Systems), volume 19, pages 131–138, 2006.

Dear reviewer,

We are glad that the reviewer appreciates our attempt, and sincerely thank the reviewer for the constructive comments. As suggested, we have additionally added some clarifications about our framework and included further experiments, such as visual comparisons experiments and running efficiency. Please let us know if you have other questions or comments.

Since half of the allocated time for reviewer-author discussion has already elapsed, we sincerely look forward to your reevaluation of our work and would very appreciate it if you could raise your score to boost our chance of more exposure to the community. Thank you very much!

Best regards,

The authors of TIF

We are deeply honored to receive your response and your recognition of our work, and we sincerely appreciate your thoughtful feedback.

[Q1] Correlation of Coarsen Graph to the Ground-truth

Reviewer hGDZ has acknowledged our viewpoints and analysis on this aspect and has provided us with positive feedback and a higher score. For further details, we kindly invite you to refer to our discussion with Reviewer hGDZ. Below, we will take Figure 8 as an example and provide a detailed analysis of the entire process to elaborate correlation between the coarsened graph at each layer and the ground-truth interpretation. Additionally, We have added a visual structure of the ground truth used at each granularity level in the MultipleCycle dataset. These additional clarifications have been updated on Pages 18-20, Figure 8.

• Firstly, the MultipleCycle dataset corresponds to different ground truths at different levels of granularity. As shown in Figure 8, the input graph is a sample from the MultipleCycle dataset, and its category is classified as "Hybrid Cycle." Specifically:
  • Its first-level structure is set as a cycle structure based on the ground truth at this granularity level, which determines its cycle attribute.
  • Its second-level structure is built on the first-level structure, configured as a mixed combination of cycle and non-cycle structures according to the ground truth at this granularity level. The clockwise sequence is cycle, non-cycle, cycle, and cycle, which determines its mixed attribute.
• Secondly, we can observe that the finer graphs display the second-level structure of the input graph. The model selects the third finer graph, which best reflects the structural information of the input graph (shows a second-level structure proceeding clockwise as cycle, non-cycle, cycle, and cycle). From the perspective of ground truth, the model selects the finer graph that is closest to the ground truth structure and layout of the input graph at this granularity level.
• Subsequently, we can observe that the coarser graphs clearly capture the first-level structure of the input graph, which is the cycle structure. The model selects the fourth coarser graph (corresponding to the second-level structure depicted in the finer graph from the previous layer), which best represents the structural information of the input graph, as the root node of the TIF. From the perspective of ground truth, the model selects the finer graph that is closest to the ground truth structure and layout of the input graph at this granularity level, while also most accurately preserving the ground truth structural information from the previous granularity level.
• Finally, at the root node of the TIF, the prediction is performed, and the model successfully identifies the category of the data as "Hybrid Cycle." From an interpretability perspective, the TIF tree model effectively captures and explains the key attributes of the MultipleCycle dataset at two distinct granularity levels.
  • The second-level granularity, which characterizes the attributes of being purely cycle, purely non-cycle, or a mixed combination of cycle and non-cycle structures.
  • The first-level granularity, which identifies whether the structure is cycle or non-cycle.
• From the perspective of ground truth, The relationship between each coarsened graph and the ground truth lies in the fact that each coarsened graph in the TIF model strives to represent the critical structures constructed by the ground truth at the granularity level that the layer aims to explain for the input graph. That is, the coarsened graph obtained at each level by TIF corresponds to the ground truth at that level of granularity.

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[Q2] Broad Applicability

• Our method demonstrates broad applicability. In the initial manuscript, we conducted experiments on the COLLAB social network dataset, evaluating prediction performance using metrics such as accuracy and F1 score, as well as explanation performance metrics including explanation accuracy, path consistency, and path importance. The experimental results show that, compared to other baselines, our model attained optimal performance across all explanation metrics and the second-best accuracy on the COLLAB dataset.
• From an interpretability perspective, when the dataset is input as a graph into the model, the model uses the coarsened graph at each layer to explain the relational structures between different scales of social clusters, such as collaborators, groups, teams, and domain communities in the social network. For example, the model uses finer graphs with the scale of "collaborators" as the reference unit to show the social relationships between collaborators. The model uses coarser graphs with the scale of "domain communities" as the reference unit to show the social relationships between domain communities. Finally, the model outputs the results of the classification task.

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Ref:

[1] Sahil Verma, Varich Boonsanong, Minh Hoang, Keegan Hines, John Dickerson, and Chirag Shah. Counterfactual explanations and algorithmic recourses for machine learning: A review. In ACM Computing Surveys, volume 56, number 12, pages 1–42, 2024. ACM New York, NY.

[2] Rui Wang, Xiaoqian Wang, and David I. Inouye. Shapley Explanation Networks. In International Conference on Learning Representations, 2021.

[3] Zhitao Ying, Dylan Bourgeois, Jiaxuan You, Marinka Zitnik, and Jure Leskovec. GNNExplainer: Generating explanations for graph neural networks. In NeurIPS (Advances in Neural Information Processing Systems), 2019.

[4] Hao Yuan, Haiyang Yu, Jie Wang, Kang Li, and Shuiwang Ji. On explainability of graph neural networks via subgraph explorations. In ICML (International Conference on Machine Learning), 2021.

[5] Hao Yuan, Jiliang Tang, Xia Hu, and Shuiwang Ji. XGNN: Towards model-level explanations of graph neural networks. In SIGKDD (ACM Knowledge Discovery and Data Mining), 2020.

[6] Zaixi Zhang, Qi Liu, Hao Wang, Chengqiang Lu, and Cheekong Lee. ProtGNN: Towards self-explaining graph neural networks. In AAAI (Association for the Advancement of Artificial Intelligence), 2022.

[7] Aosong Feng, Chenyu You, Shiqiang Wang, and Leandros Tassiulas. KerGNNs: Interpretable graph neural networks with graph kernels. In Proceedings of the AAAI Conference on Artificial Intelligence, 2022.

[8] Jun Yin, Chaozhuo Li, Hao Yan, Jianxun Lian, and Senzhang Wang. Train Once and Explain Everywhere: Pre-training Interpretable Graph Neural Networks. In NeurIPS (Advances in Neural Information Processing Systems), 2023.

[9] Junchi Yu, Tingyang Xu, Yu Rong, Yatao Bian, Junzhou Huang, and Ran He. Graph Information Bottleneck for Subgraph Recognition. In ICLR (International Conference on Learning Representations), 2020.

[10] Siqi Miao, Mia Liu, and Pan Li. Interpretable and generalizable graph learning via stochastic attention mechanism. In ICML (International Conference on Machine Learning), 2022.

[11] Yongduo Sui, Xiang Wang, Jiancan Wu, Min Lin, Xiangnan He, and Tat-Seng Chua. Causal attention for interpretable and generalizable graph classification. In SIGKDD (ACM Knowledge Discovery and Data Mining), 2022.

[12] Yuwen Wang, Shunyu Liu, Tongya Zheng, Kaixuan Chen, and Mingli Song. Unveiling Global Interactive Patterns across Graphs: Towards Interpretable Graph Neural Networks. In SIGKDD (ACM Knowledge Discovery and Data Mining), pages 3277–3288, 2024.

[13] Hao Hu, Wei Hu, An-Di Guo, Linhui Zhai, Song Ma, Hui-Jun Nie, Bin-Shan Zhou, Tianxian Liu, Xinglong Jia, Xing Liu, et al. Spatiotemporal and direct capturing global substrates of lysine-modifying enzymes in living cells. Nature Communications, 15(1):1465, 2024.

[14] Yansheng Zhai, Xinyu Zhang, Zijing Chen, Dingyuan Yan, Lin Zhu, Zhe Zhang, Xianghe Wang, Kailu Tian, Yan Huang, Xi Yang, et al. Global profiling of functional histidines in live cells using small-molecule photosensitizer and chemical probe relay labelling. Nature Chemistry, pages 1–12, 2024.

[W3] & [Q1] (1) Excessive Hyperparameters, (2) Dynamic Hardening Mechanism Increases the Complexity of the Method.

Sorry for the confusion. (1) The initial manuscript contained errors in the number and values of the hyperparameters, which we have now corrected. Our model uses only three parameters (α1–α3) across different datasets, and their values remain consistent without any additional adjustments (we have provided our code in the original supplementary material, as detailed in lines 654, 445, and 620 of encoders.py). This demonstrates the robustness of our method, which does not require hyperparameter tuning. These clarifications have been updated in Table 4 (Page 16) of the revised manuscript, as follows:

Table 4 (Page 16): The statistics of hyper-parameters setting.

(2) We initially designed the dynamic hardening mechanism and included it in the appendix of the initial manuscript to explore better interpretability. However, during subsequent experiments, we found that it did not achieve the desired effect. None of the experiments used this module, but we forgot to remove it from the appendix. In the revised manuscript, we have removed this content from the appendix.

[W4] (1) Lack of Mathematical Proof for Approach Effectiveness， (2) the Optimality of the Tree Structure.

(1) Sorry for the confusion. Existing interpretability methods primarily stem from two aspects: human-cognitive logic interpretability [3-12] and mathematical interpretability [1-2].

In this work, we primarily follow the paradigm of human-cognitive logic interpretability. We observe that existing methods often adopt an inflexible approach that captures graph connectivity at a single specific level, whereas real-world graph tasks typically involve relationships at multiple granularities (e.g., relevant interactions in proteins range from functional groups to amino acids and even to protein domains [13-14]). To address this limitation, we propose TIF, which has been extensively validated through a series of quantitative and qualitative experiments designed in strict adherence to the standards established in existing works [3-12], demonstrating the effectiveness of our approach.

(2) Sorry for the misunderstanding. In the initial manuscript of the main text (Pages 9-10) and Appendix C (Pages 20-22), we provided a series of ablation studies. These studies included analyses from perspectives such as compression ratio, path count, and routing structure. Based on the experimental results, it can be concluded that the tree structure currently used by TIF is indeed the optimal tree structure.

[W2] Running Efficiency.

Thanks. As suggested, we have additionally analyzed the theoretical complexity and compared the time consumption of our proposed framework with several interpretable baselines.

(1) We calculated the theoretical complexity of the proposed TIF framework. For time complexity, updating node embeddings requires $O(N \cdot d^2)$ for dense graphs and $O(E \cdot d)$ for sparse graphs, while clustering and perturbation modules add $O(M \cdot N \cdot d \cdot c)$. Overall, the time complexity is $O(N \cdot d^2 + M \cdot N \cdot d \cdot c)$ (dense graphs) or $O(E \cdot d + M \cdot N \cdot d \cdot c)$ (sparse graphs). For space complexity, dense graphs require $O(N^2)$ for adjacency storage, while sparse graphs need $O(E)$. Adding node embeddings, perturbations, and routing parameters results in $O(N^2 + N \cdot d \cdot (1 + M \cdot c))$ for dense graphs or $O(E + N \cdot d \cdot (1 + M \cdot c))$ for sparse graphs. These clarifications have been updated in Appendix D.2 (Pages 22-23) of the revised manuscript.

(2) We present the time required to complete the training of each interpretable model. The results of the time consumption in Table 13 show that our method is only slightly less efficient than KerGNN and GIP. Given that our model outperforms KerGNN and GIP in terms of prediction and explanation performance on the vast majority of datasets, we believe that this slight additional time cost is justified. These additional results have been updated in Appendix D.2 (Pages 22-23, Table 13) of the revised manuscript, as follows:

Table 13 (Page 23): Time consumption of different methods. The table shows the time required (in seconds) to finish training for each interpretable model on various datasets. “*” indicates the method requires an additional pre-training process which takes nearly 72 hours.

We are glad that the reviewer appreciates our work as a useful contribution to the community. We have carefully revised the manuscript according to your constructive suggestions. Below we address the main points raised in the review.

[W1] Std Value.

Sorry for the confusion. Due to formatting and space constraints, we omitted the std values in some tables in the initial manuscript. We have now provided the std values for all experimental data. The std values of Table 1 have been provide in Table 12 (Page 22, Appendix D.1) of the revised manuscript. The std values of Table 2 have been provide in the original Table 2. All revised tables are presented as follows:

Table 12 (Page 22): Comparison of different methods in terms of classification accuracy (%) and F1 score (%) along with their corresponding stds.

Table 2 (Page 9): Comparison of different methods in terms of explanation accuracy.

Dear reviewer,

We are glad that the reviewer appreciates our attempt, and sincerely thank the reviewer for the constructive comments. As suggested, we have additionally added some clarifications about our framework and included further experiments, such as those on running efficiency. Please let us know if you have other questions or comments.

Since half of the allocated time for reviewer-author discussion has already elapsed, we sincerely look forward to your reevaluation of our work and would very appreciate it if you could raise your score to boost our chance of more exposure to the community. Thank you very much!

Best regards,

The authors of TIF