Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms

Dear Reviewer,

we thank you for the constructive and thoughtful feedback. We’re especially grateful for recognizing the clarity of our exposition, the soundness of our method, and the convincing experimental results. Below, we respond to all concerns.

Question regarding illustration and intuition

Thank you for the helpful suggestion. To clarify, we are not computing the local Euler characteristic, but rather the local Euler Characteristic Transform (l-ECT). Unlike the raw Euler characteristic, l-ECT scans the local neighborhood in multiple directions, computing Euler characteristics of sublevel sets. This directional process, known to be invertible under mild assumptions, provides a rich fingerprint of local geometry and topology. While the Euler characteristic is a topological invariant, the collection of directional profiles captures detailed geometric structure. We'll add more explanations and illustrations in the revision.

“No intuition about why/when l-ECT works. No baseline comparison with XGBoost on the node features.”

Regarding intuition, see the response above; we also plan on illustrating the computation better. Regarding the baseline, we’re happy to include an additional comparison against XGBoost for the respective node classification tasks.

Request for XGBoost on raw node features

We agree that this baseline is important and are happy to include the corresponding results in our revision. In the meantime, Table 6 already shows that l-ECT features are informative: as the number of directions used in the transform is reduced, classification performance often degrades substantially. This suggests that the l-ECT encodes nontrivial structural information not already present in the raw features, and that this information contributes meaningfully to the model’s performance.

Question about spatial alignment using l-ECT

Thank you for raising this. While the experiment in Section 5.2 does serve as a sanity check, it also demonstrates that l-ECT provides a scalable alternative to alignment methods like Procrustes, which rely on costly SVD. Our approach scales linearly with n, l, and m, and allows control over approximation quality via m and l. We'll revise the text to better explain this motivation and highlight the practical benefits of l-ECT-based alignment.

Question on the definition of simplicial complexes

You're right—thanks for catching this. We meant geometric, not abstract, simplicial complexes, and will revise the text to reflect this consistently.

Question on l-ECT and point clouds

To clarify, we don’t compute a single Euler characteristic, but use the local Euler Characteristic Transform (l-ECT), which builds a vector from directional filtrations of the local neighborhood. Point clouds are treated as edge-free geometric graphs (0D complexes in R^n. Since the ECT is invertible, it serves as a rich fingerprint of local structure (in this case the point cloud). We’ll revise the text to make this clearer.

Request of homophily score reporting

Thank you for the suggestion. We will add a standard homophily measure (edge homophily ratio) for each dataset and indicate whether each dataset is homophilic or heterophilic. For example, the WebKB datasets exhibit low edge homophily (~0.2) and are thus considered heterophilic, while citation graphs like Cora are homophilic (e.g., ~0.8).

Request for using bold for all results within error margins

Thank you for pointing this out. We will update all result tables to use boldface for all values within one standard error of the best performance. We believe this makes the comparisons clearer and more meaningful.

“"Planetoid datasets" -- what is "planetoid"?”

“Planetoid” refers to the data introduced in the benchmark (Yang et al., 2016), containing the citation graph datasets Cora, Citeseer, and Pubmed. The dataset is introduced in l.353 of our manuscript. We will clarify this.

Questions about Table 7

Thank you for pointing this out. Table 7 uses the ranking setup from Platonov et al. (2023), with lower mean ranks indicating better performance. We added l-ECT post hoc using the same protocol. FSGNN, while strong, was excluded from our benchmarking due to its specialized design. As the comparison focuses on heterophilic graphs, such methods are favored—but our goal is to show that l-ECT performs well without architectural tuning, which we’ll clarify in the revision.

We are grateful for your recognition of our contributions and your close reading of the manuscript. If the improvements we propose address your concerns, we would be very grateful if you would consider raising your score. Thank you once again for your constructive and thoughtful review, we are happy to address any other questions you might have!

Best regards,

The Authors

Dear Reviewer,

we sincerely thank you for the thoughtful and detailed feedback! We are glad that you found our contributions novel and our empirical and theoretical results sound. We especially appreciate your engagement with the proofs and your recognition of l-ECT’s usefulness for node classification. Below, we address your comments point by point.

The examples of interpretability are not provided.

Thank you for raising this point. A discussion on the interpretability of l-ECT is included in Appendix A.2.1, where we show how feature importance of the underlying model naturally gives rise to interpretability. However, we agree that the main paper could more clearly showcase this aspect. In the revised version, we will better illustrate how l-ECT enables interpretation of node roles and neighborhood structures.

The details on evaluation of rotation-invariant metric is not provided.

Section 5.2 evaluates our rotation-invariant metric for aligning spatial graph representations, showing it captures structural similarity under random rotations. We’ll clarify this purpose in the text and add more details.

How do you perform differentiation since the Euler characteristic is not differentiable?

The Euler Characteristic Transform is made differentiable by embedding it into a continuous pipeline using soft (smoothed) filtrations. It doesn’t change the discrete nature of EC itself but creates a differentiable approximation around it. We will clarify this in our revision.

The performance of l-ECT on WebFB dataset is better that on common Planetoid. Do you have an explanation?

The stronger performance of l-ECT on WebKB over Planetoid datasets reflects structural differences: WebKB graphs are heterophilic, where local structure matters more than feature similarity—making l-ECT well-suited—while Planetoid graphs are homophilic, favoring message-passing GNNs. We’ll clarify this in the revision.

The proposed method outperformed standard GCN, GAT, GIN but is still worse that modern methods dedicated to hetero-graphs (see Appendix).

We agree that recent heterophily-specific methods are highly competitive on heterophilic benchmarks. However, our goal with l-ECT is not to design a specialized architecture optimized solely for heterophily, but rather to propose a general-purpose, interpretable representation that captures rich local structural information without message passing. What’s notable is that l-ECT achieves competitive performance despite being a non-aggregating and model-agnostic approach, and in many cases closes the gap with specialized models. We believe this highlights the complementary strengths of topological representations, particularly when combined with other methods. We will emphasise this aspect further in our revision and substantiate it using additional experiments.

If I understood correctly, l-ECT method is not applicable for graphs without features associated with nodes.

You're right—l-ECT, as formulated, assumes node features or geometric embeddings. This is a common assumption in graph learning, where such features are typically available and important for downstream tasks like node classification. Our method therefore aligns well with standard benchmarks and real-world settings. We'll clarify this assumption in the paper.

The definition of a simplical complex is quite clumsy mixing together geometrical and abstract simplical complexes.

Thank you for pointing this out—we completely agree. Our current presentation conflates geometric and abstract simplicial complexes. Since our method relies on geometric realizations, we’ll restrict the discussion to geometric simplicial complexes in the revised version. We’ll clarify the terminology and focus solely on the relevant setting for l-ECT. We appreciate your attention to this—it helps improve the clarity and precision of the exposition.

Hyperlinks to proofs of theorems in Appendix is not provided in main text

Thanks for this suggestion. In the revised version, we will include hyperlinks and explicit references from each theorem in the main text to its corresponding proof in the appendix.

I assume that l-ECT is defined for a geometric simplical complex, otherwise the definition (3) doesn't make sense. Why you still use term "abstract simplicial complex" in line 113? Consequently, l-ECT makes sense for graphs only if each vertex has attributes and a graph is mapped onto R^n. If I am right, this shoud be stated more directly.

This is a mistake, please see response above. We will fix this in our revision.

We are grateful for your recognition of our contributions and your close reading of the manuscript. If the improvements we propose address your concerns, we would be very grateful if you would consider raising your score. Thank you once again for your constructive and thoughtful review, we’d be happy to address any other questions you might have.

Best regards,

The Authors

Dear Reviewer,

we sincerely thank the reviewer for the careful and thoughtful evaluation of our work. We appreciate your recognition of our contributions, particularly the novelty of integrating topological data analysis into graph representation learning, and your acknowledgment of the soundness of our theoretical and empirical work. Below, we address your comments and questions in detail.

Limited exploration of computational efficiency—l-ECT computation could be expensive for large graphs.

For a fixed node $x$, the computational complexity of its $\ell ECT_k(x)$ is:

where:

• $m$ is the number of sampled directions,
• $l$ is the number of filtration steps,
• $|N_k(x)|$ is the number of vertices (or simplices) in the $k$-hop neighborhood of $x$.

In other words, the complexity scales linearly with neighborhood size and sampling resolution. In practice, as shown in Appendix A.2.1, even a small subset of directions suffices for high accuracy, reducing runtime considerably. For large graphs, parallelization of l-ECT computation provides scaling of the procedure. We will include a thorough discussion on computational complexity and improved implementations in our revision.

Relationship between l-ECT parameters and performance is not thoroughly analyzed

In Appendix A.2.1, we provide an ablation study on the number of sampled directions, showing how performance varies when using a smaller portion of directions. That said, we agree that further exploration with respect to other parameters such as the number of filtration steps l would be valuable. We are happy to include these additional ablations in the revised version to provide a more complete picture of the trade-offs involved!

LINKX (Lim et al., 2021) - A method that also bypasses message passing for heterophilous graphs WRGAT (Suresh et al., 2021) - Addresses heterophily with rewiring techniques ACM-GCN (Luan et al., 2022) - Reports state-of-the-art results on heterophilous benchmarks Papers on graph topological operators like GraphGPS (Rampavsek et al., 2022)

We appreciate the references and agree these are valuable works that provide additional context. We will discuss these references in our revised manuscript for an additional comparison!

Actually, I think the experiments should pay more attention to the special properties the proposed method can demonstrate.

This is an insightful observation. Our motivation for including heterophilous benchmarks was to highlight the method’s robustness in the absence of homophily, where traditional GNNs often degrade. We agree that l-ECT’s real strength lies in capturing local structure independent of homophily assumptions, yield broad utility. In the revision, we will reframe the narrative to better emphasize the general-purpose nature of l-ECT.

How does the computational complexity of computing l-ECTs scale with graph size and neighborhood size? This seems critical for practical applications.

Please refer to the response above.

Could you clarify your notion of graph embedding using node features? How do you handle cases where the embedding might not preserve edge relationships?

The embedding is created by assigning a node to the point in Euclidean space which is specified by its respective node feature vector. Edges are drawn between nodes if and only if there's an edge in the original graph. Since nodes may appear multiple times, edges might get lost by using this embeddings (see the paragraph before Thm.1). Therefore, in practice we use virtual edges to prevent such cases (i.e. an embedded node may occur multiple times as a neighbor). We will clarify this subtlety in our revised paper, thanks for spotting that!

What is the theoretical justification for l-ECTs working well on heterophilous graphs beyond the empirical results?

The core limitation of message passing in heterophilous settings lies in its aggregation mechanism, which tends to blur dissimilar feature signals. Our method bypasses aggregation entirely, instead constructing a vectorized topological summary of a node’s neighborhood. From a theoretical standpoint, Thm. 1 shows that l-ECT_1 encodes sufficient information to reconstruct the feature vectors of a node’s neighbors without aggregation. This property is critical in heterophilous settings where the difference between features (not their average) is informative.

How do you address the scenario where nodes have identical feature vectors but different structural roles in the graph?

Please see our response above.

We appreciate your detailed review and the encouragement regarding our theoretical and empirical contributions. If our revisions adequately address your concerns, we would be very grateful if you would consider raising your overall score. Thank you again for your time and constructive feedback! Please let us know if you have any other questions or concerns.

Best regards,

The Authors

Dear Reviewer,

we sincerely thank you for the thoughtful and constructive feedback. We are pleased that you found our paper to be well-written and our proposed method—the Local Euler Characteristic Transform (l-ECT)—to be novel and well-motivated. Below, we address your main concerns point-by-point and outline the changes we will make in the revised version.

The Texas and Wisconsin datasets are relatively small, and previous studies [1] have indicated that the Chameleon and Squirrel datasets contain significant amounts of duplicate data, which undermines their reliability. So the above datasets are not suitable for evaluating heterophilic GNNs any more, and I suggest that authors should consider more datasets proposed by [1] such as minesweeper, tolokers, questions. [1] Platonov et al. A critical look at the evaluation of GNNs under heterophily: Are we really making progress? ICLR 2023.

Thank you for pointing this out. We agree with your assessment of Chameleon and Squirrel and appreciate the recommendation. While our focus was to provide a broad comparison with commonly used benchmarks in heterophilic GNN evaluation, we recognize the importance of newer, cleaner datasets. In the revised version, we will include results on datasets like Minesweeper, Tolokers, and Questions as proposed in [1].

More heterophilic GNN baselines such as spectral-based models [2] [3] [4] and spatial-based models [5] [6] should be considered in the section 5. Besides, GraphSAGE is also a powerful baseline for heterophilic graphs. [...]

We appreciate the reviewer’s recommendation and agree that these are strong and relevant baselines in the context of heterophily-specific architectures. However, we wish to emphasize that our approach is not only proposed as a specialized architecture for heterophilic graphs, but rather as a general-purpose representation method that provides a robust and interpretable alternative to neighborhood aggregation. Our focus is on illustrating the effectiveness of l-ECT as a complementary or standalone representation, rather than competing directly with task-specific model architectures. That said, we agree that GraphSAGE is a widely adopted and illustrative baseline for generalization across homophilic and heterophilic settings. We will include GraphSAGE in our revised experiments, and clarify the scope of our contribution with respect to architecture design for heterophilic graphs.

In Table 7 of Appendix A2.3, all the listed models should be accompanied by their corresponding references.

Thank you for spotting this oversight. We will revise Table 7 to include full citations for each model.

We thank the reviewer again for acknowledging the novelty, motivation, and clarity of our work. We believe that the l-ECT offers a fundamentally new perspective on graph representation learning—particularly by preserving geometrical-topological information at the local scale, a signal that is often diluted by aggregation-based models. If our revisions adequately address your concerns, we would be grateful if you would consider raising your overall score. We are happy to address any additional questions.

Best regards,

The Authors