Paper: Effective Structural Encodings via Local Curvature Profiles (ICLR-2024)

Reviewer comment (Writing):
Please provide definitions for variable names, such as 'd_max.' Currently, there are many variables in the article that are not explained, which can be challenging for newcomers to understand.

Author response:
We thank the reviewer for pointing this out. We have added explanations for all previously unintroduced variables.

Reviewer comment (Reproducibility):
In spite of keeping settings and optimization hyperparameters consistent among different settings, the authors should still provide the corresponding parameter configurations. This would aid in experiment reproducibility and, as a result, make the results more robust.

Author response:
We have extended appendix section A.3 and included relevant hyperparameter choices to make our results reproducible without considering other papers.

Reviewer comment (Experiments):
In Section 3.1, the authors mention, 'We believe that the curvature information of edges away from the extremes of the curvature distribution, which is not being used by curvature-based rewiring methods, can be beneficial to GNN performance.' I consider this assertion somewhat speculative, and I did not find any subsequent experiments that substantiate this claim. Would it be possible to include relevant ablation experiments to support this hypothesis?

Author response:
We consider the experiments in section 4.2 to be very strong evidence for the idea that the curvature information of edges away from the extremes of the curvature distribution can be useful for GNN performance. Using the LCP clearly outperforms methods that use the ORC to rewire the graph on all datasets.

Reviewer comment (Experiments):
In Section 3.1, the authors define LCP as 'five summary statistics of the CMS.' I would appreciate a more detailed motivation for this particular definition. Additionally, it would be beneficial to include relevant ablation experiments that showcase the impact of removing specific summary statistics to demonstrate their significance in influencing the final results.

Author response:
Our use of these summary statistics is motivated by their use in the Local Degree Profile (LDP) [2]. As our ablations in Appendix A.4 show, min and max seem to be the most useful summary statistics.

Reviewer comment (Reproducibility):
Could the authors please elaborate on how the combination of LCP encoding and position encoding is implemented in Section 4.2.2? I couldn't find any details regarding the actual implementation.

Author response:
For the experiments in the main text, we concatenated the positional or structural encodings to the node feature vectors in the original datasets as a preprocessing step. When using two different encodings together, we concatenated both to the original feature vector. We have added a clarification on this in section 4.1. We also ran ablations on some of the datasets considered where instead of only concatenating encodings and thereby increasing the dimension of the node features, we linearly projected the concatenated node features down to the original node feature dimension (following [1]). This had no clear effects on performance, however.

Reviewer comment (Writing):
Some parts of the introduction are a bit dense and may be challenging for readers not deeply familiar with the field. A clearer presentation of the background and motivation could benefit a wider audience.

Author response:
We will improve the clarity of the writing in the introduction and background and related works sections. If the reviewer has concrete suggestions on which concepts a wider audience might not be familiar with, we would be grateful.

Reviewer comment (Experiments):
Including experiments on a more diverse set of datasets and domains would be better.

Author response:
We have added four additional datasets in appendix sections A.7 and A.8: three heterophilious node classification datasets [1] and one graph regression task [2]. These datasets contain both social networks (e.g. Amazon ratings) and molecules (Zinc), i.e. vary in the domain and graph topology. We find that the LCP outperforms all other encodings considered and increases performance for all three models (GCN, GIN, GAT).

Reviewer comment (Experiments):
How well does LCP generalize across different domains, and what factors might influence its applicability in practical scenarios?

Author response:
Our experiments include social networks and networks relevant to the natural sciences (e.g. molecules). While we find that the LCP leads to considerable performance gains across domains, it seems that it is generally more useful for graph-level tasks (classification, regression) than for node-level tasks.

Reviewer comment (Reproducibility):
Are there any computational bottlenecks when implementing LCP in large-scale graph datasets, and what strategies or optimizations could be considered to address these issues?

Author response:
The main computational bottleneck that comes with using the LCP is that it scales cubically with the maximum degree of a node in the graph. For example, for the Tolokers dataset that we have added, computing the LCP is infeasible because of the high average degree in the graph (it takes more than six hours, which we consider a timeout).

We can address this issue by using the combinatorial approximations presented in appendix A.1.3 instead of the ORC itself. These approximations scale linearly in the max degree, so computing the (approximate) LCP for Tolokers, for example, now only takes a few seconds. In addition, the LCP can be implemented with a different curvature notion (Forman’s curvature, short: FRC), which is more scalable than ORC. Experiments for this can be found in Table 5.