Paper: Curvature MPNNs : Improving Message Passing with Local Structural Properties (ICLR-2024)

Reviewer comment (Theory):
The writing of the paper, especially the first 7 pages, contains many mistakes. Specifically, the math presented in the paper has many errors (definition of $W_1$, sparsifying the graph improving the diameter, and other statements such as that).

Author response:
* We agree that the diameter of the graph can become infinite. We have modified the sentence accordingly. 
* We have fixed all the others mentioned mistakes and formatting issues.

Reviewer comment (Presentation):
There are many formatting issues in the paper.

Author response:
* We have fixed all the others mentioned mistakes and formatting issues.

Reviewer comment (Theory):
On page 3, the paper states that the formation of curvature in Topping et al. is to improve expressiveness. Clarification is requested because the reviewer’s understanding was that it was to improve computability.

Author response:
The curvature proposed in [1] is more expressive than Forman Curvature since it  takes into account triangles and cycles of size 4 Section 3 equation 3. Furthermore it’s can be use to improve computability on very large graphs w.r.t Ollivier Curvature.

Reviewer comment (Writing):
The definition of the two-hop $\beta^+$ is unclear and needs to be stated.

Author response:
* The two-hop neighbourhood is defined using a neighbourhood of size 2 only on edges with certain curvatures. In the case of two-hop positive curvature, the information is broadcast both to neighbours connected by edges with positive curvature and to neighbours of neighbours connected in the same way by edges with positive curvature i.e follows a chain of positively curved edges of size 2. 
We specify this answer in the section 3.4 for greater clarity.

Reviewer comment (Presentation):
There is confusion about Figure 2D being described as the 2hop *positive message passing*. The meaning of the dashed lines is unclear. The message also appears to be propagated further for the regular message passing framework, which contradicts the caption of Figure 2.

Author response:
* We are modifying Figure 2 and the caption to make it easier to read.

Reviewer comment (Theory):
For one-hop message passing, the paper incorrectly states that sparsifying the graph **reduces** the diameter of the graph. This is false; for example, in the graph used, removing either the positive or negatively curved edges disconnects the graph, which makes the diameter infinite.

Author response:
* We agree that the diameter of the graph can become infinite. We have modified the sentence accordingly.

Reviewer comment (Experiments):
Did you try to use curvature values (positive and negative) directly as weight in message-passing framework, instead of using them to eliminate some edges?

Author response:
* No. However, we plan to use curvature inside an attention mechanism to take into account both the features and the local structural property of the graph.

Reviewer comment (Writing):
The overall writing and grammatic should be checked again.

Author response:
* We also fixed the broken English sentences.

Reviewer comment (Experiments):
Regarding the experiments: It is not clear how the experiments relate to the rest of the work. For instance, do any of these datasets actually suffer from oversquashing?

Author response:
* Long-range dependencies are more relevant in heterophilic datasets because local information is not sufficient.That's why reducing bottlenecks is so important for facilitating long-distance information exchange.
This observation is largely confirmed by [1] and therefore, methods that aim to reduce bottlenecks  are tested on heterophilic datasets.

Reviewer comment (Writing):
The paper is very badly written. Many sentences do not parse well.

Author response:
We also fixed the broken English sentences.

Reviewer comment (Experiments):
Could you explain how one should choose between message passing between positive vs. negative curved neighbourhoods? Or does one simply try all possible combinations and pick the best?

Author response:
When the dataset is homophilic, the best strategy is to propagate information inside the communities i.e. using positively curved edges (Table 1 and Table 3 shows that this strategy has been empirically confirmed on the 4 homophilic datasets). For heterophilic datasets, the experimentations has shown that the best results can generally be obtained when the strategy is choosing according to the higher value of the curvature constrained edge homophily  (c.f. see appendix A.2 and A3).