Making Classic GNNs Strong Baselines Across Varying Homophily: A Smoothness–Generalization Perspective

Q1: What is the difference between the “smoothness-generalization dilemma” and the known problem of oversmoothing?

We appreciate the reviewer’s thoughtful question and the opportunity to clarify this point.

The classical over-smoothing problem refers to a phenomenon predominantly studied in homophilic graphs, where repeated message passing over high-order neighborhoods leads to node representations becoming overly similar or even indistinguishable. This convergence diminishes discriminative power and degrades performance, particularly in deeper GNNs.

In contrast, the smoothness-generalization dilemma proposed in our work is a broader and more general phenomenon. It encapsulates a fundamental trade-off between enforcing smoothness and maintaining strong generalization across diverse graph structures, including both homophilic and heterophilic graphs, and across both low-order and high-order neighborhoods.

Specifically, we show that in heterophilic graphs, even low-order neighborhoods can exhibit complex neighborhood class distributions. In such cases, smoothing acorss low-order hops, even at shallow layers, can distort informative signals, harming generalization. Furthermore, under distributional shifts induced by noise or sparsity, such poor generalization hurts performance, regardless of whether over-smoothing occurs.

In summary, over-smoothing can be regarded as a special case of the broader smoothness-generalization dilemma across varying homophily and neighborhood orders.

Q2: Oversmoothing always implies a drop in performance, why is the tie to OOD generalization particularly interesting?

We thank the reviewer for raising this important question.

While it is well established that over-smoothing leads to performance degradation, our work reveals a deeper underlying mechanism, namely, the tension between smoothness and generalization, which governs this behavior and extends beyond classical interpretations.

(1) A unified perspective on varying homophily and oversmoothing. Traditionally, over-smoothing and homophily variation have been treated as separate challenges in GNN design: the former associated with high-order aggregation in homophilic graphs, and the latter with poor representation in heterophilic graphs. Our formulation unifies both phenomena under the broader lens of the smoothness-generalization dilemma, which captures the fundamental trade-off between enforcing smoothness and achieving robust generalization under neighborhood class distribution shifts.

This perspective explains why:

• In high-order neighborhoods, smoothness impairs generalization due to over-smoothing.
• In low-order, heterophilic settings, smoothness also harms generalization, even though classical over-smoothing does not occur.

By identifying a shared root cause (e.g., the conflict between smoothness and generalization of task-relevant signals under structural variability), our framework offers theoretical insights that span different homophily regimes and neighborhood orders, providing practical guidance for architecture design.

(2) We further emphasize that the negative impact of over-smoothing is not solely due to the convergence of node representations (e.g., via A^kX). Instead, the core issue lies in the inability to balance discriminative expressiveness with generalization.

In theory, one might attempt to recover expressiveness after smoothing (e.g., through large transformation weights W^(i)). However, our analysis shows that while such compensation may improve training performance, it often worsens generalization, particularly in graphs of varying homophily, where minor structural perturbations can cause significant shifts in local distributions [1].

This explains a widely observed empirical phenomenon: GNNs can perform well on the training set despite over-smoothing, yet exhibit sharp performance drops on the test set. Our formulation provides a theoretical explanation for this gap, demonstrating that generalization failure—not representational collapse alone—is the crux of the issue. This makes the connection to OOD generalization both novel and essential for understanding the practical limitations of current GNNs.

Q3: What parts of this dilema are novel and what parts have been previoulsy studied?

We appreciate the reviewer’s question and the opportunity to clarify the novel contributions of the dilemma.

While the issue of oversmoothing in high-order neighborhoods, particularly in homophilic graphs, has been previously investigated, our work extends beyond this established line of inquiry by identifying a general smoothness–generalization trade-off that emerges not only in high-order settings but also across both low- and high-order neighborhoods, and under varying degrees of homophily and structural disparity.

Notably, we show that performance degradation can occur even in the absence of classical oversmoothing, uncovering new failure modes of GNNs that are not explained by existing theories. This observation is particularly important for graphs of varying homophily, where conventional wisdom about neighborhood similarity does not hold, yet degradation still arises through a different mechanism tied to the message passing itself.

Our contribution lies in reframing and extending oversmoothing and varying homophily within a broader and more general theoretical framework, allowing us to explain performance degradation at different neighborhoods and varying homophily under a single analytical lens. This unification enables a consistent understanding of GNN behavior across different regimes of homophily, depth, and structural variation.

We believe this reframing offers important insights for both theoretical understanding and practical GNN design.

W1: ...does not convey the amount of existing work connecting oversmoothing to generalization, risk or decreases in test accuracy (Keriven 2022, Wu et al. 2023, Oono & Suzuki 2020 to name a few)...it should be clearer that previous works have studied how oversmoothing hurts performance in GNNs and why the distinction of focusing on OOD generalization is important...

We sincerely thank the reviewer for this insightful comment.

We fully agree that our current draft could better contextualize our contribution within the existing body of work connecting oversmoothing to generalization performance, and we will revise the manuscript accordingly to explicitly cite and discuss the relevant studies, including Oono & Suzuki (2020), Keriven (2022), and Wu et al. (2023).

Our work builds upon and extends this line of research in the following key ways:

• Theoretical Advancement: We adopt the analytical framework of Oono & Suzuki (2020), as discussed in our preliminaries, and derive a tighter upper bound that explicitly links smoothness to generalization performance, offering a clearer and more refined understanding of the trade-off involved.
• Extension to Varying Homophily: While prior works focus on oversmoothing primarily in homophilic and high-order settings, we demonstrate that performance degradation also arises under varying degrees of homophily, including low-order neighborhoods, which cannot be captured by classical oversmoothing analyses alone. This significantly broadens the scope of prior studies.
• Architectural Contribution: Unlike all the three works, which focus on theoretical analysis without architectural innovation, we propose a unified IGNN framework that is empirically validated. Our design incorporates three variants tailored to mitigate the smoothness–generalization trade-off under diverse homophily regimes.
• Broader Perspective: Keriven (2022) and Wu et al. (2023) focus on oversmoothing without considering structural disparity or varying homophily. Wu et al. (2023), in particular, restricts its analysis to attention-based GNNs and task-specific test risk without offering architectural strategies. In contrast, our IGNN framework provides a generalizable design principle that addresses smoothness and generalization across different architectures and homophily conditions, thereby offering a unified perspective for future GNN development.

We will ensure that these distinctions are made clearer in the revised manuscript.

W2: While this is clearly conveyed in the Appendix (Table 9), I believe it should be made clear in Section 5 and perhaps in the introduction. Mentioning some existing methods that follow each design principle could also help convey the intuition to readers familiar with previous methods.

We appreciate the reviewer’s helpful suggestion regarding the clarity.

While Table 9 in the Appendix provides a comparison that illustrates how existing architectures satisfy one or two of the proposed design principles, we agree that this distinction should be more prominently and explicitly presented in the revised main paper. We will revise Section 5 to clearly highlight that c-IGNN is the first architecture to simultaneously satisfy all three principles, in contrast to prior methods that typically meet only a subset.

We appreciate the reviewer’s thoughtful feedback, which will help improve the clarity of our presentation.

We thank the reviewer for the insightful comments. Please let us know if you have any further questions. We’d be happy to clarify.

References:

[1] Haitao Mao, Zhikai Chen, et. al. Demystifying structural disparity in graph neural networks: Can one size fit all? Advances in neural information processing systems, 36, 2024

We sincerely thank the reviewer for the thoughtful and constructive comments.

1. Discussion of Related Works

We have carefully examined [1, 2] and found they did not talk about generalization. This insteresting finding makes us curious about how they jointly solve heterophily and oversmoothing without the help of generalization we used in our dilemma. Below are the details.

• Neural Sheaf Diffusion in [1] introduces a novel topological formulation that attributes both heterophily and oversmoothing to the underlying "geometry" of the graph, using the language of sheaves. Although their framework successfully mitigates oversmoothing by allowing richer structural encoding via sheaves, the authors themselves note (Page 10) that their work “does not address the generalization properties of sheaves.”

Interestingly, [1] observes that “Sheaf Convolutional Networks (SCNs) are generally not constrained to decrease the Dirichlet energy” in page 7, which indirectly hints at improved generalization. This connection, however, is not made explicit. As the Dirichlet energy is tightly coupled with smoothness [3], the observation suggests that SCNs might inherently avoid enforced smoothness, allowing room for better generalization. Nevertheless, this interpretation remains implicit and underexplored in their theoretical treatment.

In summary, while [1] shares a high-level concern with oversmoothing and heterophily, their framework is algebraic-topologically motivated and does not engage with the generalization aspect central to our dilemma.

• Reverse Diffusion GNNs in [2] adopt a different strategy by reversing the graph diffusion process to recover distinguishable node representations by learning the states in the past. This approach effectively addresses oversmoothing and heterophily by countering the overly smilar representations introduced by forward aggregation and allowing long-distance nodes to interact with each other in the reverse process. However, the motivation in [2] is largely architectural, and the connection to generalization is not disscussed.

Their success can be seen as aligned with our theoretical insight: namely, that excessive smoothness from higher-order aggregation suppresses generalization. The reverse diffusion technique can be interpreted as alleviating the rigidity of forward aggregation, thus indirectly supporting generalization across varying neighborhood orders.

Taken together, [1] and [2] offer important complementary perspectives. However, neither formulates nor explicitly analyzes the smoothness-generalization dilemma as a unified cause behind oversmoothing and heterophily. Our framework fills this gap by exposing how the trade-off between smoothness and generalization governs performance.

2. How the "smoothness-generalization dilemma" will be introduced in the revision

In the updated version of the paper, we will:

• Expand the Related Work section to include a dedicated subsection on oversmoothing, explicitly discussing and citing the works the reviewer pointed out in the rebuttal and their relation to our framework w.r.t. genralization and heterophily.
• Clarify the novelty of our dilemma by contrasting it with existing lines of research. Prior works either:
  • Study oversmoothing and generalization together, but leaving out the discussion of varying homophliy;
  • Address heterophily and oversmoothing jointly through architectural innovations, while leaving the implications for generalization unexamined.
• Position our dilemma as a unifying theoretical lens that explains how oversmoothing, poor generalization, and heterophily are interconnected manifestations of the same underlying trade-off. By disentangling this interplay across varying homophliy, hop orders and structural disparity, we aim to establish a comprehensive theoretical paradigm.

This in-depth exchange has greatly enriched the understanding of the dilemma. Initially, our primary focus was on resolving varying homophily across hops, treating oversmoothing as a secondary issue at high-hop orders. Upon revisiting the broader literature, we now recognize that our analysis may offer a unifying explanation across different regimes and design spaces for oversmoothing.

We sincerely appreciate the reviewer’s suggestions, which have helped clarify the broader implications of our analysis. We hope that these discussions can address the concerns and are happy to incorporate any further feedback, believing that these revisions will significantly strengthen the paper.

[1] Bodnar, C., et al. Neural sheaf diffusion: A topological perspective on heterophily and oversmoothing in gnns. NeurIPS 2022.

[2] Park, M., et al. Mitigating oversmoothing through reverse process of gnns for heterophilic graphs.

[3] Zhou, Kaixiong, et al. "Dirichlet energy constrained learning for deep graph neural networks." NeurIPS 2021.

W1: some recent approaches that aim to cater to both homophily and heterophily but are missing...FAGCN [1] and \omega GAT [2]**...**CO-GNNs[3]...

Thank you for the valuable suggestion. We have now included the results of FAGCN [1], ω-GAT [2], and Co-GNN [3] as follows, which compares the top-10 methods in table 3 (Overal performance in the manuscript) including Polynormer and GOAT as suggested by other reviewers.

These methods achieve mid-range average ranks overall. We believe they are still worth discussing due to the following reasons:

(1) FAGCN and ωGAT are both variants of the a-IGNN family, which achieves adaptive filtering capabilities. As shown in our experimental section, a-IGNN variants (e.g., GPRGNN, OrderedGNN, DAGNN, ACMGCN, etc.) exhibit significantly diverse performance, which we attribute to their customized attention-based designs. This highlights the sensitivity of their results to hand-crafted attention mechanisms.

(2) Co-GNN, on the other hand, is a node-level dynamically adaptive message-passing method.It demonstrates unique advantages in neighbor selection and long-range tasks. For instance, in the roman-empire dataset, a long-chain language-token graph generated from word sequences and syntactic dependencies, Co-GNN outperforms many baselines due to its fine-grained adaptivity, while the graph transformer Polynormer shows great advantages probably due to the success of the transformer architecture in language tasks.

We will incorporate the results and cite these methods in the revised version accordingly.

W2: ..some statements and figures are confusing (See Questions 1 and 3)...

We thank the reviewer for the helpful suggestions on improving the clarity of specific statements and visual elements. Below, we provide point-by-point responses and will revise the corresponding text and figures to improve clarity in the final version.

Q1: What is the source of Fig 1 (c) and (d) and what does the zig-zag-like pattern in them signify?

We appreciate the reviewer’s attention to Fig. 1 (c) and (d). These plots are designed to offer visual intuition for the core insight of our work.

Fig. 1(c) illustrates the emergence of a dilemma between smoothness and generalization in classic GCN message passing across varying levels of homophily of difference hops.

Fig. 1(d) illustrates the alleviation of this dilemma through decoupled adaptive smoothness and generalization, demonstrating how an ideal architecture would balance smoothness and generalization dynamically.

These figures are conceptual illustrations intended to reflect our theoretical findings. We will revise the captions to make these intentions clearer.

Q2: By 'smoothness', do the authors refer to the smoothing induced by the transformation step or the aggregation step

This is an excellent question. In our theoretical analysis, we consider smoothness as influenced by both the aggregation step and the transformation step.

In particular, the smoothness indicator d_M(H^(l)), as discussed in Corollary 3.2 and Theorem 4.1, is bounded by: the second largest eigenvalue of the adjacency matrix (capturing the aggregartion effect), and the maximum singular value or Lipschitz constant of the transformation matrix W^(l) (capturing the effect of the transformations).

Therefore, our analysis integrates both sources of smoothness. We will revise the explanation to more clearly reflect this dual contribution.

Q3: "Consequently, either oversmoothing or poor generalization will occur at large k". I believe the whole point was a smoothness-generalization trade-off, i.e. oversmoothing would lead to poor generalization, so how is this an either-or situation?

We appreciate the reviewer’s thoughtful question and the opportunity to clarify this point.

While it is true that oversmoothing typically leads to poor generalization, our theoretical framework reveals a more nuanced dilemma:

• If the model attempts to counteract oversmoothing (i.e., to recover discriminative representations from the over-smoothed A^kX) by increasing the spectral norm of W^(i), this leads to a larger Lipschitz constant, which in turn worsens the generalization.
• Conversely, if the model limits the norm of W^(i) to maintain a low Lipschitz constant and preserve generalization ability, it cannot effectively reverse the over-smoothed A^kX, resulting in poor discriminability of node embeddings.

In this sense, the phrase emphasizes that attempting to fix one inevitably worsens the other. This constitutes the core of the smoothness–generalization trade-off we identify: mitigating oversmoothing through amplification degrades generalization, while preserving generalization risks severe embedding collapse.

We will revise the phrasing in the final version to make this interplay clearer and avoid the false implication of mutual exclusivity.

Q4: Could dirichlet energy be used to showcase the trade-off between smoothness and generalization empirically and how it varies for IGNN?

Thank you for this insightful suggestion. Indeed, as shown in Lemma 1 of [1], the Dirichlet energy at the k-th layer of a GCN can be upper and lower bounded by the eigenvalues of the (augmented) normalized Laplacian and the singular values of the transformation matrices. This implies a theoretical connection between Dirichlet energy and the smoothness-generalization we found in our framework.

While we focus on a different smoothness measure in this work, the Dirichlet energy may indeed serve as an alternative empirical indicator of the dilemma and may help further validate and extend our findings. We consider this a promising direction for future research and will mention this possibility in our discussion section.

Q5: Do the architectures like GAT-sep and GATE-sep ... also incorporate all three principles?

We thank the reviewer for this important question.

No, architectures such as GAT-sep and GATE-sep do not satisfy all three of our proposed design principles. Specifically, while they implement ego-neighbor separation by isolating the ego features H^(0) from aggregated multi-hop features H^(K), they do not realize inceptive message passing across intermediate neighborhood hops H^(1)~H^(k). This limits their ability to achieve the fine-grained neighborhood-wise generalization that our design principle emphasizes.

It is worth noting that ego-neighbor separation has indeed become a common technique in heterophilic GNNs, first formalized in H2GCN, and subsequently adopted by many follow-up works such as GPR-GNN, ACM-GCN, and even some recent graph transformer implementations. However, our experiments demonstrates that ego-neighbor separation alone is not sufficient to achieve optimal performance. Several models that incorporate this technique still underperform compared to those that fully embrace more principles of inceptive message passing.

In summary, while ego-neighbor separation is benefit for heterophilic settings, GAT-sep and GATE-sep do not fulfill the full set of principles proposed in our framework.

We thank the reviewer for the constructive comments. We’re happy to address any remaining concerns.

References:

[1] Zhou, Kaixiong, et al. "Dirichlet energy constrained learning for deep graph neural networks." Advances in neural information processing systems 34 (2021): 21834-21846.

We are sincerely grateful for your considerate and insightful feedback, as well as for raising the score. Your thorough review and the time devoted to reviewing our work have meaningfully enhanced its quality. All suggestions will be carefully incorporated into the revised version.

W1: the complexity of IGNN should be carefully evaluated and discussed.

We appreciate the reviewer’s suggestion for the complexity. Due to space constraints in the main paper, we provided model complexity, parameter count, and runtime analysis in Appendix B. Below, we summarize the key findings for convenience. For details, please refer to Appendix B.1.

• r-IGNN: The residual connection does not significantly change the complexity compared to GCN. If the representation of the previous hop also has a transformation in the residual connection (which we do not apply in our implementations), then it will require more parameters.
• a-IGNN: The model adaptively determines α_v^(k) for each node, which slightly reduces the parameter count (O(K · 2F)). Its per-layer complexity is lower than others, but still scales with the number of edges and nodes.
• c-IGNN: The explicit multi-hop aggregation increases computational cost compared to GCN. The complexity grows with K, making it more expensive as the number of hops increases. However, it enjoys hop-wise distinct generalization and overall generalization, which holds significance in GNN universality across varying homophily.
• Fast c-IGNN (see Appendix B.1): By decoupling aggregation into preprocessing, it shifts the expensive aggregation operations outside training. This makes it scalable for large graphs. Among these models, Fast c-IGNN achieves the best scalability by precomputing multi-hop aggregation. In contrast, a-IGNN and r-IGNN require more computational resources due to their recursive neighborhood aggregation.

We hope these analyses sufficiently address the reviewer’s concern.

W2: authors should provide comprehensive tuning results.

We thank the reviewer for highlighting the importance of hyperparameter tuning.

(1) While Table 11 in Appendix E.2.2 provides the final configurations, we fully agree that reporting the tuning process is also important. To support reproducibility, we have already released the detailed search space definitions in our anonymous repository as presented in Appendix E.2.2 (line 840). Based on this, researchers can readily reproduce the tuning process using standard hyperparameter optimization frameworks such as Optuna [1], in combination with our publicly available codebase.

(2) In response to the reviewer’s suggestion, we will additionally include ready-to-run Optuna tuning scripts in the revised version of the code repository. Due to the NeurIPS 2025 anonymity policy, we are currently unable to update the anonymous repository during the rebuttal period. We apologize for this temporary limitation, but we will ensure these scripts are included in the revised version.

Q1: Concerning the hope-wise analysis, the two evaluated datasets are heterophily datasets, are there any homophily datasets or larger scale heterophily datasets?

We thank the reviewer for the thoughtful question.

(1) We would like to clarify that the datasets used in the hop-wise analysis already cover both homophilic and heterophilic regimes (line 386-392). Specifically:

• The Photo dataset has a high homophily ratio of 0.8272, representing a typical homophilic graph.

• The Squirrel dataset has a low homophily ratio of 0.2072, representing a heterophilic graph.

This combination allows us to evaluate the behavior of different IGNN variants across contrasting structural regimes. As shown in Figure 4, our variants consistently show a first increase then steady performance trend across a wide range of hops (from 0 to 64), demonstrating strong adaptability in both settings.

(2) In response to the reviewer's suggestion regarding larger-scale heterophilic datasets, we have additionally showcased a hop-wise analysis on Pokec, a real-world social network dataset with heterophily (homophily ratio ≈ 0.44).

The results exhibit a notable increase in performance that stabilizes thereafter, highlighting the effectiveness in improving hop-wise and overall generalization.

Q2: why is GCN+SN not evaluated? And why GCN+SN+RN not evaluated

We appreciate the reviewer’s thoughtful questions.

We did not include GCN+SN or GCN+SN+RN in our ablation studies because the SN module is designed to operate in conjunction with the IN structure. Specifically, without the IN architecture, the transformation layers are shared or decoupled across all hops, and thus SN cannot effectively disentangle the transformations of different neighborhood orders. As a result, without IN, applying SN in isolation or in combination with RN would not preserve its intended design or functionality.

However, to assess the contribution of SN in a meaningful context, we evaluated its addition to the IN and IN+RN settings, as shown in Table 5. In both cases, we observe consistent performance gains, validating SN’s utility in adapting to varying homophily and improving generalization.

In short, SN is not a standalone module but rather one that is structurally dependent on the IN design. We will clarify this architectural dependency and our evaluation choices more explicitly in the revised version.

We appreciate your time and are open to any follow-up questions.

References:

[1] Akiba, Takuya, et al. "Optuna: A next-generation hyperparameter optimization framework." Proceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining. 2019.

W1: ...the paper does not include efficiency comparisons...

We thank the reviewer for pointing out efficiency comparisons. We apologize for not making this aspect more prominent in the main manuscript due to space constraints, as we have provided detailed analyses of model complexity, parameter counts, and runtime comparision in Appendix B, and summarize the key findings of efficiency comparisons here for convenience.

An empirical analysis of training runtimes across the top models in overall performance is conducted. We measured the average training time over 100 epochs under two settings:

• Squirrel (heterophilic, full-batch): hops (2, 8, 16, 32)
• OGB-arxiv (homophilic, full-batch): hops (2, 10).

Squirrel (seconds):

OGB-arxiv (seconds):

The top 5 most efficient models are highlighted in bold. These results demonstrate that our IGNN variants, especially (fast) c-IGNN with a preprocessing and caching technique to decouple expensive aggregation operations from training (line 740-753), consistently achieve competitive or superior training efficiency.

We hope these additional analyses sufficiently address the reviewer’s concern.

W2: suggest...adopt dataset splits aligned with prior work...roman-empire and cham-f...

We thank the reviewer for this valuable suggestion.

(1) Justification for Split Strategy: Our choice was guided by our theoretical emphasis on generalization, an aspect highly sensitive to dataset splits. Prior work [4] has demonstrated that different split strategies can exhibit substantial variation in structural distributions, which in turn can significantly affect generalization behavior. To mitigate these confounding effects, we adopted a unified 10× random split scheme with a 48%/32%/20% train/validation/test ratio [5,6,7,8]. This reduces variance across datasets stemming from heterogeneous splitting policies in earlier work.

(2) Evaluation on Standard Splits: In response, we have also re-evaluated the top 5 baselines with the same parameters on the public splits of roman-empire and cham-f as follows. The results confirm that the observed performance remains consistent with those from our splits. The similarity in partition ratios contributes to the consistency.

• cham-f |splits|<ours, 48%/32%/20%>|<public [1], 46%/32%/22%>|our rank|public rank| |-|-|-|-|-| |SGFormer|44.44±3.01|43.30±3.92|4|3| |UniFilter|46.07±4.74|43.76±4.00|2|2| |OrderedGNN|42.13±3.04|42.57±3.82|7|6| |SIGN|44.66±3.46|43.02±2.98|3|4| |IncepGCN|43.31±2.18|42.98±2.75|5|5| |r-IGNN|42.47±3.06|41.64±2.62|6|7| |a-IGNN|40.45±3.80|41.49±3.78|8|8| |c-IGNN|47.13±5.18|46.50±3.77|1|1|

• roman-empire |splits|<ours, 48%/32%/20%>|<public [1], 50%/25%/25%>|our rank|public rank| |-|-|-|-|-| |SGFormer|75.71±0.44|75.74±0.55|7|7| |UniFilter|74.90±0.91|75.10±0.69|8|8| |OrderedGNN|82.65±0.91|83.39±0.33|2|2| |SIGN|81.56±0.57|79.46±0.68|3|5| |IncepGCN|80.97±0.49|80.73±0.42|4|3| |r-IGNN|80.67±0.47|80.34±0.40|5|4| |a-IGNN|78.01±3.12|79.02±3.00|6|6| |c-IGNN|86.07±0.60|86.54±0.54|1|1|

In the revised version, we will report results under both split protocols.

W3: ...omits several related works. Notably, [2] and [3] should be included...

We thank the reviewer for highlighting these important works. We report their performance in comparison with the top 10 baselines as follows. Their inclusion does not alter the overall conclusions of our study.

Notably, Polynormer shows a great advantage in roman-empire which was not seen in other datasets. We checked it and found it was a long-chain graph converted from words. This aligns with the strengths of transformers in natural language processing. As we focus on general graphs and IGNNs' avg_rank still show consistent advantages, we leave the language graphs to future studies.

We will revise the manuscript to include proper citations and discussions.

W4: the connection between these theoretical insights and the model design is not sufficiently clear...

We thank the reviewer for the opportunity to clarify the connection between our theoretical analyses and the architectural design.

(1) Theoretical Objectives: Our theoretical investigation highlights the smoothness-generalization dilemma under varying homophily across hops, demanding (a) resolve the trade-off, (b) adaptive smoothness that aligns with varying homophily, (c) strong generalization across diverse structural conditions.

(2) Architectural Realization: Our IGNN introduces three key principles

1. IN: Decoupling dilemmas across hops (Lines 284–288). While fully decoupling smoothness and generalization within message-passing is intractable due to their coupled factors (A,k,W), we address this by decoupling the dilemmas across hops. The Inceptive Aggregation (IN) enables each hop to regulate its own smoothness-generalization trade-off independently, without interference from other hops. This partial decoupling ensures hop-specific adaptation.
2. RN: Enabling adaptive smoothness (Section 5.2.1). To ensure that each hop can flexibly address its own homophily, we introduce three RN modules. We theoretically show that these RN modules, when built atop IN, endow each hop with adaptive smoothness (Theorem 5.1), a key requirement for varying homophily.
3. SN: Enhancing generalization (Section 5.2.2). Recognizing that different hops encounter distinct generalization challenges due to structural disparities (line 188-207), we further introduce SN. By isolating the transformation matrices across hops, SN enables independent generalization capacities per hop (distinct Lipschitz constant). Our theoretical results demonstrate that SN leads to a shrunk overall Lipschitz constant (Theorem 5.3), implying better generalization for the entire model.

This design is thus a direct and principled instantiation of our theoretical insights, ensuring that the dilemma is mitigated in a hop-aware and structure-sensitive manner.

Minor: It is recommended that the authors explicity inelude the intermediate variables in Table 2...

Thank you for your suggestion. We will carefully revise it to ensure better clarity.

Thank you again for your constructive comments. Please let us know if you have any further questions. We’d be happy to clarify.

[4] Haitao Mao, et al. Demystifying structural disparity in graph neural networks: Can one size fit all? NeurIPS 2024

[5] Pei, Hongbin, et al. GEOM-GCN: GEOMETRIC GRAPH CONVOLUTIONAL NETWORKS. ICLR 2020.

[6] Yan, Yujun, et al. Two sides of the same coin: Heterophily and oversmoothing in graph convolutional neural networks. ICDM 2022.

[7] Song, Yunchong, et al. "Ordered GNN: Ordering Message Passing to Deal with Heterophily and Over-smoothing." ICLR 23.

[8] Zhu, Jiong, et al. Beyond homophily in graph neural networks: Current limitations and effective designs. NeurIPS 2020.

Thank you very much for your thoughtful and constructive feedback, as well as for raising your score. We deeply appreciate the time and care you invested in reviewing our work, which has meaningfully improved its quality. We will incorporate the suggested changes into the revised version.