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# System prepended metadata

title: ICLR 2026 Response

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# ICLR 2026 Response
# Author Final Remarks (영문; characters: 4,953/5,000)
We thank the reviewers for their thoughtful feedback. Below, we briefly summarize the motivation and main contributions of our work and describe how we have addressed the key concerns raised during the review process to improve the manuscript.

---
### **Motivation & Design Rationale**
To clarify our motivation, we summarize the key points below as reflected in the revised manuscript. We hope that this framework will further support future QR-based UQ research in GNNs.

**```Limitations of Conformal Prediction (CP) in graph settings```**
- Message passing in graphs breaks exchangeability, and the post-hoc nature of CP prevents it from modeling graph dependencies during training, leading to fundamental limitations for CP-based methods.

**```Advantages of Quantile Regression (QR) for uncertainty estimation```**
- QR estimates conditional quantiles directly during training, eliminating the need for post-hoc calibration and enabling flexible modeling of asymmetry and heterogeneity—both of which are important for graph-structured data.

**```Absence of QR-based GNN-UQ methods and limitations of existing approaches```**
- QR-based UQ for GNNs has not been systematically explored, so we implemented the QR baselines.
- Although these QR models perform well with MLPs, they became unstable under graph dependencies, and RQR showed persistent quantile crossing. This motivated the adjusted RQR$^{adj.}$ which adds an ordering penalty for fair comparison.

**```Necessity and role of the QpiGNN design```**
- To address these limitations, we introduce QpiGNN, which uses a dual-head architecture and a joint loss tailored to graph settings.
- By separating prediction and uncertainty, QpiGNN avoids representational interference and reduces quantile instability under graph dependencies.

---
### **Contributions & Strengths**
Below are the main contributions of our work, several of which were also noted positively by reviewers.

**```S1. Clarity```**
- The paper clearly presents the problem setup and the motivation for the quantile-free formulation, and it differentiates our approach from related work—particularly QpiGNN—in a transparent manner.

**```S2. Motivation```**
- We highlight UQ in GNNs as an important yet understudied problem, identify the structural limits of quantile-based and conformal methods, and explain why a quantile-free approach naturally fits graph-structured data.

**```S3. Novelty```**
- Our dual-head quantile-free architecture separates prediction and uncertainty estimation, mitigating non-crossing issues and improving GNN stability, while providing a simple and practical UQ method that operates without quantile supervision or resampling.

**```S4. Theory```**
- We ground our method theoretically by interpreting the joint loss via Lagrangian relaxation, clarifying the coverage–width trade-off, and providing convergence, sensitivity, and both asymptotic and finite-sample guarantees under realistic assumptions.

**```S5. Experiments```**
- We evaluate QpiGNN across diverse synthetic and real datasets, showing consistent performance under noise and various distribution or graph-structure shifts. Ablations on architecture and sensitivity further support its robustness.

---
### **Reviewer Concerns & Responses**
We summarize the main concerns raised by the reviewers and describe how we addressed them in the revised manuscript.

**```C1. Distinction from RQR```**
- QpiGNN is not a simple RQR variant; it addresses key sources of instability in QR-based UQ on graphs, including residual correlation and loss of exchangeability.
- Its dual-head, disentangled design is fundamentally different from RQR’s residual-modeling approach. Implementing RQR revealed quantile crossing, motivating the adjusted RQR$^{adj.}$-GNN. The revision clarifies these differences.

**```C2. Theoretical Validity under GNN Dependencies```**
- Our theory avoids independence assumptions and relies only on a bounded-difference condition for message passing.
- It aligns with McDiarmid-type guarantees, and we show coverage stability of $1/N + \delta_G$ across diverse graphs.
- We note that the bounded-difference term may vary in dense or high-degree graphs.

**```C3. Experimental Enhancements```**
- The revision highlights the existing dual-head ablation, Bayesian baseline, and real-data ablation more clearly.
- We also added L1–L2 penalty comparisons, an Evidential Regression baseline, and analyses under varying target coverage.

**```C4. Scalability and Generalization```**
- QpiGNN retains the linear scalability of standard message-passing GNNs and runs stably on large graphs.
- Its dual-head quantile-free design is task-agnostic and extends to other GNN tasks by replacing only the encoder. 
- We also note dynamic graph UQ as a future direction.

---
We thank the reviewers once again for their valuable feedback and hope that our contributions and revisions have been clearly and satisfactorily addressed.

Best regards,

Authors

# Author Final Remarks (국문)
리뷰어 여러분의 심도 있는 피드백과 건설적인 논의에 감사드립니다. 아래에서는 본 연구가 해결하고자 한 핵심 동기와 주요 기여를 간략히 정리하고, 리뷰 과정에서 제기된 핵심 우려들에 대해 어떻게 대응하고 논문을 개선했는지 AC께 명확히 전달드리고자 합니다.

### From Motivation to QpiGNN Design or Motivation and Design Rationale
본 연구의 동기를 명확히 전달하기 위해 다음과 같이 정리하며, 해당 내용은 수정본에 반영하였습니다. 

```1. 그래프 환경에서 Conformal Prediction(CP)의 한계```
* 그래프에서는 메시지 패싱으로 인해 교환 가능성 가정이 성립하기 어려우며, CP는 post-hoc 방식이므로 학습 과정에서 그래프 종속성을 반영할 수 없습니다. 이러한 구조적 제약은 CP 기반 접근의 적용성에 근본적 한계를 드러냅니다.

```2. Quantile Regression(QR)을 활용한 불확실성 추정의 장점```
* QR은 학습 과정에서 조건부 분위수를 직접 추정하므로 post-hoc 보정이 필요 없으며, 분포의 비대칭성·이질성을 유연하게 반영할 수 있습니다. 이는 그래프 데이터에서 구조적 불확실성을 모델링하는 데 중요한 역할을 합니다.

```3. QR 기반 GNN UQ 연구의 부재와 기존 방법의 한계```
* QR기반의 GNN-UQ 방법은 기존 문헌에 체계적으로 존재하지 않아, 이에 대한 실증적 분석을 위해 QR baseline을 GNN에 직접 구현했습니다 (SQR-GNN과 RQR-GNN). 
* 그 결과, MLP에서는 우수한 QR 모델들이 그래프 종속성 하에서는 불안정해지고 구조적 제약에 취약하다는 점을 확인했습니다. RQR의 경우 quantile crossing 문제가 발생하여, 공정한 비교를 위해 ordering penalty를 추가한 RQR$^{adj.}$도 별도로 구성했습니다.

```4. QpiGNN 설계의 필요성과 역할```
* 이러한 한계를 해소하기 위해 우리는 그래프 특성에 적응적인 dual-head 아키텍처와 disentangled joint loss를 설계한 QpiGNN을 제안합니다. 이 구조는 예측과 불확실성 추정을 동일한 표현 공간에서 동시에 학습시키는 기존 QR 계열 모델과 달리, 두 기능을 명확히 분리하여 표현 충돌과 quantile instability를 근본적으로 줄이고, 그래프 환경에서 quantile 안정성을 개선하도록 설계되었습니다. 
* 우리의 연구가 향후 QR 기반 GNN-UQ 연구의 기초를 다지는 역할을 하기를 기대합니다.

---
### Contributions
아래는 본 논문의 주요 기여이며, 리뷰 과정에서 여러 리뷰어가 긍정적으로 평가한 부분들입니다.

**```S1. Clarity```** 
문제 설정과 quantile-free formulation의 동기가 명확하게 정리되어 있으며, 관련 연구—특히 OpiGNN과의 차이점—가 분명하게 제시되어 있습니다. 또한 synthetic·real 데이터셋을 균형 있게 구성해 전체적인 서술이 읽기 쉽다는 평가를 받았습니다.

**```S2. Motivation```** 
GNN에서의 UQ가 중요한 미해결 문제임을 명확히 제시하고 있으며, 기존 quantile 기반·conformal 접근의 구조적 한계를 짚어 그래프 환경에서 quantile-free 접근이 자연스럽다는 점을 설득력 있게 설명하였습니다.

**```S3. Novelty```** 
Dual-head quantile-free 구조를 통해 non-crossing 문제를 효과적으로 완화하고, quantile supervision이나 resampling 없이도 단순성과 실용성을 갖춘 새로운 방법을 제시합니다. 또한 예측–불확실성 분리를 통해 oversmoothing 문제까지 부분적으로 완화합니다.

**```S4. Theory```** 
Joint loss를 Lagrangian relaxation 관점에서 해석해 coverage–width trade-off를 명확히 설명하며, 비볼록 최적화 맥락의 수렴·민감도 분석을 포함해 이론적 근거를 제시합니다. 더불어 asymptotic coverage와 finite-sample concentration 결과도 현실적인 가정 아래 제공합니다.

**```S5. Experiments```**
다양한 synthetic·real 실험과 베이스라인을 포함한 체계적 비교를 수행하였으며, noise·distribution shift·graph structure shift 등 다양한 조건에서 일관된 성능을 보입니다. 또한 아키텍처 선택과 민감도에 대한 ablation을 통해 방법의 강건성을 지원합니다.

---
### Reviewer Concerns & Responses
아래에서는 리뷰어들이 제기한 주요 우려에 대해 명확한 해설과 함께, 이에 대응해 논문에 반영한 주요 수정 사항을 정리합니다. 자세한 내용은 개별 리뷰어 답변에서 확인하실 수 있습니다.

**```C1. RQR과의 차별성에 대한 우려```**
- QpiGNN은 RQR의 단순 변형이 아니라, 그래프 종속성 환경에서 QR 기반 UQ가 불안정해지는 핵심 요인(메시지 패싱으로 인한 잔차 상관, 교환 가능성 붕괴)을 해결하기 위해 설계된 quantile-free GNN 접근입니다.
- Dual-head + disentangled loss 구조는 residual modeling 방식의 RQR과 개념적으로 구분되며, RQR은 GNN에 적용된 선행 사례가 없어 우리가 baseline을 위해 직접 구현했습니다. 적용 과정에서 quantile crossing 문제가 일관되게 발생하여, 비교의 공정성을 위해 ordering penalty를 포함한 RQR$^{adj.}$-GNN을 별도로 구성했습니다.
- 수정본에서는 이러한 구조적·개념적 차이를 보다 명확히 드러내도록 관련 설명을 보강했습니다.

**```C2. 이론이 GNN 종속성을 충분히 반영하는지에 대한 우려```**
- 수정본에서는 제안한 이론이 독립성 또는 교환가능성 가정에 의존하지 않고, 메시지 패싱으로 인한 국소적 변화 전파를 제한하는 bounded-difference 안정성 조건만을 요구함을 명확히 하였습니다.
- 이는 현대 GNN 분석에서 널리 활용되는 McDiarmid-type finite-sample 보장과 정합적이며, Appendix에서는 empirical coverage가 $1/N + \delta_G$ 범위 내에서 안정성을 갖는다는 결과를 제시했습니다. Homophily, heterophily, 시공간 그래프 등 다양한 구조에서 동일한 경향이 실험적으로 확인됨도 보였습니다.
- 아울러 dense graph나 매우 높은 degree를 갖는 특수 구조에서 bounded-difference term이 달라질 수 있다는 한계도 명확히 기술했습니다.

**```C3. 실험적 보완 필요성```**
- 수정본에서는 기존에 포함된 dual-head ablation, Bayesian baseline, real-data ablation을 더 분명하게 드러나도록 구성했습니다. 
- 또한 리뷰어들의 요청에 따라 다음을 추가했습니다: L1 vs. L2 penalty 비교, Evidential Regression(ER) baseline, Baseline target coverage 변화 분석 등. 이러한 보완을 통해 QpiGNN의 설계적 선택과 안정성에 대한 근거를 강화했습니다.

**```C4. 확장성 및 일반화 가능성```**
- QpiGNN은 표준 메시지 전달 GNN과 동일한 연산 경로를 사용하며, 추가 반복 계산이나 보조 모듈이 없어 GraphSAGE와 유사한 선형 확장성을 유지합니다. 실제로 12K nodes와 200K+ edges 규모의 대규모 그래프에서도 BayesianNN 및 MC Dropout과 유사한 자원 사용량으로 안정적으로 동작하는 것을 확인했습니다.
- 수정본에서는 QpiGNN의 scalability와 task generalization을 보다 명확히 기술했습니다. 또한 dual-head quantile-free 구조는 task-agnostic하게 설계되어 인코더만 교체하면 분류·링크 예측 등 다양한 GNN 과제로 확장 가능합니다. Dynamic graph UQ는 의미 있는 향후 연구 방향으로 별도 언급했습니다.

리뷰어 여러분의 소중한 피드백에 다시 한 번 감사드리며, 본 연구의 기여와 개선 사항을 AC께서 명확하게 파악해 주시기를 바랍니다.

Best regards,
Authors

# 논문 수정 내용
- 논문 다시 읽어보기
- 그래프 UQ 강조를 위한 abs, intro 수정?
- RQR과 QpiGNN 간의 개념적 차이를 강조할 수 있게 추가 수정?

- ER 실험 추가 (0)
- L1 vs L2 실험 추가 (0)
- Ablation 그림 수정 (0)
- real ablation 내용 추가 (0)
- Baseline coverage/penalty -> hyperparameter 분석 appendix 추가 (0)
    -- UQ의 그래프 회귀에 대한 흥미로운 문제 강조: QpiGNN/RQR은 페널티에 둔감 (0)
    -- 원고에서 이러한 한계를 명확히 밝히고 잠재적인 확장 방안을 논의 (0)
- 동적 그래프 UQ 연구 -> future work 추가 (0) 
- dense graph나 매우 높은 degree를 가진 특수 그래프에서는 bounded-difference term이 달라질 수 있음을 limitations -> discussion에 문장 추가 (0)
- UQ에서 일반적으로 과소 적용을 failure로 모습 입장에서 평가함 -> discussion에 내용 추가 (0)
- RQR의 파레토 최적 관점에서 성능 평가 -> quantitative 결과의 footnote로 내용 추가 (0)
- 식 (4)의 추가 항은 그래프 구조를 활용하는 메커니즘이라기보다는 실용적인 해결책 -> 내용 추가 (0)
- 덧셈 페널티가 안정적인 동작을 제공합니다. 목표 범위가 좁을 때 곱셈 항이 안정적일 수 있다는 점을 인지함을 나타냄 (0)
- 유한 표본 안정성 보장과 점근적 LLN 논의를 명확히 구분하고 실제 그래프에서 유계 차분 가정을 뒷받침하는 경험적 증거를 강조
- 부록 B.5의 목적은 QpiGNN이 평가되는 영역에서 농도 기반 추론이 적절함을 입증하는 것 (0)
-  RQR과 QpiGNN 간의 개념적 차이는 단순히 구조적인 문제가 아니라, 그래프 의존성 하에서 분위수 없는 목적함수의 근본적인 차이 (0)
- 듀얼 헤드 기능적 동기를 더 잘 강조 (0)
- Eq (3) 또는 (4)에서 Eq (5)로 변경하는 장점; (4)로 학습된 OpiGNN(또는 (5)로 학습된 RQR)의 성능을 보여주는 ablation (0)
- SQR 붕괴는 주로 이종 그래프에서 발생한다는 점 (0) 


#
# Summary of ICLR response (5,000 characters에 맞춰야함 -> 댓글 여러개 가능!)
## Official Review by Reviewer Cv3a (4/3)
### Weakness 1 
While the task is interesting from the perspective of quantile prediction, the paper positions itself as an uncertainty quantification work. Therefore, comparisons with other established uncertainty quantification methods for regression—such as **evidential regression** or **Bayesian neural networks**—would provide a stronger baseline context.
> 분위수 예측 면에서는 흥미롭지만, 불확실성 정량화 분야에서 널리 쓰이는 대표적인 경쟁 방법들(예: Evidential Regression, Bayesian Neural Networks)과의 비교가 있으면 더 강력한 baseline context를 제공할 수 있음
> 답변: 이미 BayesianNN, MC Dropout, CF-GNN 등 핵심 UQ baseline을 모두 포함. Evidential Regression은 GNN setting을 전제로 설계된 방법이 아니라 직접 비교의 실효성이 낮음. 우리 연구의 비교군은 GNN 환경에서 그래프 회귀에서의 UQ라는 맥락에서 선택
> Evidential regression[4]은 Normal–Inverse-Gamma 분포의 ((\mu, v, \alpha, \beta))를 예측해 aleatoric·epistemic 불확실성을 추정하지만, 목표 coverage를 직접 조절하는 방식은 아닙니다. 맞을까요? 
이에 따라 본 연구에서는 예측 분산 $\sigma^{2}=\frac{\beta}{\alpha-1}$을 기반으로 Gaussian quantile을 적용해 대칭 구간을 구성해 비교하였습니다. 다만 evidential regression은 (\sigma^2 \to 0)으로 수렴하는 과신 문제가 잘 알려져 있고, 저희 실험에서도 지나치게 좁은 구간이 나타났습니다. 이를 완화하기 위해 기존 연구에서 사용하는 작은 분산 하한과 완만한 스케일 조정을 적용하여 $\sigma_{\text{adj}} = s \cdot \max(\sigma,\ \sigma_{\min})$ 분산 붕괴를 방지했습니다. 이 과정을 통해 evidential regression을 interval-prediction 기준에서 공정하게 비교할 수 있도록 하였습니다. 논문에 있는 결과와 취합한 비교 결과는 다음과 같습니다.
| Model                 | Basic     | Gaussian  | Uniform   | Outlier   | Edge      | BA         | ER         | Grid       | Tree       |
| --------------------- | --------- | --------- | --------- | --------- | --------- | ---------- | ---------- | ---------- | ---------- |
| BayesianNN            | 1.00/30.1 | 1.00/2.98 | 1.00/3.00 | 1.00/295  | 1.00/3.06 | 1.00/3.08  | 1.00/3.01  | 1.00/3.01  | 1.00/3.00  |
| MC Dropout            | 0.99/0.32 | 0.55/0.20 | 0.65/0.26 | 0.58/0.06 | 1.00/0.30 | 0.67/0.26  | 0.76/0.23  | 0.33/0.16  | 0.64/0.20  |
| CF-GNN                | 0.92/1.90 | 0.91/2.90 | 0.90/3.04 | 0.93/1.92 | 0.92/1.78 | 0.90/68.27 | 0.90/17.15 | 0.94/3.18  | 0.93/0.97  |
| Evidential Regression | 1.00/1.24 | 1.00/1.85 | 1.00/1.53 | 0.97/0.88 | 1.00/1.07 | 1.00/6.89  | 0.72/1.86  | 1.00/11.92 | 1.00/11.21 |
| QpiGNN(optimal)       | 0.90/0.30 | 0.95/0.64 | 0.93/0.62 | 0.90/0.49 | 0.94/0.54 | 0.98/0.48  | 0.98/0.63  | 0.99/0.93  | 0.96/0.39  |
### Weakness 2
**The motivation for designing this framework specifically for graphs** is underdeveloped. There is no graph-specific adaptation in the architecture or loss beyond using a GNN backbone, and the theoretical analysis assumes weak dependence among neighboring nodes without addressing how real-world graph correlations may **affect coverage guarantees**.
> 프레임워크를 그래프에 특화하여 설계해야 하는 동기가 충분히 제시되지 않았음. GNN 백본을 사용한 것 외에는 구조나 손실 함수에서 그래프에 특화된 조정이 없으며, 이론적 분석에서도 인접 노드들 간의 약한 의존성을 가정할 뿐 실제 그래프의 상관관계가 커버리지 보장에 어떤 영향을 미치는지 다루지 않음.
> 답변: dual-head 구조가 GNN의 메시지 패싱 특성과 결합되어 node-wise uncertainty를 더 잘 포착하게 만드며, dual-head 구조는 단순한 구조적 선택이 아니라 메시지 패싱 때문에 불확실성이 oversmoothing되는 GNN 고유 문제를 해결하기 위한 그래프 특화 설계임을 강조. 우리 모델의 dual-head 구조는 단순한 일반적 설계가 아니라, GNN에서 메시지 패싱으로 인해 발생하는 over-smoothing과 uncertainty dilution을 완화하기 위한 그래프 특화. 단일-head가 불확실성을 이웃과 동일하게 만드는 문제는 비그래프 모델에서는 나타나지 않으며, ablation에서도 dual-head만이 안정적인 calibration–width 성능을 보임. 또한 그래프 구조적 교란을 포함한 실험들은 QpiGNN이 구조 변화에도 coverage를 유지함을 보여주며, 이는 이론적 가정이 실제 그래프 상황에서도 유효함을 뒷받침. 이러한 결과는 본 모델이 그래프 데이터의 종속성과 구조적 shift를 다루기 위해 설계되었음을 명확히 한다.
### Weakness 3
The paper’s practical application and generalization capability remain somewhat unclear. While results show improved coverage, it is uncertain how well the approach scales to **large or dynamically evolving graphs**, or how it performs in non-regression graph tasks such as **classification or link prediction**.
> 이 논문의 실제 활용성과 일반화 능력은 다소 불분명. 이 접근법이 대규모 그래프나 동적으로 변화하는 그래프에 얼마나 잘 확장되는지, 그리고 분류나 링크 예측 같은 비회귀 작업에서 어떻게 작동하는지는 확실하지 않음
> 답변: Discussion의 task extention이 이미 존재함. 5.4의 structual shift 상황을 실험적으로 다뤘음. 시뮬레이션으로 보고 추후 확장할 수 있는 연구임(동적 그래프 상황)

## Response to Reviewer Cv3a’s Weaknesses
Thank you for acknowledging the clarity of our formulation and the strength of our quantile-free approach. We appreciate your thoughtful and constructive feedback. We address each concern below and will revise the manuscript accordingly to ensure these points are clearly articulated.

---
> **W1. Missing Baseline UQ Methods**

We thank the reviewer for noting that, since our work is positioned as UQ rather than quantile prediction, comparisons with established regression-UQ methods are necessary. We address this point as follows.

```UQ vs. quantile prediction```\
Our method is a UQ approach because it directly learns prediction intervals without relying on quantile targets or post-hoc calibration. This enables QpiGNN to overcome limitations of existing quantile-based and conformal methods in GNNs. Another reviewer also noted that UQ for GNNs is underexplored yet practically critical, aligning with our motivation.

```Regression-UQ baselines```\
We note that our evaluation includes the UQ baselines most relevant for graph regression—BayesianNN[1], MC Dropout[2], and CF-GNN[3]—which naturally align with message-passing architectures and offer meaningful points of comparison. 

Following the reviewer’s suggestion, we additionally include Evidential Regression (ER)[4] as a baseline. While ER predicts a distribution rather than intervals, we adapt it into a stabilized interval-prediction form for fairness.

Across all datasets, QpiGNN achieves substantially tighter intervals while maintaining competitive coverage, demonstrating its effectiveness relative to existing UQ approaches. We will clarify this baseline rationale and present ER results in the revised manuscript.

| Model                 | Education | Election  | Income    | Unemploy. | Twitch    | Chamelon  | Crocodile | Squirrel  | Anaheim   | Chicago   |
| --------------------- | --------- | --------- | --------- | --------- | --------- | --------- | --------- | --------- | --------- | --------- |
| BayesianNN            | 1.00/2.96 | 1.00/2.98 | 1.00/2.97 | 1.00/2.98 | 1.00/3.07 | 1.00/2.95 | 1.00/3.07 | 1.00/2.97 | 1.00/2.94 | 1.00/2.99 |
| MC Dropout            | 0.40/0.11 | 0.48/0.18 | 0.45/0.09 | 0.41/0.09 | 0.91/0.15 | 0.47/0.02 | 0.46/0.01 | 0.31/0.02 | 0.50/0.11 | 0.34/0.07 |
| CF-GNN                | 0.90/3.10 | 0.91/0.94 | 0.91/2.92 | 0.89/2.61 | 0.89/2.34 | \-        | \-        | \-        | 0.90/2.82 | 0.91/2.26 |
| ER | 1.00/4.37 | 1.00/6.90 | 1.00/3.12 | 1.00/4.80 | 0.99/1.33 | 0.97/1.08 | 1.00/1.56 | 0.97/0.80 | 1.00/2.09 | 1.00/3.06 |
| QpiGNN(optimal)       | 0.99/0.59 | 0.98/0.77 | 0.99/0.44 | 1.00/0.73 | 0.94/0.36 | 0.96/0.23 | 0.97/0.16 | 0.96/0.18 | 0.93/0.40 | 0.98/0.36 |

---
> **References**\
[1] Kendall, Alex, et al. "What uncertainties do we need in bayesian deep learning for computer vision?." In NeurIPS, 2017.\
[2] Gal, Yarin, et al. "Dropout as a bayesian approximation: Representing model uncertainty in deep learning." In ICML , 2016.\
[3] Huang, Kexin, et al. "Uncertainty quantification over graph with conformalized graph neural networks." In NeurIPS, 2023.\
[4] Amini, Alexander, et al. "Deep evidential regression." In NeurIPS ,2020.

--- 
> **W2. Insufficient Graph-Specific Motivation**

We appreciate the reviewer’s point that our motivation for a graph-specific UQ framework should be more explicit. We will strengthen this discussion by highlighting two central issues:

```Limitations of CP-based graph UQ```\
While CF-GNN is an important UQ approach for graph data, it is fundamentally built upon CP and therefore relies on the assumption of exchangeability between the calibration and test sets—an assumption structurally fragile in graphs due to message-passing dependencies. 

Our experiments intentionally break exchangeability (degree splits, community splits), and QpiGNN maintains stable coverage. This motivates a UQ framework that is inherently graph-aware and exchangeability-free.

| Split Type | PICP | MPIW |
| ---------- | -------- | ----- | 
| Random     | 0.98–1.00   | 0.40–0.97    | 
| Degree     | 0.94–1.00   | 0.44–0.87    |
| Community  | 0.95–1.00   | 0.46–0.86    | 

```Why existing QR methods fail on graphs```\
Existing QR methods such as SQR and RQR were developed for i.i.d. regression with MLPs, and they do not transfer reliably to graph settings. Message passing introduces structural dependencies that fundamentally change how QR losses behave.

- **Single-head coupling problem:** 
When prediction and uncertainty are generated from a single head, message passing entangles the two signals. This often causes interval collapse or over-smoothed intervals, especially in heterophilic graphs where node-wise heteroskedasticity must be preserved. For RQR, we even observed that an additional ordering penalty was required just to maintain valid interval ordering.
- **Node-wise loss is incompatible with graph dependence:**
Standard QR losses are entirely node-wise and assume independent samples. Under message passing, this leads to unstable gradients because coverage and width interact within a single objective. As a result, training frequently becomes unstable or collapses when applied to GNNs.

These issues are consistently observed in our experiments: while QpiGNN maintains stable coverage even under structural perturbations, existing QR methods either collapse or produce intervals that are excessively wide.

| Perturbation Type | QpiGNN                    | SQR-GNN               | RQR$^{adj.}$-GNN      |
| ----------------- | ------------------------- | --------------------- | --------------------- |
| Feature Noise $\uparrow$   | 0.89–0.92 / 0.66–0.83 | 0.65–0.76 / 0.69–0.81 | 0.84–0.85 / 0.76–0.78 |
| Target Noise $\uparrow$    | 0.96–1.00 / 0.44–1.05 | 0.49–0.98 / 0.52–0.99 | 0.95–0.98 / 0.87–1.32 |
| Edge Dropout $\uparrow$    | 0.84–0.91 / 0.21–0.23 | 0.53–0.82 / 0.44–0.50 | 0.86–0.89 / 0.80–0.84 |

```Graph-Adaptive Design in QpiGNN```\
To address the limitations discussed above, QpiGNN combines two graph-adaptive design components.

- **Dual-head architecture:** 
QpiGNN separates prediction and uncertainty into two distinct message-passing pathways, preventing the entanglement that often occurs when both signals are propagated through a single head. This design alleviates oversmoothing and preserves node-wise heteroskedasticity even on heterophilic or noisy graphs. Our ablation studies further confirm that the dual-head architecture provides substantially more stable coverage and interval widths compared to single-head variants.

- **Joint loss formulation:** 
Whereas RQR entangles coverage and width within a single conditional loss—leading to instability under graph dependence—QpiGNN explicitly decomposes the objective into a coverage term, a width term, and a violation term. The coverage term drives the model toward the target coverage level, the width term encourages compact intervals, and the violation term provides fine-grained correction for missed intervals. This disentangled structure mitigates gradient interference caused by local correlation patterns induced by message passing, enabling more stable learning of node-level uncertainty.

Together, the dual-head architecture and the disentangled loss constitute graph-adaptive design choices that directly address the unique structural dependencies present in graph data and enable reliable uncertainty quantification under these conditions.

> **W3. Unclear Practical Scope and Generalization**

We appreciate the reviewer’s point that the practical applicability and generalization of our approach—particularly to large or dynamically evolving graphs, as well as to non-regression tasks like classification or link prediction—should be clarified. We address this concern as follows.

```Scalability to large or dynamic graphs```\
While our experiments do not directly include very large or dynamic graphs, several factors support QpiGNN’s scalability in such settings.

- Our real-world datasets already contain graphs with up to 12K nodes and over 200K edges (e.g., Crocodile, Squirrel), and QpiGNN shows training time, memory usage, and parameter counts comparable to BayesianNN and MC Dropout. This indicates that the method operates efficiently without additional computational overhead even on sizable graphs.
- Appendix I shows that QpiGNN maintains the same $\mathcal{O}(E d + N d^2)$ complexity as GraphSAGE, since it does not require repeated forward passes or auxiliary GNN modules. This yields the same near-linear scalability as standard message-passing GNNs, suggesting no structural barriers to scaling to larger graphs.

| Dataset     | #Nodes | #Edges  | Model      | Train Time (sec) | CPU (MB)  | Params (K) |
|-------------|--------|---------|------------|------------------|-----------|------------|
| Crocodile | 11,631 | 180,020 | BayesianNN | 21.34            | 7601.32   | 1695.87    |
|             |        |         | MC Dropout | 18.37            | 7601.61   | 1695.81    |
|             |        |         | QpiGNN | 14.87        | 7601.44 | 1695.87 |
| Squirrel  | 5,201  | 217,073 | BayesianNN | 5.36             | 1966.30   | 411.39     |
|             |        |         | MC Dropout | 2.92             | 1966.29   | 411.33     |
|             |        |         | QpiGNN* | 5.28         | 1966.31 | 411.39  |

Although dynamic graph UQ remains challenging and relatively unexplored, QpiGNN’s message-passing structure is compatible with temporal GNNs such as TGAT, making extension to evolving graphs feasible. We appreciate the reviewer for highlighting this important future direction.

```Generalization to other graph tasks```\
QpiGNN naturally generalizes beyond node regression, as discussed in the task extensions subsection. The dual-head quantile-free design applies directly to other graph tasks by reusing the same prediction–width mechanism on task-specific embeddings, requiring no architectural changes beyond substituting the base encoder.

- For graph-level regression, pooled graph embeddings $\mathbf{h}\_G = \rho(\mathrm{GNN}(\mathbf{X}, \mathcal{E}))$ are mapped to prediction and width via $\hat{y}\_G = \mathbf{W}\_{\mathrm{pred}}\mathbf{h}\_G + b\_{\mathrm{pred}}$ and $\hat{d}\_G = \mathrm{Softplus}(\mathbf{W}\_{\mathrm{diff}}\mathbf{h}\_G + b\_{\mathrm{diff}})$, yielding intervals $[\hat{y}\_G - \hat{d}\_G,\ \hat{y}\_G + \hat{d}\_G]$.

- For link prediction, edge embeddings $\mathbf{h}\_{uv} = \phi([\mathbf{h}\_u \Vert \mathbf{h}\_v \Vert \mathbf{h}\_u \odot \mathbf{h}\_v])$ are processed analogously to produce calibrated intervals $\hat{y}\_{uv} \pm \hat{d}\_{uv}$.

While QpiGNN is designed for regression, the same uncertainty mechanism can be adapted to classification—for example, by forming predictive sets or margin-based confidence intervals using class-logit outputs. These extensions demonstrate that QpiGNN is task-agnostic and not restricted to a single problem setting.

##
## Official Review by Reviewer kbuh's (6/3)
### Weakness 1
The theoretical coverage guarantee relies on the assumption that the local dependencies introduced by GNN message passing decay rapidly enough to allow for the use of concentration inequalities. Given the complex, often high-homophily nature of real-world graphs, this assumption of weak dependence on graph structure is often violated, potentially making the finite-sample guarantees less robust than claimed.
> 이론적 커버리지 보장은 GNN 메시지 패싱으로 인해 생성되는 국소적 의존성이 충분히 빠르게 감소한다는 가정 위에 성립하지만, 실제 그래프는 복잡하고 homophily한 경우가 많기 때문에, 그래프 구조에 대한 이러한 ‘약한 의존성’ 가정은 자주 깨집니다. 따라서 유한 표본에서의 보장이 논문에서 주장하는 것만큼 견고하지 않을 가능성이 있음
> 답변: 1) 가정이 지나치게 강하지 않다 2) 실제 데이터(weak-dependence를 위반한다고 볼 수 있음)에서도 성립한다 3) 실험으로 보완했다
### Weakness 2
The key architectural decision is the dual-head decoupling of $\hat{y}$ and $\hat{d}$. While intuitively helpful for mitigating oversmoothing, the paper lacks a clear ablation study to formally prove its necessity. Specifically, an experiment comparing the proposed dual-head architecture against a single-head architecture that outputs both $\hat{y}$ and $\hat{d}$ should be included to validate this crucial design choice.
> dual-head 필요성을 입증하는 ablation이 없다
> 답변: 이미 해당 실험 논문에 있으며, Reviewer nVMe는 It provides insightful ablations regarding various architectural choices.을 강점으로 언급함.
## Response to Reviewer kbuh's Weaknesses
Thank you for recognizing the strength of our dual-head, quantile-free design and the principled trade-off captured by our Lagrangian formulation. We address the remaining concerns below and will revise the manuscript to emphasize these clarifications.

---
> **W1. Coverage guarantees vs. graph dependence**

We appreciate the reviewer’s concern about the validity of weak-dependence assumptions on real-world graphs, particularly those with strong homophily or long-range correlations. We agree that such structures can influence finite-sample behavior, and clarifying the exact scope of our theoretical results is important.

Our guarantees, however, rely on a stability-style bounded-difference property induced by message passing—rather than on independence or rapidly decaying correlations. The weak-dependence assumption is used only in the asymptotic LLN argument, and we will make this distinction explicit.

```Finite-sample concentration does not require independence```\
Our concentration results use McDiarmid’s inequality, which only requires that modifying a single node causes a controlled, bounded change in the output. This does not assume independence among nodes or fast correlation decay; it simply requires that message passing does not amplify a local perturbation into a graph-wide instability. This form of stability is standard in modern GNN analyses.

```Bounded-difference property derived for empirical coverage```\
In Appendix B.5, we prove that empirical coverage satisfies $|\widehat{C}(G) - \widehat{C}(G')| \le \frac{1}{N} + \delta_G$, where $\delta_G$ is a small graph-dependent term capturing the limited indirect influence induced by message passing. This follows from the localized, Lipschitz-like stability of GNNs and is consistent with established theory in prior work [1–3].

```Empirical validation across diverse graph structures```\
To assess whether this stability holds beyond theory, we evaluated QpiGNN on ten graphs spanning strong homophily, high heterophily, and spatial–temporal networks. In all cases, QpiGNN maintained calibrated coverage, compact interval width, and robustness under structural perturbations and non-exchangeable splits. These results support that the stability assumption is practically reasonable even when dependence is nontrivial.

We will update the manuscript to clearly separate the finite-sample stability guarantee from the asymptotic LLN discussion and to highlight the empirical evidence supporting the bounded-difference assumption on real-world graphs.

> **References**\
[1] Gama, Fernando, Alejandro Ribeiro, et al. "Stability of graph neural networks to relative perturbations." ICASSP, 2020. \
[2] Verma, Saurabh, et al. "Stability and generalization of graph convolutional neural networks." KDD, 2019. \
[3] Sankar, Aravind, et al. "Beyond localized graph neural networks: An attributed motif regularization framework." ICDM, 2020.

---
> **W2. Dual-head vs. single-head ablation**

We appreciate the reviewer for stressing the importance of validating the dual-head architecture. We fully agree, and our revised manuscript will highlight this comparison more clearly.

The paper includes an ablation study comparing:
- the proposed dual-head model,
- a fixed-margin variant, and
- a single-head model jointly predicting ($\hat{y}, \hat{d}$).

The results consistently show that the single-head model fails to produce calibrated intervals, while the dual-head design yields accurate coverage and smaller interval widths.

The single-head variant often collapses to underconfident or degenerate interval widths, indicating that the uncertainty and prediction outputs interfere with each other when jointly optimized. In contrast, the dual-head design preserves heteroskedasticity and avoids oversmoothing, producing well-calibrated intervals across all synthetic graph types.

| Ablation | Basic | Gaussian | Uniform | Outlier | Edge | BA | ER | Grid | Tree |
| -- | -- | -- | -- | -- | --- | -- | -- | -- | -- |
| Dual Head w/o fixed $\hat{d}$| 0.91/0.31 | 0.92/0.53 | 0.88/0.43 | 0.88/0.47 | 0.93/0.43 | 0.98/0.58 | 0.98/0.45 | 0.98/0.86 | 0.96/0.39 |
| Dual Head w/ fixed $\hat{d}$ | 0.94/0.20 | 0.51/0.20 | 0.51/0.20 | 0.68/0.20 | 0.97/0.20 | 0.84/0.20 | 0.73/0.20 | 0.38/0.20 | 0.67/0.20 |
| Single Head | 0.46/0.07 | 0.25/0.09 | 0.25/0.10 | 0.71/0.23 | 0.58/0.07 | 0.78/0.19 | 0.32/0.11 | 0.26/0.14 | 0.06/0.02 |

We will explicitly surface this ablation in the main text and clarify why decoupling prediction and uncertainty is architecturally critical for GNN-based UQ.

##
## Official Review by Reviewer nVMe (2/4)
### Major Weakness 1
The main weakness is the originality / novelty of the approach: It differs from related work like RQR in two ways: i) It uses MLP-based heads for predicting the mean and interval width instead of convolution-based, and ii) uses a different loss for training. The contribution of i) is very incremental to me (the authors even acknowledge that dual-head architectures have been used in Bayesian regression). For ii), the work does not conclusively show the merits of changing the objective from Eq (3) or (4) to Eq (5). In particular, I did not find ablations that showcase the performance of OpiGNN trained with (4) (or RQR trained with (5)) to justify the new objective.
### Major Weakness 2
The authors also claim that their novel loss Eq (5) disentangles coverage and interval width, but the same statement can be made about the existing loss of Eq (3) (the latter term, weighted by $\lambda/2$ penalizes interval width, while the first penalizes miscoverage). It is unclear to me from both a theoretical and empirical (see point above) perspective where the merits of the new loss in Eq (5) lie.
### Major Weakness 3
The authors report a comparison to the baselines only at a coverage of $\alpha=0.9$. On real data, both SQR and RQR only slightly miss the coverage requirement, but at significantly smaller intervals. At the same time, OpiGNN achieves coverage at a probability that is significantly above the target, potentially at the cost of too conservative intervals.
- While I acknowledge that having a lower coverage than required is the worse failure mode, neither of the models seems to be well calibrated in terms of realizing the minimal interval width at the desired coverage requirement.
- It would be interesting to see if the baselines can realize different trade-offs at different penalties for the interval width (e.g., a slightly lower penalty may just push them into the regime of meeting the 90% coverage consistently but still providing narrower intervals).
- A more useful depiction of these results would be a Pareto plot similar to Figure 6. There, it would be more apparent that neither the baselines nor OpiGNN realizes the optimal intervals for $\alpha=0.9$.
### Major Weakness 4
The theoretical arguments of the paper are not very involved and show rather obvious claims. E.g., Prop. 1 basically just re-states the weak law of large numbers. To do so, it makes strong assumptions that are not obviously justified to me (e.g., that means and intervals converge to the targets in probability).
- Importantly, neither of the claims is specific to OpiGNN and can be transferred to the baselines SQR and RQR as well. The theoretical findings, therefore, do not support the architectural contribution of the paper.
### Minor Weakness 1
Some minor claims are not fully supported by experiments:
- 141: SQR is supposedly unstable and miscalibrated: The results show that, with the exception of three (heterophilic?) datasets, it just slightly underachieves the desired coverage, but provides smaller intervals instead.
- 161: Why should (4) be a penalty that explicitly targets miscalibration in GNNs? This seems to be applicable to non-GNN architectures. In particular, I do not see how this formulation directly targets the graph domain.
### Minor Weakness 2 ** 확인 필요**
OpiGNN seems to be insensitive to the desired coverage $\alpha$ (see Figure 3, and the close to 100% coverage in Table 1). How do baselines and OpiGNN perform at different coverage levels (a similar ablation to Figure 3 that includes baselines at suitable penalty strenghts would be insightful here).
### Minor Weakness 3 ** 확인 필요 **
Remark 1: The authors argue that each neighbour contributes at most O(1 / deg(v)). However, this does not imply that each neighbour contributes O(1/N), as 
1/dev(v) > 1/N, and therefore, if something is smaller than the inverse of the degree, it does not imply that it is smaller than 1/N. Furthermore, I do not see why the influence over different layers should behave additively (even if the provided O(1/N) bound was correct).
### Minor Weakness 4
All ablations are conducted on synthetic data. I believe that ablations on real data is more insightful to highlight the advantages and limitations of the approach in practice.
### Question 1
In L. 273, you claim that L1 regularization is advantageous over L2 regularization because of its robustness against outliers. Do you have any experimental evidence that this is relevant for the datasets studied here?
### Question 2
In Figure 6, where do RQR and SQR lie here? Where does OpiGNN trained with (3) lie here? These kinds of plots would be interesting to have as a supplement for Table 1.
### Question 3 ** ** 확인 필요 **
In Appendix B.5, how does the robustness toward single-node perturbations (which also holds for any of the other GNN-baselines) justify the concentration arguments as claimed? Can you elaborate on this?
### Question 4
Figure 4 shows that there is no smooth trade-off between interval width and coverage, at least in terms of the interval width penalty. Instead, there seems to be two regimes, one with a fixed coverage close to 100%, and a second regime with rapidly collapsing coverage. Why is that? Is there a way to make OpiGNN realize a broader range of desired coverage / interval-width trade-offs?
## Response to Reviewer nVMe's Major Weaknesses 
Thank you for the positive feedback. We address the remaining concerns below and will revise the manuscript to clarify these points.

---
> **Major W1. Novelty Concern**

Thank you for the comment. We clarify below the structural issues that arise when applying QR to GNNs, and we will revise the manuscript to make this point more explicit.

```Dual-head MLP contribution```\
We appreciate the reviewer’s comment and clarify that QpiGNN does not use MLP-based heads. The model uses a GNN encoder followed by two linear heads, so the distinction between “MLP-based” and “convolution-based” architectures is not applicable in our setting.

Importantly, the dual-head design in QpiGNN is not a minor variant of heteroscedastic or Bayesian regression. In those settings, the mean and variance are optimized jointly through a single Gaussian NLL objective, and the resulting gradient coupling is generally compatible with i.i.d. models.

In GNNs, however, message passing induces strong structural dependencies, and this gradient coupling leads to graph-specific failure modes—most notably entanglement of prediction and width, oversmoothing, gradient interference, and interval collapse. These issues arise precisely because both signals propagate through a shared encoder.

QpiGNN’s dual-head design directly separates these signals, and together with the disentangled objective, it stabilizes training under graph dependence.

```Necessity of changing the objective```\
The shift from the RQR objective to our joint objective is necessary due to a structural mismatch between QR-style losses and message-passing in GNNs. In QR losses, coverage and width are tightly coupled, which is harmless in i.i.d. MLP settings but becomes unstable under graph dependencies. Message passing causes prediction and width gradients to interfere, leading to interval collapse, oversmoothing, and inconsistent coverage. Our joint objective is designed specifically to avoid these graph-induced failure modes.

Keeping the architecture fixed, we replaced our objective with the RQR loss. Across all datasets:
- QpiGNN delivers stable, near-target coverage with compact intervals.
- QpiGNN-RQR loss consistently collapses—coverage falls to 0.19–0.50 and intervals often blow up (>0.8).

|Model|Basic|Gaussian|Uniform|Outlier|Edge|BA|ER|Grid|Tree|
|--|--|--|--|--|--|--|--|--|--|
|QpiGNN(λ=0.5)|0.89/0.30|0.92/0.55|0.88/0.43|0.89/0.47|0.94/0.39|0.98/0.49|0.98/0.63|0.98/0.87|0.96/0.39|
|QpiGNN(λ=optimal)|0.90/0.30|0.95/0.64|0.93/0.62|0.90/0.49|0.94/0.54|0.98/0.48|0.98/0.63|0.99/0.93|0.96/0.39|
|QpiGNN-RQRLoss(λ=0.1)|0.92/0.84|0.93/0.61|0.90/0.71|0.94/0.46|0.93/0.85|0.19/0.85|0.50/0.83|0.87/0.62|0.27/0.81|

|Model|Education|Election|Income|Unemploy.|Twitch|Chameleon|Crocodile|Squirrel|Anaheim|Chicago|
|--|--|--|--|--|--|--|--|--|--|--|
|QpiGNN(λ=0.5)|0.99/0.57|0.98/0.77|0.99/0.41|1.00/0.74|0.59/0.08|0.51/0.03|0.92/0.08|0.73/0.07|0.92/0.39|0.97/0.36|
|QpiGNN(λ=optimal)|0.99/0.59|0.98/0.77|0.99/0.44|1.00/0.73|0.94/0.36|0.96/0.23|0.97/0.16|0.96/0.18|0.93/0.40|0.98/0.36|
|QpiGNN-RQRLoss(λ=0.1)|0.90/0.53|0.89/0.52|0.90/0.37|0.84/0.46|0.95/0.72|0.94/0.30|0.97/0.24|0.92/0.25|0.83/0.44|0.82/0.41|

---
> **Major W2. Loss Function Justification**

We thank the reviewer for raising this important point. We agree that the distinction between Eq. (3) and our proposed Eq. (5) should be clarified more explicitly.

```Why Eq. (3) Does Not Disentangle Coverage and Width```\
Although Eq. (3) contains separate “miscoverage” and “width” terms, they are multiplicatively coupled: as the model narrows the interval, the residual-based miscoverage penalty shrinks proportionally. As a result, undercoverage is not properly penalized, and the model cannot target a specific coverage level. This issue is mild in i.i.d. MLP regression (the setting RQR was designed for) but becomes severe under graph message passing, where neighborhood aggregation entangles gradients for the upper/lower bounds and causes oversmoothing and calibration instability.

```Why Our loss true disentanglement```\
Eq. (5) resolves this structural problem by separating coverage and width into additive terms: a coverage term that directly enforces $\hat{c} \approx 1-\alpha$ and an independent width term controlled by (\lambda). Because the gradients do not interfere, Eq. (5) provides stable and interpretable control of the coverage–width trade-off and admits a clean Lagrangian interpretation that supports our theoretical analysis. 

To isolate the effect of the loss itself, we trained QpiGNN with the RQR loss (in Response to Major W1) while keeping the architecture fixed. Across all synthetic and real-world datasets, the RQR objective led to collapsed or highly unstable coverage (e.g., 0.19–0.50 on synthetic graphs) and inflated or erratic interval widths, while Eq. (5) produced consistent, well-calibrated intervals. This confirms that the instability arises from Eq. (3)’s structural coupling—not from architectural differences.

---
> **Major W3. Coverage–Width Tradeoff**

We appreciate the reviewer’s thoughtful comments regarding calibration, interval width, and the need to evaluate trade-offs beyond a single coverage level. We fully agree with the points raised and address them below.

```Baseline coverage evaluation```\
We additionally evaluated SQR and RQR under multiple target coverage levels to examine whether the baselines can realize different coverage–width trade-offs. 

Although SQR adjusts its quantile inputs according to the target coverage, the actual coverage remained consistently below the desired level on most datasets and collapsed severely on non-homophilous graphs. RQR showed a similar pattern: changing the target coverage altered the interval widths slightly, but the achieved coverage barely moved and often stayed well below the target. This behavior reflects the structural coupling between coverage and width in its loss formulation.

|RQR|Education|Election|Income|Unemploy.|Twitch|Chameleon|Crocodile|Squirrel|Anaheim|Chicago|
|--|--|--|--|--|--|--|--|--|--|--|
| target coverage = 0.80 | 0.78/0.37 | 0.77/0.40 | 0.78/0.26 | 0.76/0.35 | 0.62/0.42 | 0.67/0.09 | 0.72/0.03 | 0.68/0.10 | 0.65/0.22 | 0.72/0.30 |
| target coverage = 0.85 | 0.83/0.41 | 0.82/0.44 | 0.83/0.28 | 0.80/0.39 | 0.68/0.46 | 0.73/0.09 | 0.75/0.03 | 0.75/0.10 | 0.68/0.23 | 0.80/0.32 |
| target coverage = 0.90 | 0.87/0.32 | 0.89/0.47 | 0.86/0.22 | 0.87/0.33 | 0.30/0.03 | 0.37/0.01 | 0.44/0.01 | 0.22/0.01 | 0.88/0.32 | 0.87/0.21 |
| target coverage = 0.95 | 0.94/0.65 | 0.93/0.60 | 0.94/0.42 | 0.90/0.51 | 0.83/0.56 | 0.92/0.17 | 0.86/0.05 | 0.94/0.19 | 0.83/0.62 | 0.91/0.51 |

|SQR|Education|Election|Income|Unemploy.|Twitch|Chameleon|Crocodile|Squirrel|Anaheim|Chicago|
|--|--|--|--|--|--|--|--|--|--|--|
| target coverage = 0.80 | 0.78/0.28 | 0.75/0.32 | 0.78/0.19 | 0.79/0.35 | 0.09/0.02 | 0.11/0.01 | 0.15/0.01 | 0.09/0.01 | 0.67/0.35 | 0.73/0.23 |
| target coverage = 0.85 | 0.82/0.30 | 0.78/0.36 | 0.83/0.21 | 0.83/0.38 | 0.12/0.03 | 0.11/0.01 | 0.16/0.01 | 0.10/0.01 | 0.72/0.38 | 0.76/0.24 |
| target coverage = 0.90 | 0.88/0.32 | 0.89/0.47 | 0.86/0.22 | 0.87/0.33 | 0.30/0.03 | 0.37/0.01 | 0.44/0.01 | 0.22/0.01 | 0.88/0.32 | 0.87/0.21 |
| target coverage = 0.95 | 0.88/0.36 | 0.85/0.43 | 0.87/0.23 | 0.88/0.44 | 0.12/0.03 | 0.14/0.02 | 0.20/0.01 | 0.09/0.01 | 0.74/0.43 | 0.77/0.27 |

```Baseline penalty evaluation```\
We conducted additional experiments to test whether SQR and RQR can achieve better coverage–width trade-offs through penalty tuning. The results consistently show that neither method can do so.

- SQR requires manually choosing two quantiles, and these quantiles do not correspond to any principled mechanism for achieving target coverage. Hence SQR cannot be calibrated, nor can it explore a coverage–width trade-off in a controlled way.

- RQR cannot decouple coverage and width (structural limitation): We performed λ-sweep experiments on multiple datasets: Changing λ results in almost no change in coverage, because Eq.(3) couples width and miscoverage into a single dependent term. Even with our modified RQR (which adds an ordering constraint absent in the original paper), RQR frequently fails to reach the target coverage. 

|Model|Syn.|US.|Twitch|Wiki.|Trans.|
|-|-|-|-|-|-|
|RQR(λ=1)|0.86/0.66|0.89/0.44|0.91/0.42|0.87/0.13|0.87/0.40|
|RQR(λ=0.5)|0.86/0.64|0.88/0.42|0.91/0.41|0.88/0.07| 0.86/0.38 |
| RQR(λ=0.1) | 0.86/0.66 | 0.88/0.44 | 0.91/0.42 | 0.88/0.13 | 0.87/0.40 |

Together, these results demonstrate that baseline models cannot obtain calibrated intervals simply by tuning hyperparameters.

---
> **Major W4. Theory Justification**

We thank the reviewer for the comments on the theoretical section. We clarify below the specific role of Proposition 1 in our framework and why its implications apply to Eq. (5), and we will revise the paper to make these points more explicit.

We agree that Proposition 1 relies on standard tools, but its purpose is to clarify how the loss in Eq. (5) behaves under the graph-induced dependencies created by message passing. Because message passing produces correlated samples, it is not obvious a priori that empirical coverage will remain stable; Proposition 1 formalizes the conditions under which coverage consistency holds for our explicit miscoverage objective.

Importantly, this argument does not extend to SQR or RQR. These methods do not include a coverage term in their objectives and therefore provide no mechanism to ensure the concentration of empirical coverage—even asymptotically. This matches our empirical findings: SQR and RQR exhibit persistent over- or under-coverage on real graphs, whereas QpiGNN reliably converges to the desired target.

Thus, the intent of Proposition 1 is not to introduce a novel theorem, but to justify why Eq. (5) yields stable calibration under graph dependence—something that SQR and RQR are not designed to guarantee.

## Response to Reviewer nVMe's Minor Weaknesses 
> **Minor W1. Insufficient Support**

We thank the reviewer for pointing out these issues regarding the empirical support for our claims and the motivation behind Eq. (4). These are helpful observations, and we clarify both points below while ensuring that the revision makes the underlying reasoning more explicit.

```Stability of SQR```\
While SQR performs reasonably on some homophilic datasets, it becomes highly unstable on heterophilic graphs—such as Chameleon, Squirrel, and Twitch—where coverage drops to the 0.2–0.4 range. The smaller intervals in these cases arise from severe under-coverage rather than efficiency. Thus, our statement that SQR exhibits dataset-dependent miscalibration is directly supported by the empirical results.

```Role of Eq. (4) and its connection to GNNs```\
The ordering penalty in Eq. (4) is not intended as a general-purpose correction but addresses a failure mode that occurs specifically when RQR is applied to GNNs. Message passing couples the lower and upper bounds when they share a single representation, which leads to frequent quantile crossing and interval collapse—issues we did not observe in MLP-based settings. Eq. (4) simply enforces the interval ordering that the original RQR loss cannot reliably maintain under graph-induced dependencies.

To confirm this, we compared (i) a single-head model with Eq. (4) and (ii) a dual-head model trained with the original RQR loss. The two produced nearly identical predictions, indicating that Eq. (4) does not change the learning objective; it only prevents structural failure within GNNs. Without such ordering control, the model collapses to trivial constant-width intervals and cannot be meaningfully evaluated.

---
> **Minor W2. Coverage Sensitivity**

The observed insensitivity of OpiGNN to the target coverage level α is not a structural limitation of the model. It is an artifact of evaluating all datasets with a fixed λ.

When λ is held constant, the width-regularization term can dominate in certain regimes, leading to mildly conservative intervals that make the model appear less responsive to α. However, when λ is tuned per-dataset or per-α, OpiGNN adjusts appropriately and the over-coverage disappears.

By contrast, SQR and RQR remain largely insensitive to both α and λ. Their achieved coverage barely changes because their objectives inherently couple width and miscoverage, preventing any meaningful coverage–width trade-off. This behavior is consistent with the structural limitations of their loss formulations and highlights the intended advantage of our disentangled objective.

We appreciate the reviewer for raising this point and will clarify it in the revision, along with potential extensions—such as smoother surrogate penalties—that could further enhance α-sensitivity and enable more continuous trade-offs.

---
> **Minor W3. Flawed Influence Bound**

We thank the reviewer for the careful reading and agree that our original Remark 1 was ambiguous. We clarify our position as follows.

We do not rely on the incorrect implication $O(1/\deg(v)) \Rightarrow O(1/N)$. The term $O(1/\deg(v))$ was intended only to express bounded and decreasing per-neighbor influence under normalized message passing, not to claim an explicit $1/N$-scale bound. We have revised the text to avoid this unintended implication.

The influence across layers is not assumed to be strictly additive. Our argument uses a standard Lipschitz-based perturbation bound, not linear additivity. The statement was meant as intuition; we agree that the wording overstated the idea and have corrected it.

Importantly, the theoretical result does *not* depend on either disputed step. The coverage consistency proof requires only: bounded per-neighbor influence, and weak dependence across nodes. These conditions hold for normalized GNNs without requiring the stronger (and incorrect) $O(1/N)$ interpretation or additive layer influence. Thus, while we appreciate the reviewer’s correction, the main theoretical claim remains valid and unaffected.

---
> **Minor W4. Real-data ablations**

Thank you for raising this point. We agree that real-data ablations are important. As shown in Appendix K (Table 10), we include ablations on both synthetic and real datasets. We apologize for the confusion caused by the caption, which incorrectly referred only to synthetic data, and we will update it accordingly.

## Response to Reviewer nVMe's Questions
> **Q1. L1 vs L2 robustness**

Thank you for the question. We tested this directly by replacing the L1 width penalty in Eq.(5) with an L2 version.

L2 consistently produced much wider intervals—despite similar or even overly conservative coverage—because it amplifies heavy-tailed residuals arising from heterophily, structural noise, and message passing. L1 remained stable across all datasets and achieved the target coverage with substantially tighter intervals.

|Model|Basic|Gaussian|Uniform|Outlier|Edge|BA|ER|Grid| Tree|
|--|--|--|--|--|--|--|--|--|--|
|L1|0.90/0.30|0.95/0.64|0.93/0.62|0.90/0.49|0.94/0.54|0.98/0.48|0.98/0.63|0.99/0.93|0.96/0.39|
|L2|0.90/0.81|0.94/0.69|0.91/0.75|0.94/0.63|0.90/0.79|0.17/0.94|0.87/0.87|0.99/1.01|0.31/1.01|

|Model|Education | Election | Income | Unemploy. | Twitch | Chamelon | Crocodile | Squirrel | Anaheim | Chicago |
|--|--|--|--|--|--|--|--|--|--|--|
| L1 | 0.99/0.57 | 0.98/0.77 | 0.99/0.44 | 1.00/0.73 | 0.94/0.36 | 0.96/0.23 | 0.97/0.16 | 0.96/0.18 | 0.93/0.40 | 0.98/0.36 |
| L2 | 0.99/0.68 | 0.99/0.79 | 0.99/0.43 | 0.98/0.86 | 0.97/0.75 | 0.97/0.70 | 1.00/0.69 | 0.97/0.53 | 0.94/0.52 | 0.94/0.45 |

---
> **Q2. Baseline placement request**

Thank you for the suggestion. We will add Pareto-style plots to clarify how the methods compare on the coverage–width trade-off. 

The tables show that SQR gives narrow but under-covered intervals, RQR is competitive only on a few datasets, and the Eq.(3) OpiGNN variant is dominated by the full Eq.(5) model. The full OpiGNN consistently achieves target coverage with much tighter intervals, aligning with the Pareto-optimal region.

|Model|Basic|Gaussian|Uniform|Outlier|Edge|BA|ER|Grid| Tree|
|--|--|--|--|--|--|--|--|--|--|
| QpiGNN- Eq. (5)    | 0.90/0.30 | 0.95/0.64 | 0.93/0.62 | 0.90/0.49 | 0.94/0.54 | 0.98/0.48 | 0.98/0.63 | 0.99/0.93 | 0.96/0.39 |
| RQR           | 0.90/0.82 | 0.88/0.53 | 0.90/0.68 | 0.90/0.36 | 0.93/0.83 | 0.78/0.76 | 0.88/0.77 | 0.72/0.48 | 0.85/0.46 |
| SQR           | 0.85/0.33 | 0.88/0.50 | 0.88/0.51 | 0.90/0.10 | 0.91/0.32 | 0.81/0.72 | 0.75/0.60 | 0.78/0.53 | 0.80/0.26 |
| QpiGNN-Eq. (3)    | 0.92/0.84 | 0.93/0.61 | 0.90/0.71 | 0.94/0.46 | 0.93/0.85 | 0.19/0.85 | 0.50/0.83 | 0.87/0.62 | 0.27/0.81 |

|Model|Education | Election | Income | Unemploy. | Twitch | Chamelon | Crocodile | Squirrel | Anaheim | Chicago |
|--|--|--|--|--|--|--|--|--|--|--|
| QpiGNN-Eq. (5)    | 0.99/0.59 | 0.98/0.77 | 0.99/0.44 | 1.00/0.73 | 0.94/0.36 | 0.96/0.23 | 0.97/0.16 | 0.96/0.18 | 0.93/0.40 | 0.98/0.36 |
| RQR           | 0.87/0.32 | 0.89/0.47 | 0.86/0.22 | 0.87/0.33 | 0.30/0.03 | 0.37/0.01 | 0.44/0.01 | 0.22/0.01 | 0.88/0.32 | 0.87/0.21 |
| SQR           | 0.88/0.32 | 0.89/0.47 | 0.86/0.22 | 0.87/0.33 | 0.30/0.03 | 0.37/0.01 | 0.44/0.01 | 0.22/0.01 | 0.88/0.32 | 0.87/0.21 |
| QpiGNN-Eq. (3)    | 0.90/0.53 | 0.89/0.52 | 0.90/0.37 | 0.84/0.46 | 0.95/0.72 | 0.94/0.30 | 0.97/0.24 | 0.92/0.25 | 0.83/0.44 | 0.82/0.41 |

---
> **Q3. Concentration Justification**

Thank you for the comment. Appendix B.5 is not meant to claim that QpiGNN is uniquely robust. Its purpose is to verify that a key assumption for applying concentration arguments—the approximate bounded-difference condition—actually holds in our graph setting.

Because message passing creates nontrivial dependencies, classical McDiarmid-type assumptions cannot be taken for granted in GNNs. Appendix B.5 tests this directly and shows that perturbing a single node (or its local neighborhood) changes the empirical coverage $\hat{c}$ by only $1/N + \delta_G$, with $\delta_G$ consistently small. This demonstrates that (\hat{c}) behaves as a low-sensitivity functional even under graph correlations, which is exactly the condition needed to justify our concentration-based analysis. 

We agree that other GNNs may show similar stability; the experiment is not intended to show exclusivity. Its role is to justify that concentration-based reasoning is appropriate in the regimes where QpiGNN is evaluated. We will clarify this in the revised appendix.

---
> **Q4. Trade-off Discontinuity**

Thank you for the insightful question. The two-regime pattern in Figure 4 arises from the interaction between the joint loss and the discrete nature of empirical coverage. When $\lambda_{\text{width}}$ is small, the coverage penalty dominates and intervals expand, yielding a high-coverage regime. As $\lambda_{\text{width}}$ increases, intervals shrink until a critical point where many violations occur simultaneously, causing a sharp coverage drop. Message passing amplifies this transition, so a fully smooth trade-off curve is not expected. Still, Appendix B.6 shows that tuning $\lambda_{\text{width}}$ yields stable intermediate trade-offs in practice.

A smoother coverage–width trade-off is indeed a meaningful extension. Replacing the hard miscoverage indicator in Eq.(5) with a smooth surrogate would yield continuous penalties and smoother behavior, which we will note as future work.

## Official Comment by Reviewer nVMe (1/2)
Thank you for the very detailed response; I will try to address all your points.
### Major
### Dual-head MLP contribution
I am not convinced by these -- to me -- somewhat handwavy arguments. The architectural novelty is adding / replacing the final layer of the GNN with two linear heads. I do not dispute that this is effective in practice (i.e., improves coverage), but to me it still appears to be an incremental tweak to existing solutions. The architectural difference between OpiGNN and other GNN-based architectures is that the final layer of OpiGNN is an MLP / linear layer with 2 outputs, while the other architectures are fully convolutional. Is that understanding right?
### Loss Function
I understand the argument that OpiGNN's loss is additive and decouples coverage and width into different summands. Disregarding the expectation over nodes, I see strong similarities between the two summands of both loss functions:

Width-Term (similar to me):
- Eq (3): $\lambda(y^u - y^l)^2$
- Eq (5): $\lambda(y^u - y^l)$

Coverage-Term (upper and lower penalty additive in Eq (5) instead of multiplicative in Eq (3))
- Eq (3): $(\alpha + 2\lambda -1(covered))(y-y^l)(y-y^l)(y-y^{u}p)$ 
- Eq(5): 1(not - covered - lower)|y-y^l|+ 1(not-covered -upper)|y-y^u|

Additionally, Eq (5) introduces: $MSE(P(covered), 1-\alphs$

Is this additional term the key difference? If so, how are the other changes above justified? I do not quite understand your argument about multiplicative coupling and why it is worse than additive coupling.

From your results, I see that for $\lambda = 0.1$, the RQR loss sometimes leads to miscoverage but also has a smaller interval width. To me, these results point more to the fact that $\lambda = 0.1$ is not as well chosen as e.g. $\lambda = 0.5$ for OpiGNN's objective.

Especially on real datasets (comparing with OpiGNN 
), I also do not see the clear-cut consistent advantage of your loss function: on Chameleon, Squirrel, Income, (almost) Election, and Education, the RQR loss produces the favorable result.

### Baselines at different coverage
Thank you for this insightful additional evaluation. For SQR, the case is pretty clear that this method fails consistently under different coverage requirements, and I agree with the collapse on heterophilic data.

For RQR, the case seems less clear: While still the desired coverage is consistently not met, setting the coverage requirement to 95%, the results compare favorably to OpiGNN at a requirement of 90%.

I still want to clarify that I believe this is problematic: RQR seems to undercover, while OpiGNN overcovers. But, in practice, I could just set the requirement for RQR to 95% to obtain tighter intervals at a coverage of 90%. Again, my point here is that from a Pareto-optimal perspective, neither OpiGNN nor RQR outperforms the other. Arguably, RQR's failure mode is more severe in practice, however.

I believe that it is valuable to have a model that at least ensures coverage consistently (even at too large intervals), but I believe this perspective needs to be highlighted in the paper to fully understand where OpiGNN lies in the literature.

### Baseline Penalty
Thank you for that evaluation. It seems that, similar to OpiGNN, RQR is more or less insensitive to the penalty magnitude. Put differently, both models do not allow to smoothly control the trade-off between coverage and interval width through the penalty term.

This potentially reveals an interesting problem in UQ for regression on graphs.

Again, I believe this limitation (that both baselines but also OpiGNN) suffers from needs to be highlighted explicitly.

### Theory Justification
I better understand the purpose of Prop. 1 now, thank you! However, I disagree with:

- Equation (3) does not include a coverage term: As written above, the first term penalizes miscoverage explicitly. Therefore, I can make a similar argument about why RQR converges to the desired coverage.

- While SQR and RQR undercover, OpiGNN overcovers: Both models do not empirically converge to the desired coverage of 90%.

## Official Comment by Reviewer nVMe (2/2)
### Minor
### Stability of SQR
I partially agree: On heterophilic datasets, SQR collapses -- but on homophilic datasets this claim is excegerated. The "data-dependent" disambiguation is also not present in the paper. To me, this claim, therefore, is too strong: It should be adapted to "on heterophilic graphs" or something like that.

### Eq (4)'s Connection to GNNs
I understand and also do not dispute that Eq (4) empirically is effective in addressing a problem that arises in GNNs. This problem does not occur in i.i.d. settings and, therefore, does not need to be addressed. However, Eq. (4) still does not exploit any structural components. This is a somewhat nitpicky issue, but Eq. (4), to me, is more a hotfix to this issue (that can, in general, arise in any domain) and not a principled solution for the graph domain. Again, this is a minor concern, and slightly different wording fully resolves this already.

### Coverage Sensitivity
"However, when λ is tuned per-dataset or per-α, OpiGNN adjusts appropriately and the over-coverage disappears." -- Figure 3 seems to show exactly that for different coverage requirements, the choice of $\lambda$ still has little effect, and OpiGNN overcovers at close to 100% consistently.

### Flawed Influence Bound
Thank you for the clarification: Can you elaborate on the "standard Lipschitz-based perturbation bound"? What is the fully revised reasoning that justifies this claim?

### Questions
### L1 vs. L2
Thank you, this experiment resolves the question.

### Baselines in Pareto-Plot
Thank you for adding these. This is insightful. It shows the empirical advantage of Eq. (5)'s loss over Eq. (3)'s loss (even though the motivation is still not fully clear to me, see above). From this table, I maintain my assessment that RQR is not consistently outperformed by OpiGNN in terms of realizing a trade-off between coverage and interval width (see above).

### Concentration Justification
Thank you for that clarification. I agree with your reasoning. The paper benefits from clarifying that this analysis is not OpiGNN specific but also can be applied to, e.g., RQR.

### Trade-Off Discontinuity
Thank you for that insightful explanation. I see that this is a challenging problem to solve. It is, of course, unreasonable to expect one paper to solve every issue that arises. However, as frequently highlighted in my response, the core issue here is that OpiGNN also realizes (arguably a preferable) case of miscalibration compared to RQR: OpiGNN overshoots in terms of coverage and sacrifices interval width -- RQR undercovers but has tighter intervals.


Overall, I highly appreciate the author's efforts for clarification. My main concern regarding novelty is not resolved at this point. Some other points are also not fully clarified yet.
> novelty에 대한 concern이 이번에 안풀렸다는 입장

In terms of a broader picture, OpiGNN is an architecture that is biased toward overcoverage and larger intervals -- compared to the state-of-the-art baseline RQR, which slightly undercovers but makes tighter predictions. 
This point is understated in the paper; I still disagree with claims that OpiGNN is a strictly superior architecture.
> RQR이 좁은 구간을 생성하는가..? 동의하기 어렵지만 리뷰어가 주장.
> tightness입장에서 undercoverage가 overcoverage보다 더 나을 수 있다는 입장인 것 같다.
> QpiGNN이 strictly superior architecture이라고 주장하는 것 보다는, motivation(CF-GNN -> QR-based -> QpiGNN)을 설명하는 방향으로?




## Response to Reviewer nVMe's Comments (1/2)
Thank you for the detailed follow-up feedback. I appreciate the careful attention to each point and the clarification of what remains unclear. I hope the responses below help address your remaining concerns.

---
> **Overall: QpiGNN Motivation**

Our motivation comes from the structural limitations of CP-based methods, whose exchangeability assumption rarely holds in graph-structured data due to message-passing dependencies. Although SQR and RQR perform well in standard regression, QR-based UQ has not been explored for GNNs. 

To address this, we implemented the first QR-GNN baselines (SQR-GNN and RQR-GNN) and found that standard QR becomes unstable under graph dependencies. These issues led to the design of QpiGNN, with a dual-head architecture and disentangled loss, as a graph-adaptive approach. We hope this work lays a foundation for future research on QR-based UQ for GNNs.

---
> **C1. Dual-head MLP contribution**

Thank you for restating your understanding; we appreciate the chance to clarify.
- Although the dual-head structure may look like a simple split of a single head, our point is not architectural novelty in form but functional motivation. Most commonly used GNN architectures—including those employed in our baselines—already incorporate a linear or MLP layer as the final prediction head, so the contrast with “fully convolutional” architectures may not fully capture the relevant distinction.
- The key issue is that message passing entangles prediction and uncertainty signals, making standard QR methods unstable on GNNs. QpiGNN’s dual-head design explicitly separates these signals, enabling stable training where methods like SQR and RQR otherwise tend to collapse.

We will revise the manuscript to better highlight this functional motivation, and we hope this addresses the concern.

---
> **C2. Loss Function**

Thank you for your follow-up questions regarding the loss formulation. We provide a concise clarification below.

```Additional term```\
The term (target coverage − current coverage)$^2$ in QpiGNN serves as a global coverage-feedback mechanism, helping the model stay near the target coverage. RQR lacks such global feedback, which contributed to higher sensitivity to graph-induced instabilities (e.g., quantile crossing, interval collapse).

```Multiplicative coupling```\
We do not claim that multiplicative coupling is inherently inferior. However, in GNNs, message passing increases representation variability, and multiplicative terms $(y−y^l)(y−y^u)$ amplify it quadratically—often causing instability on heterophilic graphs or datasets with wide target ranges.

Additive penalties, by growing only linearly, avoid this amplification and provided much more stable behavior across datasets. We acknowledge that multiplicative terms can be stable when the target range is narrow, and we will clarify this nuance in the revision.

```RQR’s strength```\
We acknowledge RQR as a strong QR method and therefore implemented it as the first GNN baseline. However, due to repeated instability, PICP/MPIW could not be computed in many runs. We introduced a minimal ordering penalty ($\gamma^order$)to obtain a stable baseline RQR$^{adj.}$-GNN. QpiGNN is a gradual development that builds on RQR’s strengths while addressing graph-specific challenges. 

---
> **C3. Baselines at different coverage**

Thank you for this insightful observation. As you pointed out, from a Pareto-optimal perspective, neither OpiGNN nor RQR strictly dominates the other, and we agree that RQR can produce tighter intervals when calibrated at a higher target coverage.

Our intention was simply to highlight that OpiGNN achieves stable—and slightly conservative—coverage across graph settings, which is often preferable to undercoverage in graph UQ. We also acknowledge the strong performance of our RQR-GNN baseline and will clarify these points in the revision.

---
> **C4. Baseline Penalty**

Thank you for highlighting this perspective. As noted, this reflects a broader challenge for QR-based UQ in graph regression and is a limitation shared by both baselines and OpiGNN. We will discuss this more explicitly in the revised manuscript. Thank you again for the valuable feedback.

---
> **C5. Theory Justification**

We are glad that the purpose of our proposition is clearer, and we appreciate the opportunity to clarify the remaining points.
- Regarding Equation (3), we agree that it contains a sample-wise miscoverage penalty that could, in principle, guide RQR toward the desired coverage. Our earlier comment was intended from the perspective of global coverage feedback, and we appreciate the reviewer’s clarification.
- We also agree that neither undercoverage nor overcoverage reflects strict convergence to the target level. The graph UQ literature generally views undercoverage as the more severe failure mode, which is why we followed that convention. Nevertheless, we will clarify this distinction more explicitly in the revised manuscript.

## Response to Reviewer nVMe's Comments (2/2)
Thank you for the follow-up. I hope the responses below address your concerns.

---
> **C1. Stability of SQR**

Thank you for the comment. We agree that SQR’s collapse mainly occurs on heterophilic graphs and will adjust the wording accordingly.

---
> **C2. Eq (4)'s Connection to GNNs**

Thank you for the comment. The additional term in Eq. (4) was introduced because applying RQR directly to GNNs made it difficult to compute evaluation metrics due to quantile crossing. It is indeed a practical workaround rather than a mechanism that leverages graph structure. We will revise the wording to reflect this more accurately.

---
> **C3. Coverage Sensitivity**

Thank you for the helpful clarification. We agree that, as shown in Figure 3—where both the target coverage and λ were varied—OpiGNN still tends to overcover, and we acknowledge that our earlier description may have been misleading.

This behavior is mainly due to the current loss design: the violation term penalizes undercoverage but does not symmetrically penalize overly conservative intervals, which can naturally lead to higher coverage. We appreciate the reviewer for highlighting this.

We will clarify this limitation in the manuscript and discuss potential extensions to achieve tighter control around the target coverage. Thank you again for raising this valuable point.

---
> **C4. Flawed Influence Bound**

Thank you for the follow-up question. We are glad to clarify what we mean by the “standard Lipschitz-based perturbation bound.”

In the revised reasoning, we only use a basic stability property: if each message-passing layer is $L$-Lipschitz, then a local perturbation can grow by at most $L^k$ over $k$ layers. This ensures bounded influence without requiring any specific decay rate or additive accumulation across layers.

These assumptions are standard in GNN stability analyses, and the corrected argument no longer relies on the unintended implications of the original remark. We will update the manuscript to reflect this more clearly. Thank you again for encouraging a more precise explanation.

---
> **C5. L1 vs. L2**

Thank you for confirming—glad to hear that the experiment resolves the question.

---
> **C6. Baselines in Pareto-Plot**

We’re glad the additional results were helpful. We agree that, without placing more weight on undercoverage than overcoverage, it is difficult to claim that QpiGNN strictly outperforms RQR in the coverage–width trade-off. 

Our original interpretation followed the common practice in graph UQ of treating undercoverage as the primary failure mode, while slight overcoverage is generally viewed as conservative but acceptable [1–3]. We acknowledge that this perspective was not explained clearly enough in the manuscript. With the additional results now included, we will revise the paper to make this viewpoint more explicit and provide a more balanced discussion of the trade-off between QpiGNN and RQR.

---
> **C7. Concentration Justification**

Thank you, we will clarify this in the revision.

---
> **C8. Trade-Off Discontinuity**

As noted in our response to C6, we agree that without assigning greater importance to undercoverage, it is difficult to assert that QpiGNN strictly outperforms RQR in the coverage–width trade-off.

Our original objective, following standard practice in UQ, was to avoid undercoverage and then minimize interval width among models that satisfy the target coverage[2]. From this perspective, QpiGNN provides narrower intervals among the models that achieve acceptable calibration, which motivated our interpretation.

That said, we understand the reviewer’s broader perspective. We will expand the discussion in the revised manuscript to acknowledge both viewpoints and to clarify how different interpretations of miscalibration affect the perceived trade-off. We hope that this perspective will be valuable for future research on graph-based uncertainty quantification.

---
Overall, We hope that our additional clarifications have helped address the reviewer’s concerns. 

As discussed in our motivation, we believe our work provides a meaningful first step toward systematic QR-based UQ for GNNs. The reviewer’s insightful comments help clarify the positioning of our contribution, and we will reflect these perspectives in the revision to present QpiGNN’s role more clearly and balanced.

---
> **References**\
[1] Kendall, Alex, et al. "What uncertainties do we need in bayesian deep learning for computer vision?." In NeurIPS, 2017.\
[2] Huang, Kexin, et al. "Uncertainty quantification over graph with conformalized graph neural networks." In NeurIPS, 2023.\
[3] Romano, Yaniv, et al. "Conformalized quantile regression." In NeurIPS, 2019.

##
## Official Review by Reviewer xHf3 (6/3)
### Weakness 1 (Graph dependence)
The coverage guarantees rely on weak dependence assumptions, but the treatment remains at a high level. There’s no rigorous dependence model or formal graph-specific concentration analysis beyond citing WLLN.
### Weakness 2 (Comparison with conformal prediction theory)
While CP methods are used as baselines, the paper does not deeply analyze why QpiGNN succeeds under violations of exchangeability. A theoretical comparison would strengthen the claims
### Weakness 3 (Positioning)
While the quantile-free formulation is original in GNNs, it builds closely on RQR. The conceptual gap between RQR and QpiGNN could be articulated more strongly
## Response to Reviewer xHf3's Weaknesses 
Thank you for recognizing the strength of our motivation, the importance of UQ in GNNs, and the contributions of our quantile-free design, theory, and empirical evaluation. We hope the following points resolve the remaining concerns. We will update the manuscript to ensure these points are clearly emphasized.

---
> **W1. Graph dependence**

We agree that our initial discussion of graph dependence was high-level and appreciate the opportunity to clarify this aspect. Our guarantees rely on a stability-based argument rather than independence or exchangeability assumptions, and we will make this more explicit.

```Graph-aware bounded-difference property```\
In standard k-hop message-passing GNNs with normalized aggregation, changing a single node feature affects only a limited k-hop neighborhood, and each aggregation step is Lipschitz. This yields $\big| \hat{c}(X) - \hat{c}(X^{(v)}) \big| \le O(1/N) + O(1/|\mathcal{N}(v)|)$, which directly provides the bounded-difference term $\tfrac{1}{N} + \delta_G$.

This bounded-difference stability is the *only* condition required for our concentration guarantees—no mixing, sparsity, or independence assumptions are needed. This stands in contrast to graph-based CP methods that rely much more heavily on exchangeability.

```Empirical validation across graph types```\
Although the theory does not specify a dependence model for each real-world graph, we empirically verify the assumed stability on diverse graph families (homophilic, heterophilic, spatial–temporal). QpiGNN consistently maintains calibrated coverage and stable interval widths. We will clarify that our theory + broad empirical evidence jointly support the applicability of QpiGNN under graph dependence.

We appreciate the reviewer’s point and will strengthen both the theoretical explanation and its connection to practical graph structures.

---
> **W2. Comparison with conformal prediction theory**

We agree that contrasting QpiGNN more clearly with CP-based methods strengthens the contribution. Our central point—which we will emphasize more explicitly—is that QpiGNN’s guarantees depend on stability (bounded differences), not exchangeability, making its theoretical basis fundamentally different from CP.

```Why QpiGNN works without exchangeability```\
CP-based methods require node-level exchangeability for valid coverage. In contrast, QpiGNN learns prediction and interval width jointly through a quantile-free objective and dual-head architecture. Concentration of the empirical coverage follows directly from the bounded-difference property above, not from i.i.d. or exchangeability assumptions. This explains why QpiGNN remains valid under arbitrary graph-dependent sampling schemes.

```Practical evidence under non-exchangeable splits```\
QpiGNN maintains stable coverage across random, degree-based, and community-based splits—settings that directly violate exchangeability and where CP-based baselines lose guarantees. The results illustrate that our stability-based reasoning—not exchangeability—drives QpiGNN’s reliability. We will streamline the presentation of these findings in the revision.

| Split Type | PICP | MPIW |
| ---------- | -------- | ----- | 
| Random     | 0.98–1.00   | 0.40–0.97    | 
| Degree     | 0.94–1.00   | 0.44–0.87    |
| Community  | 0.95–1.00   | 0.46–0.86    | 

We will make this theoretical distinction explicit to clarify why QpiGNN succeeds where CP methods have inherent limitations.

---
> **W3. Positioning**

We appreciate the reviewer’s feedback and agree that the conceptual distinction between QpiGNN and RQR should be articulated more strongly.

Although both methods share a quantile-free motivation, the quantile-free principle behaves fundamentally differently under relational dependence, and QpiGNN is designed to operate in this regime.

```RQR assumes sample-wise independence```\
RQR models residual distributions under an standard regression setting. When applied to graph data, residuals become correlated through message passing, distorting the learning objective and often yielding entangled or invalid intervals. In fact, we had to introduce an additional ordering penalty $\gamma_{\text{order}} \cdot \text{ReLU}(\hat{\mathbf{y}}^{\text{low}} - \hat{\mathbf{y}}^{\text{up}})$ to ensure fair comparison.

```QpiGNN is explicitly designed for relational dependency```\
Our dual-head, graph-aware architecture decouples prediction and uncertainty, maintains node-wise heteroskedasticity despite message passing, and prevents oversmoothing of interval widths. This makes the quantile-free objective identifiable and stable even when predictions depend on the relational structure.

Thus, the conceptual gap between RQR and QpiGNN is not merely architectural; it stems from the fundamentally different behavior of quantile-free objectives under graph dependence. We will emphasize this distinction clearly in the revised manuscript.