__ Summary and Contributions__: ===AFTER AUTHOR FEEDBACK===
The responses successfully address my concern. I believe this work is novel and can make a valuable contribution to the literature. Hence I am willing to increase my overall score to 7.
This paper proposes a new “convolution” method PANConv and a pooling method PANPool. Instead of considering each neighbor as current GNNs, PANConv view every path linking the message sender and receiver as the elemental unit in message passing process. This strategy can somehow avoid diluting information when aggregate information from distant node. It is shown that PANConv+PANPool can outperform current methods on graph-level tasks.

__ Strengths__: (1) Intuitively, it is reasonable that considering every path as unit in message passing can probably alleviate the “Bottleneck” problem [1]. The motivation of PANConv is clear and meaningful.
(2) The proposal of PANPool is convincing and the corresponding visualization (Figure 2) is great.
(3) The experiments can demonstrate the effectiveness of PAN on graph-level tasks.
[1] Alon, U., & Yahav, E. (2020). On the Bottleneck of Graph Neural Networks and its Practical Implications. arXiv preprint arXiv:2006.05205.

__ Weaknesses__: (1) Although the motivation is claimed in this paper, it is not clear how this method can learn better representations than existing message passing methods. More intuitive descriptions, analysis examples, and/or theoretical supports should be included to show the superiority of the proposed method.
(2) It would be more convincing if experiments on node classification tasks are utilized to show the effectiveness of the proposed convolution method. As we know, node classification is the most straight-forward usage of learned node representations.
(3) More necessary ablation studies about PANConv and PANPool should be conducted, or we do not really know which part contributes to the improvement.

__ Correctness__: The claims and method are correct and well described.

__ Clarity__: In general, this paper is well written and clear.
As mentioned before, it would be better if more intuitive descriptions, analysis examples, and/or theoretical supports should be included to show the superiority of the proposed method.

__ Relation to Prior Work__: Yes. This paper describes the related work very consistently and the difference between previous work is well demonstrated.

__ Reproducibility__: Yes

__ Additional Feedback__: As shown in “Weaknesses”, more analysis and claim about why the proposed convolution can perform better than existing message passing methods is necessary for improvement. It would be more convincing if experiments on node classification and more ablation studies can be included.

__ Summary and Contributions__: This paper proposes a general framework for graph classification and regression tasks. The core idea, the path integral, which is borrowed from the quantum physics, is insightful and solid in theory. More importantly, this work links the communities of both physics and graph neural networks. Furthermore, this paper introduces a new dataset, which can serve as another benchmark.

__ Strengths__: The proposed models in this paper, PAN convolution and PANPool is solid in theory. Given the physical ideas behind, PAN might be a powerful tool in analyzing many graph data, like biological, chemical, and physical systems. The authors also prove that many existing GNN methods (e.g. GraphSAGE, GAT, and GCN) are the special case of PAN. PAN is more accurate and stable compared with previous methods, which is demonstrated by the extensive experiments (Table 1 and Figure 4). Besides, the new dataset proposed in this paper can serve as another benchmark.

__ Weaknesses__: The idea of PAN has been presented at the ICML workshop [1]. But that is not a big problem cause [1] focuses node-level classification, but this paper focuses graph-level classification and regression tasks instead and proposes a new model PANPool. Besides, the theoretical analysis is much more detailed.
[1]Ma Z , Li M , Wang Y . PAN: Path Integral Based Convolution for Deep Graph Neural Networks[J]. 2019.

__ Correctness__: The core idea is insightful. The experimental results also demonstrate the claims.

__ Clarity__: The organization of the paper is good. The paper is well written and the language is rigorous and standardized.

__ Relation to Prior Work__: This work clearly discusses the differences from the previous work in the section of related work. Moreover, the authors prove that many existing GNN models can be viewed as special cases of PAN convolution (line 206~211).

__ Reproducibility__: Yes

__ Additional Feedback__: Post rebuttal
After reading the authors rebuttal, I think this is a qualified paper for NeurIPS and keep my rating unchanged.

__ Summary and Contributions__: This paper proposed a novel mechanism of graph convolution and graph pooling based on a discrete analogy of the path integral formulation used in Physics. The resultant convolution operator leverages different powers of adjacency matrices in a simple way. Authors further propose a graph pooling method based on the normalization constant in the path-integral formulation which captures the subgraph centrality to some extent. Experiments on several graph classifications tasks and a newly proposed point distribution recognition task (along with a new dataset) show that the proposed method achieves very impressive results.
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I read the response from authors and other review comments. My concerns are largely resolved. I still believe this is a solid contribution to the community and would like to keep my original rating.

__ Strengths__: 1, I like the connection between the path integral and graph convolution. The proposed framework indeed provides a simple yet elegant way of leveraging multi-scale information in the network/graph. Moreover, this framework permits a simple graph pooling formulation. From my understanding, the proposed method is easy to implement and just adds little overheads to the vanilla GCN.
2, The related work is adequately discussed.
3, The paper is clearly written. I enjoy reading the paper. The intuitive examples about the normalization strategy and the pooling indeed help the understanding.

__ Weaknesses__: 1, One “concern” or suggestion I had is about the expressiveness of the proposed PANConv operator. Besides the regular weights, the only learnable “weight” for the n-th power of the adjacency matrix is a scalar, i.e., e^{-E(n)/T}. From a pure machine learning perspective, I think it would be better to have more capacity on the weight by using a vector or even matrix parameterization (the entries are output by some neural networks which not only takes path length n as input but also some feature from A^n) of the weight although I am not sure about its physical interpretation.
2, It would be great to run some baselines without using the proposed pooling so that one can understand how important the pooling is.
3, Although Hybrid PAN is proposed, I did not find any experimental verification on its usefulness in the paper.
4, In section 5.1, there is no validation set mentioned for the graph classification task. It is a bit surprising that L = 1 or 2 already performs so well. It is unclear from the description whether PANConv of all layers share the same cutoff L. If that is the case, the same cutoff L at top layer may have a different effect than the same L at bottom layer since bottom layers already contribute to the diffusion of the node information. Therefore, I feel like the comparison of L not only depends on the graph type but also the architecture (e.g., depth).
5, In section 5.2, since one of the contributions is this dataset, I would suggest authors to introduce some of the important aspects of it like the statistics. By just looking at the figure 3, it seems that the number of nodes under each category is quite different and may be exploited by the model easily. Later, I checked the supplementary file and found the average number of nodes under three categories are almost the same which relieved my worry. But it would be great to spend some space in the paper to showcase the statistics like table 1 in the supplementary file.
6, The newly proposed PointPattern dataset seems very easy which may not be a good benchmark. Do you have any thoughts to increase the level of difficulty?
7, Figure 3 is a bit small.

__ Correctness__: From my understanding, the caims and the method are correct.

__ Clarity__: The paper is clearly written.

__ Relation to Prior Work__: It is clearly discussed regarding the differences from previous work.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: ===AFTER AUTHOR FEEDBACK===
The author feedback addresses my comments on hyperparameters and node classification. I still stand by my original review otherwise, and believe this is a solid paper that belongs in NeurIPS
This paper proposes a graph convolution method, PAN inspired by path integrals. In a PAN convolution step, each node is influenced by all paths between that node and its neighbors, weighted by the path length. When viewed as a random walk of information along the graph, this operation corresponds to a walk that maximizes entropy rather than a generic random walk, as is the case with GCN. The weight matrix associated with PAN can be used to downsample the graph based on subgraph centrality as well. Experiments show that PAN outperforms several other common graph convolution methods graph classification and regression tasks where graphs represent molecules. They also show PAN's superior performance on a new artificial dataset they created.

__ Strengths__: The algorithm presented in this paper is novel and well motivated with meaningful connections to existing graph theory/network concepts. Learning from graph data with deep networks is currently a very relevant and active topic in the Neurips community. The empirical results are significant and support the the argument that path number and length are useful quantities with which to weight neighbor node contributions.

__ Weaknesses__: This method as presented and implemented still has the limited power and expressiveness of message passing GNNs which take weighted averages over neighboring nodes.

__ Correctness__: The claims and method appear to be correct. The only potential issue that I noticed, is that there is no validation set for the graph classification benchmarks, and it's unclear how the hyperparameters were chosen for the different networks.

__ Clarity__: The paper is clear and well written.

__ Relation to Prior Work__: Yes, Section 4 contextualizes this paper among existing graph convolution work.

__ Reproducibility__: No

__ Additional Feedback__: The paper states that model hyperparameters for the graph classification task are provided in the supplementary material. I do not see them there. Please add them.
All of the experiments with the exception of the proposed synthetic dataset are graph classification/regression on molecule data, but PAN should apply just as naturally to network data and node classification. Why did you choose only moceules? If PAN also performs very well at node classification on network graph data, that would be impressive.
I'm also curious about using PAN with edge features, our more complex learned energy functions. Have you experimented with anything along this line?