Thank you for your constructive feedback and comments, which have greatly helped us improve the clarity and depth of our manuscript.

Weaknesses

Weakness 1:Older papers

Answer 1: Our baseline methods include several recent approaches, such as DMP by Zhu et al. (2023), SMPT by Li et al. (2024), and DGCL by Jiang et al. (2024). These references highlight our effort to incorporate state-of-the-art methods into our analysis, ensuring that our comparisons are both relevant and up-to-date.

Weakness 2: More theoretical than experimental

Answer 2: We appreciate your observation regarding the balance between theoretical development and experimental validation. In response, we have included visualizations of different path orders and performance comparisons of path complexes with various geometric features (Pages 9 and 10).

Our initial focus was to establish the theoretical foundation of the PCMP method, introducing a novel topological framework for molecular representation. This framework utilizes topology to depict molecular structures, serving as the basis for the message passing and pooling mechanisms. The results from our regression and classification tasks, along with ablation studies, demonstrate the practical utility and predictive efficacy of path complexes in molecular property prediction.

Weakness 3: Computational expense

Answer 3: The PCMP method constructs higher-order paths using only covalent bonds, which significantly reduces computational complexity compared to traditional methods that include both covalent and non-covalent bonds. This strategic design not only retains the model’s effectiveness but also optimizes the use of computational resources, achieving a balance between performance and resource demands.

To further enhance efficiency, the computational overhead of path complexes can be reduced by minimizing the use of high-order paths. This ensures scalability while maintaining the accuracy of predictions.

Weakness 4: Dataset description and tasks

Answer 4: In Appendix A.3, we provide a detailed introduction to our datasets, particularly regarding the predictive targets for the QM9 dataset. In our revision, we include concise descriptions of each dataset and elaborate on the specific molecular properties we aim to predict, such as the electron spatial extent (α) and the energies of the highest occupied molecular orbital (εHOMO) and the lowest unoccupied molecular orbital (εLUMO).

Questions

Question 1: Time and memory complexity

Answer 1: In the PCMP method, higher-order path construction is restricted to covalent bonds, significantly reducing the computational burden compared to approaches that consider all interactions, including non-covalent ones. The computational complexity of our algorithm is O(N^2), where N represents the number of atoms in a molecule. This quadratic complexity primarily arises from pairwise interactions in path complex formation. However, by limiting interactions to covalent bonds, the number of processed interactions remains manageable, ensuring computational efficiency.

Question2: Using 3-path complexes, higher orders path?

Answer2: Our approach considers up to three-body (3-path) interactions, effectively capturing interactions among four atoms. This design parallels considerations of dihedral and improper angles in molecular force fields, providing a robust and comprehensive representation of molecular structure. As higher-order paths introduce additional complexity, designing effective geometric features for these paths is a key focus of ongoing optimization in our model.

Question 3: Dataset

Answer3: The QM7 and QM9 datasets are related to quantum mechanics, while the Tox21, HIV, and MUV datasets pertain to physiology and biophysics. For comprehensive details about the datasets, please refer to Appendix A.3.

Thank you for your thoughtful feedback and questions, which have provided valuable insights to improve our manuscript.

Weakness

Weakness1: Unsubstantiated Claim on Force Field Mimicking.

Answer1: We appreciate the opportunity to clarify this point. Our primary objective is not to predict molecular dynamics (MD) processes directly. Instead, we draw inspiration from MD principles, leveraging geometric features derived from atomic positions to predict molecular properties. Since our model does not use atomic coordinates as inputs, it cannot predict forces or derive partial derivatives of outputs with respect to coordinates. This design aligns our approach more closely with topological representations of molecular geometry than with direct MD simulations.

Weakness 2: Computational Complexity.

Answer2: Thank you for addressing the computational aspects of our method. In our PCMP approach, the construction of higher-order paths is restricted to covalent bonds. This choice significantly reduces the computational burden compared to methods that include all possible interactions, such as non-covalent bonds.

The computational complexity of our algorithm is $O(N^2)$, where $N$ is the number of atoms in a molecule. This quadratic complexity arises from pairwise interactions during the formation of path complexes. However, by limiting high-order paths to covalent bonds only, we effectively manage the number of interactions processed, ensuring that runtime remains feasible for practical applications.

Weakness 3: Lack of Equivariance in Model Design

Answer 3: Our model ensures invariance and equivariance by employing relative geometric features, such as angles and distances, which inherently account for variations due to rotations and translations. This approach enables the model to effectively capture essential geometric dependencies required for accurate molecular property predictions without relying on absolute coordinates. Furthermore, we explicitly state the invariance of the Path Complex in Theorem 3.3 (Page 4) and provide a detailed proof in the Appendix.

Weakness 4: Interpretability

Answer 4: The interpretability of our model stems from its use of Path Complexes to capture molecular force fields and encode detailed geometric information. By incorporating higher-order geometric features, such as bond angles, dihedral angles. Path Complexes provide a structured and physically meaningful representation of molecular systems.

To further enhance interpretability, we designed a bidirectional information transfer mechanism, allowing effective communication between high-order paths (e.g., 3-paths) and lower-order paths (e.g., 0-paths). The validity of this mechanism was demonstrated through ablation studies (Page 8), showing its contribution to improved predictive performance. Additionally, in the newly added section “Impact of Geometric Features in Path Complex” (Page 10), we empirically validated the effectiveness of integrating more geometric information.

These results illustrate that the Path Complex framework not only improves performance but also provides insights into how molecular geometries influence property predictions, contributing to the interpretability of our approach.

Questions: MM force fields, path orders or feature, over-squashing, large molecules

1. For MM force field, the answer is in General Response
2. For path orders or feature question. Due to the message passing mechanism in the path complex, updating the information for an n-order path necessitates the use of both n-1 order paths and n+1 order paths, thereby fixing the sequence of path orders within the path complex. We added Figure 6 illustrates the influence of different path orders on regression and classification tasks. The section \paragraph{Impact of Geometric Feature in Path Complex}” (page 10) demonstrates how various path features affect the model’s performance.
3. Our 3-path representation provides a framework for characterizing 4-body interactions, including improper angles. By explicitly incorporating features of improper angles into the 3-path configuration, we can effectively encode their geometric properties within our graph-based model. This approach allows us to capture the structural nuances of improper angles, enhancing the model’s ability to accurately represent molecular geometries.
4. The issue of over-squashing is a significant problem worthy of investigation within the context of Graph Neural Networks (GNNs). However, in this paper, we focus solely on exploring the expressive capabilities of PWL.
5. We acknowledge the limitation of not including macromolecular structures in our study, primarily due to the significant computational demands of applying our path complex method to large molecules. Currently, our resources are more suited to smaller molecules.

We appreciate your questions and suggestions, which have helped us refine and clarify our work.

Weakness

Weakness 1: Clarity

Answer 1: ** We agree that relocating some of the more intricate details to an appendix would make the manuscript more accessible to general readers while providing additional depth for interested researchers. Accordingly, we will reorganize the manuscript to shift complex technical sections to an appendix, improving overall readability.

Weakness 2: Experimental Limitations

Answer 2: We acknowledge that a limitation of our study is the exclusion of large molecular structures, which stems from the challenges of designing geometric features based on protein positions. Currently, our focus and resources are geared towards smaller molecules. While our focus is on predicting properties of small molecules, our model still performs well on datasets like QM9, which has over a hundred thousand samples, and the MUV dataset, which is nearly as large. For our own testing, here are the specific details:

1. QM7 dataset: Running 500 epochs under our test parameters required approximately 2 hours for five iterations.
2. QM9 dataset: Due to the smaller batch size, it took about 12 hours on average to complete 500 epochs across five iterations.
3. Tox21 dataset: Completing 1000 epochs required an average of 4 hours over five iterations.
4. HIV dataset: Completing 1000 epochs required an average of 18 hours over five iterations.
5. MUV dataset: Completing 1000 epochs required an average of 39 hours over five iterations.

Questions

Question 1: Recently models, M3GNet, MACE, or EquiformerV2?

Answer 1: Thank you for raising this point. Unlike models such as M3GNet, MACE, or EquiformerV2, which focus on molecular dynamics (MD) tasks, our approach is designed for molecular property prediction based on geometric features. Since we do not input atomic coordinates directly, our model cannot predict forces or derive partial derivatives of outputs with respect to coordinates. Instead, we leverage relative geometric features to capture molecular properties inspired by MD principles.

Question 2: How about the speed compared with other models?

While direct speed comparisons with other models are challenging due to limited reporting on computational overheads in prior works, we highlight key points regarding the efficiency of the PCMP model:

1. No Pre-training Required: Unlike many models requiring pre-training, PCMP skips this step, significantly reducing preparation time and enabling rapid deployment and iterative testing.
2. Computational Overhead: Compared to models like Mol-GDL (), which do not use pre-training, PCMP is approximately twice as slow, primarily due to the additional complexity of constructing and processing path complexes, particularly higher-order paths.
3. Performance-Speed Trade-off: The additional computational cost is justified by the model's ability to capture intricate molecular interactions and geometric features, as evidenced by its superior performance on multiple benchmark datasets.

Question 3: Expand this framework to periodic systems, i.e. inorganic materials?

Answer 3: Given the structural similarities between molecular materials and inorganic materials, we believe our PCMP can be adapted to periodic systems. From a topological perspective, in addition to path complexes, the quotient graph also presents a viable option. We are optimistic about using topological techniques to handle similar geometric datasets and plan to explore this direction in future research.

Question 4: Interpretability of Path Features

Answer 4: We have added a new section titled \paragraph{Impact of Geometric Feature in Path Complex} on page 10. Figure 6 visualizes the impact of path orders on regression and classification tasks. Furthermore, in the ablation study section, we analyzed how different path orders affect the model’s predictive accuracy. Our findings indicate that higher-order paths significantly enhance the model’s ability to predict molecular properties.

Thank you for your careful review and valuable comments on our work.

Weakness

Weakness 1: Many body expansions.

Answer 1: While geometric features such as bond lengths, angles, dihedral angles, and improper angles are well-established in molecular modeling, previous studies often used both covalent and non-covalent bonds for angle calculations. By focusing solely on covalent bonds, our method aligns more closely with real molecular structures. The natural ability of path complexes to represent these geometric features ensures their utility in molecular property prediction, as demonstrated in Table 5 (Page 10).

Weakness 2: It’s a necessity to proof invariance or equivariance

Answer2: We have addressed this by adding Theorem 3.3 (Page 4), which proves the invariance of the path complex. Our model achieves invariance by employing geometric features based on relative positional information (e.g., relative angles and distances), avoiding dependence on absolute coordinates. This ensures the model’s outputs remain unaffected by basic geometric transformations such as translations and rotations.

Weakness 3: Despite claiming that the method “enables systematic exploration of connectivity and interaction for analyzing complex systems and networks,” there are no experiments on these tasks supporting this claim.

Answer3: Thank you for pointing out the overreaching claim in our paper regarding the capability of our method to analyze complex systems and networks. We acknowledge that our PCMP framework is primarily designed for data with rich geometric information, such as molecular structures, and currently lacks experimental validation for complex systems analysis. We are actively working on adapting our model to better handle diverse data requirements, including non-geometric data. We will revise our manuscript to accurately reflect the scope and current capabilities of our research.

###Weakness 4: RDKit and 3D molecular datasets for DFT-level property prediction.

Answer4: We relied on RDKit because the original data files were in SMILES format. RDKit allowed us to generate atomic coordinates for extracting geometric features. Additionally, we compared our method against state-of-the-art models such as DMP【1】, SMPT【2】, and DGCL【3】. Our primary goal was to establish a novel geometric deep learning framework rather than simulate molecular dynamics, which informed our choice of datasets (e.g., QM9 for α, εHOMO, εLUMO predictions).

Questions

Answer: Thank you for your valuable suggestions. In our study, we have indeed recognized the importance of comparing our PCMP to well-established models. Specifically, we have included comparisons of PCMP with DimeNet in our results section, particularly focusing on classification tasks for datasets such as Tox21, HIV, and MUV. In these comparisons, PCMP demonstrated superior performance in terms of ROC-AUC metrics. We acknowledge the potential insights that could be gained from further comparisons with other models like GemNet and MACE.

[1] Jinhua Zhu, Yingce Xia, Lijun Wu, Shufang Xie, Wengang Zhou, Tao Qin, Houqiang Li, and Tie-Yan Liu. 2023. Dual-view Molecular Pre-training. In Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD '23). Association for Computing Machinery, New York, NY, USA, 3615–3627. https://doi.org/10.1145/3580305.3599317

[2] Yishui Li, Wei Wang, Jie Liu, and Chengkun Wu. 2024. Pre-training molecular representation model with spatial geometry for property prediction. Comput. Biol. Chem. 109, C (Apr 2024). https://doi.org/10.1016/j.compbiolchem.2024.108023

[3] Jiang X, Tan L, Zou Q. DGCL: dual-graph neural networks contrastive learning for molecular property prediction[J]. Briefings in Bioinformatics, 2024, 25(6): bbae474. https://academic.oup.com/bib/article/25/6/bbae474/7779242

I have to comment this:

Inability to Derive Forces:

The authors state:

“Unlike direct simulations of molecular dynamics (MD) processes, our Path Complex Message Passing (PCMP) model inputs geometric features, not atomic coordinates, and thus cannot derive corresponding forces through differentiation. Therefore, our approach aligns more with topological deep learning rather than neural network potentials.”

However, there are numerous works (e.g., DimeNet, DimeNet++, GemNet, PaiNN, TorchMD-Net, ViSNet) that incorporate geometric features and successfully derive forces through differentiation. Here’s a straightforward method for achieving this:

1. Enable gradients for atomic positions by setting requires_grad=True.

2. Calculate geometric features (e.g., bond lengths, angles, etc.) based on these positions.

3. Perform message passing and compute a scalar quantity, such as energy, as the final representation.

4. Use torch.autograd.grad to compute the gradient of the energy with respect to the atomic positions. The negative of these gradients corresponds to the forces.

Given this well-established approach, the inability to compute forces cannot be considered a valid limitation or excuse. Furthermore, if this paper is truly focused on topological deep learning rather than neural network potentials, why include tasks requiring 3D structures and do not compare with geometric graph neural networks (GNNs)? If the goal is purely topological modeling, tasks based on SMILES or other 2D representations would suffice.

Many-Body Interactions:

The authors claim:

“We exclusively use covalent bonds to construct higher-order paths (e.g., 2-paths and 3-paths), capturing angles and dihedral angles that better reflect many-body interactions in molecular force fields. The computational complexity of our algorithm is , where  represents the number of atoms within a molecule.”

However, many existing works (e.g., DimeNet, PaiNN) already capture angles, and others like GemNet and ViSNet handle dihedral angles explicitly. Additionally, models such as NequIP, Equiformer, and MACE employ many-body expansions that implicitly represent these features in a more theoretically grounded manner. The claim that this method better reflects many-body interactions in molecular force fields is totally unsupported, especially since this model does not perform tasks related to molecular force fields.

Computational Complexity:

Papers like PaiNN and MACE use density-based tricks to reduce the complexity of geometric feature calculations to $O(N)$. In contrast, this paper relies on explicit calculations, which are $O(N^2)$  and are already implemented in older models such as DimeNet and GemNet, which further use radial basis functions (RBF), angular basis functions (CBF), and spherical basis functions (SBF) to obtain smooth gradient. This approach does not introduce any novel computational efficiencies.

Conclusion:

I cannot identify the significant novelty in the approach described in this paper. The authors’ claims fail to provide sufficient evidence to substantiate their purported advantages and appear inconsistent with the paper’s stated goals.

1. On Higher-Order Geometric Features:

The authors state:

“Our model’s input variables include not only angles and dihedral angles but also higher-order geometric features, such as triangle areas formed by 2-path 3-body interactions and tetrahedron volumes formed by 3-path 4-body interactions.”

However, numerous previous works, from DimeNet and GemNet to NequIP and MACE, already incorporate higher-order geometric tensors beyond 3-body interactions. These methods utilize similar or even more advanced geometric representations while maintaining their ability to predict molecular properties and forces. The claim that these features are unique to this work is unsupported and overlooks the substantial body of prior research.

2. On Gradient Calculation and Force Prediction:

The authors argue:

“It’s unreasonable to compute ’s gradient solely with respect to the coordinates x. Such a calculation would ignore the contribution of higher-order geometric features to y.”

This statement is totally problematic and misleading. The computation of forces as gradients of the energy with respect to atomic coordinates is a cornerstone of modern machine learning force fields. Models such as DimeNet, GemNet, NequIP, and MACE explicitly incorporate many-body interactions and high-order geometric tensors, which mostly contribute to the accurate prediction on both molecular properties and forces. Suggesting that taking gradient on force prediction is “unreasonable” undermines the validity of these well-established works and lacks theoretical justification.

3. On Benchmarking and Comparisons:

The authors state:

“We included tasks requiring 3D structures as a means to validate the ability of our path complex framework to capture geometric information, which is intrinsic to many molecular properties. This does not imply an intention to replicate or compete with geometric GNNs.”

If the focus is on validating the ability to capture geometric information, comparisons with state-of-the-art geometric GNNs such as DimeNet, PaiNN, GemNet, NequIP, and MACE are mandatory. Without such comparisons, conducting 3D molecular tasks is effectively meaningless. Furthermore:

• If the work is truly aimed at topological deep learning, there should be evaluations on established topological benchmarks.
• If the goal is to work on 3D molecular structures, comparisons with prevailing benchmarks and baselines in this domain are essential. The lack of performance on either front is a major limitation.

4. On Broader Predictions:

Even if force prediction is beyond the scope of this work, the model could still address molecular property prediction tasks such as energy, HOMO, and LUMO (e.g., benchmarks in QM9). The absence of performance evaluation on such standard tasks leaves the contributions of this paper unsubstantiated.

Conclusion:

All the aforementioned works (DimeNet, GemNet, NequIP, MACE, etc.):

1. Involve many-body interactions.

2. Utilize high-order geometric tensors.

3. Predict both molecular properties and forces.

I strongly recommend the authors carefully study these papers in this field before making unsupported claims. You still have time to conduct several experiments and comparison as other reviewers also suggested during the rebuttal period Moreover, failing to compare against geometric GNNs on molecular 3D tasks or against topological deep learning methods on topological benchmarks raises serious questions about the completeness of this work.

We appreciate the reviewer pointing out the well-established methods for deriving forces via differentiation with respect to atomic coordinates. However, we would like to clarify that:

On Computing Forces via Differentiation

It is true that common force-predicting models rely on computing the gradient of the final predicted value with respect to input atomic coordinates to derive the forces on each atom. However, in our model, atomic coordinates are not used as direct inputs. This design choice provides strong invariance and equivariance properties, but it comes with the limitation that forces cannot be directly predicted. The reasons are as follows:

1. Our model’s input variables include not only angles and dihedral angles but also higher-order geometric features, such as triangle areas formed by 2-path 3-body interactions and tetrahedron volumes formed by 3-path 4-body interactions. In this context, the predicted value y is a function of multiple variables beyond atomic coordinates.
2. These input variables span different scales, making it unreasonable to compute y ’s gradient solely with respect to the coordinates x . Such a calculation would ignore the contribution of higher-order geometric features to y.

In this work, we focus on predicting global graph-level properties rather than atomic forces. We acknowledge that force prediction could be achieved by modifying our PCMP framework to include atomic coordinates as inputs and adjusting the architecture accordingly. This is a potential direction for future work.

On the Rationality of Higher-Order Path Construction

Unlike previous models, we propose a novel framework for modeling higher-order information that is particularly natural for biomolecules. From a mathematical perspective, the elements in molecules—atoms, bonds, bond angles, and dihedral angles—correspond perfectly to 0-paths, 1-paths, 2-paths, and 3-paths, respectively. Together, these paths form a path complex. To the best of our knowledge, this correspondence has not been explicitly explored in prior work. Highlighting this alignment was one of our primary motivations for this study.

With the path complex representation of molecules, our framework offers several unique advantages:

1. Direct higher-order modeling: Higher-order interactions can be explicitly modeled as higher-order paths.
2. Established mathematical foundation: Path complexes have a mature mathematical foundation, enabling the use of existing mathematical tools to study biomolecules.
3. Flexible message passing: The path complex representation allows for more flexible message-passing schemes, extending beyond pairwise interactions.

These three points distinguish our approach from existing models.

Tasks Requiring 3D Structures

We included tasks requiring 3D structures as a means to validate the ability of our path complex framework to capture geometric information, which is intrinsic to many molecular properties. This does not imply an intention to replicate or compete with geometric GNNs. Instead, we aim to demonstrate the potential of path complexes to provide a new perspective on molecular representations.

We agree with the reviewer that topological tasks based on 2D representations (e.g., SMILES or molecular graphs) would also be relevant, and we plan to explore these tasks in future work.

Computational Complexity

We acknowledge the reviewer’s comment regarding computational complexity. While some models (e.g., PaiNN, MACE) utilize density-based techniques to achieve $O(n)$ complexity, our approach prioritizes the explicit calculation of geometric features to preserve the interpretability and flexibility of representing higher-order paths.

Additionally, while radial basis functions (RBF) and spherical basis functions (SBF) are widely used to smooth gradients, they are orthogonal to the contributions of our path complex framework. Our work does not aim to introduce novel computational efficiencies but instead focuses on providing a new mathematical perspective for modeling molecular systems.ng a new perspective for modeling molecular systems.

Conclusion

Our goal is not to develop a new neural network potential method. Instead, we aim to integrate path complexes with molecular force fields from a mathematical and physical perspective, providing a novel framework for predicting biomolecular properties. We acknowledge the limitations of our current approach and view them as opportunities for future improvement. We sincerely thank the reviewer for their detailed comments and will consider incorporating additional tasks and comparisons to further validate the applicability of our framework