Non-exchangeable Conformal Prediction for Temporal Graph Neural Networks

Adithya Kulkarni; Dawei Zhou; Jian Kang; Tuo Wang; Yujun Yan

REVIEW 5 major objections 5 minor 49 references

Non-exchangeable Conformal Prediction for Temporal Graph Neural Networks

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A conformal prediction framework for temporal graphs claims that a predefined coverage level can still be guaranteed despite temporal dependencies, and achieves up to a 31% reduction in prediction set size.

desk verdict NCPNet's diffusion score and efficiency-aware weights are worth a look, but the advertised coverage guarantee is not proven: the proof omits a weight-ordering condition and the deployed quantile differs from the one analyzed. read the letter →

arxiv 2507.02151 v1 pith:IDHYHMCM submitted 2025-07-02 cs.LG

Tuo Wang , Jian Kang , Yujun Yan , Adithya Kulkarni , Dawei Zhou This is my paper

classification cs.LG

keywords conformalpredictiontemporalgraphsgraphneuralnetworksnon-exchangeabilityuncertaintyquantificationweightedquantilesetscoverageguarantee

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Standard conformal prediction assumes exchangeability: a calibration set and a test point are interchangeable. In temporal graphs this fails, because nodes, edges, labels, and scores evolve and depend on one another over time. The paper introduces NCPNet, which adopts the non-exchangeable weighted-quantile bound from prior theory [4] and adapts it to temporal graphs, proving that a predefined coverage level can still be guaranteed. It couples this theory with a topological and temporal diffusion-based non-conformity score and an efficiency-aware optimization of the weights, and reports empirical coverage at the target level on four datasets while reducing prediction-set size by up to 31% on WIKI.

What carries the argument

The central machinery is the weighted quantile with learnable parameters $\Omega$, together with a soft differentiable selection of the quantile index that uses the cumulative sum of $\Omega$ against $1-\alpha$ and a softmax assignment. This is coupled with a topological and temporal diffusion non-conformity score (Eq. 11) that blends each node's raw score with the scores of its structural and temporal neighbors. The weight optimization minimizes an efficiency loss that counts, in a differentiable way, how many candidate labels fall above the soft quantile, so the quantile becomes both coverage-aware and size-aware.

What would settle it

Re-run the calibration with the fitted weights randomly permuted along the sorted score axis; if empirical coverage on a fresh test split drops below the target while coverage with the original weights stays at the target, then the guarantee depends on the unstated weight-monotonicity condition. Alternatively, directly check whether the learned $\Omega$ satisfies $\omega_{n_c+1} \ge \omega_k$ for every calibration point $k$ with $s_k < s_{n_c+1}$.

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Extended reading notes

Core claim

The paper claims that, although temporal dependencies break exchangeability, a conformal prediction set can still meet a predefined coverage level $1-\alpha$ by replacing the standard unweighted quantile with a weighted quantile whose weights are learned, and by bounding the coverage gap in terms of the total variation distance between score distributions and the weights. Concretely, Lemma 3.4 upper-bounds the gap by $\frac{\sum_{i=1}^{n_c} \omega_i d_{TV}(\phi, \phi_i)}{1 + \sum_{i=1}^{n_c} \omega_i}$, so with well-chosen weights the gap can be kept small. NCPNet realizes this by a diffusion score that mixes each node's score with those of its topological and temporal neighbors, and an efficiency-aware optimization that tunes the weights to shrink prediction sets while retaining coverage.

Load-bearing premise

The proof's inequality requires weights to be monotone in sorted score order—the test point's weight must be at least as large as the weight of any calibration point with a smaller score—and the training procedure neither enforces nor verifies this, so the coverage guarantee in Lemma 3.4 is not shown for the weights actually used.

Editorial extensions

If this is right

The same calibration procedure transfers to any temporal GNN backbone, so uncertainty quantification no longer requires exchangeable calibration data.
Empirical coverage at the target level is achieved on WIKI, REDDIT, DBLP, and IBM for TGAT, JODIE, and TGN backbones, with prediction sets up to 31% smaller on WIKI.
Weighted quantiles with learned weights make the coverage gap shrink as the diffusion score and the weight optimizer reduce the total-variation term in Lemma 3.4.
Ablation results show that both the diffusion score and the weight optimization contribute to efficiency; removing either one degrades prediction-set size.
The framework covers both exchangeable and non-exchangeable conditions, since the bound in Lemma 3.4 reduces to the standard conformal guarantee when the total-variation distance is zero.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the learned weights do not satisfy the monotonicity condition in the proof, the stated coverage guarantee in Lemma 3.4 does not cover the trained weights; checking this on the fitted models would tell whether the reported efficiency gains are supported by the theory.
The diffusion mixing parameters are fixed per dataset and grid-searched; learning them per node could further shrink sets on hubs or temporally active nodes, but that extension would need a fresh coverage analysis.
The same weighted-quantile plus diffusion recipe could be applied to other non-exchangeable structured data such as spatiotemporal traffic or temporal knowledge graphs, where a neighbor definition across time is available.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The headline claim—guaranteed coverage for conformal prediction on temporal graphs—is not backed by the paper's own math. The proof of Lemma 3.4 in Appendix B needs the inequality (ω_{n+1}−ω_k)(s_{n+1}−s_k) ≥ 0 for every k, which is a real constraint linking weight order to score order. The paper never states that condition, and Algorithm 1, which fits ω by gradient descent on an efficiency loss, does not enforce it. Without that, the bound from Barber et al. doesn't apply to the deployed procedure. To make matters worse, Eq. 12 replaces the weighted quantile with a softmax-weighted average (η = S_π B), so the miscoverage characterization in Eq. 16 doesn't match the actual prediction rule.

That said, the paper has genuine new pieces. The diffusion-based non-conformity score in Eq. 11—mixing a node's own score with topological and temporal neighbors—is a sensible way to smooth uncertainty in evolving graphs. The efficiency-aware weight optimization is a practical addition, and the experiments span four datasets and three backbone temporal GNNs, which is more than many papers in this space. Code and data are provided. Those pieces might be useful to someone building a conformal method for temporal graphs.

Where the paper falls short is the central promise. The 31% efficiency improvement on WIKI is a selected outcome of fitting ω on the calibration-training split and grid-searching λ1, λ2; it's not a prediction from the theory. The empirical coverage numbers in Table 1 look fine, but they are not evidence for the theorem. For a reader who wants a guaranteed coverage bound, this paper doesn't deliver it.

Who is this for? Someone doing applied conformal prediction on temporal graphs could try the diffusion score as a heuristic and maybe get better efficiency. But the paper as written overclaims the theory, and the core contribution—the coverage guarantee—is unsupported.

My recommendation: this deserves serious peer review, but not acceptance in its current form. The authors need to either fix the theory by adding and enforcing the required weight-ordering constraint, or drop the guarantee claim and market NCPNet as an empirical method. As submitted, the central result is unproven.

Referee Report

5 major / 5 minor

Summary. This paper proposes NCPNet, a conformal-prediction framework for temporal graph neural networks. The authors argue that temporal dependencies violate the exchangeability assumption, prove a non-exchangeable coverage-gap bound inspired by Barber et al. (2023), and then implement the method with a diffusion-based non-conformity score (module M1) and an efficiency-aware soft quantization with learned weights (module M2). They report experiments on WIKI, REDDIT, DBLP, and the IBM anti-money-laundering dataset across three temporal GNN backbones, claiming that NCPNet achieves 95% coverage with smaller prediction sets, including up to a 31% reduction in prediction set size on WIKI.

Significance. Conformal prediction for temporal graphs is a timely and relevant problem, and the paper contains several useful engineering ideas: diffusing non-conformity scores over topological and temporal neighborhoods and learning a differentiable quantile in an end-to-end manner are sensible heuristics. The authors also provide data/code and evaluate across multiple backbones and datasets, which is a strength. If the coverage guarantee were valid, this would be a meaningful advance. As it stands, however, the central theoretical claim is not supported: the proof of Lemma 3.4 requires a weight-monotonicity condition that Algorithm 1 does not enforce, the deployed soft quantile in Eq. (12) is not the weighted quantile used in the lemma, and the weights and diffusion parameters are data-dependent. The reported efficiency gains are therefore empirical properties of a tuned heuristic rather than consequences of the stated guarantee.

major comments (5)

[Appendix B, Eq. (20)] The proof of Lemma 3.4 reduces to establishing (ω_{n+1} − ω_k)(s_{n+1} − s_k) ≥ 0 for every k. This requires the weights to be ordered consistently with the sorted non-conformity scores, a monotonicity condition that is never stated in the paper and is not enforced by Algorithm 1 or Eq. (12). Since gradient descent on Eq. (13) can produce weights that violate this condition, the proof of Lemma 3.4 does not apply to the weights actually used by NCPNet, and the claimed coverage guarantee is unsupported.
[Section 4.1, Eq. (12)] The differentiable quantile η = S_π B is a softmax-weighted average of the sorted scores, not the weighted quantile Q_{1−α}(Σ_i ω_i δ_{φ}) that appears in Lemma 3.4 and in Eq. (16) of the appendix. No proposition in the paper shows that thresholding by η obeys the miscoverage characterization in Eq. (16). The theoretical guarantee therefore applies to a different prediction rule from the one that is implemented and evaluated.
[Algorithm 1 and Section 5.1] The weights Ω are optimized on D_c_train, and the diffusion parameters λ1 and λ2 in Eq. (11) are selected by grid search using calibration data. The fixed-weight, pre-specified-weights assumption under which the non-exchangeability bound is derived is violated, because the weights depend on the same calibration data used to build the predictor. As a result, the reported coverage values and the abstract's '31% reduction' are selected outcomes of the calibration procedure rather than out-of-sample predictions guaranteed by Lemma 3.4.
[Lemma 3.4 and Appendix B] The lemma states d_TV(φ, φ_i) where φ and φ_i are finite sets of scalar scores (Eqs. 7–8), but total-variation distance is defined for probability distributions; as written, the lemma is not a well-formed mathematical statement. In addition, the appendix derives a bound involving normalized weights ω̃_i with Σ_i ω̃_i = 1 and concludes with α + Σ_i ω̃_i d_TV, which does not match the claimed expression (Σ_i ω_i d_TV)/(1 + Σ_i ω_i). These formal gaps are load-bearing because Eq. (6) is the theoretical foundation of the paper.
[Proposition 3.2] The main text states that the proof of Proposition 3.2 is provided in Appendix B, but Appendix B contains only the proof of Lemma 3.4 and does not prove Proposition 3.2. Since the violation of exchangeability motivates the entire framework, the proof should be supplied, or the proposition should be presented as an illustrative argument rather than a proven result.

minor comments (5)

[Abstract and Table 1] The abstract's '31% reduction' is not directly visible in Table 1; the WIKI efficiency improvements against NEX are 34.2%, 35.9%, and 33.5% for TGAT, JODIE, and TGN, respectively. Please clarify the reference point for the 31% figure.
[Section 4.1, Eq. (12)] The quantity Γ is defined in Eq. (12) but never used afterward; either remove it or explain its role.
[Algorithm 1] The phrase 'Computer model output' appears to be a typo and should read 'Compute model output'.
[Notation throughout] The notation for weights is inconsistent between Lemma 3.4, which uses ω_i with denominator 1 + Σ_i ω_i, and Appendix B, which uses normalized weights ω̃_i with Σ_i ω̃_i = 1; the two notations should be unified and the relationship between them made explicit.
[Section 5.3, ablation study] The ablation 'NCPNet w/o s' is described as removing the topological and temporal non-conformity score, but Eq. (11) contains both a score term and diffusion terms; please state precisely which term is removed in this ablation.

Circularity Check

3 steps flagged · score 6.0 of 10

Coverage guarantee is an imported weighted-quantile bound whose proof conditions the learned weights do not satisfy; the headline 31% efficiency gain is the direct objective of the fitted parameters.

fitted input called prediction [Section 4.1 (M2), Eq. 13, Algorithm 1; Abstract]
"Based on a differential selection of the desired quantile value, to ensure both desired coverage and improved efficiency, we propose a method to optimize the weighted parameters based on coverage and efficiency considerations... Consequently, the efficiency loss is defined as: L_Efficiency = Σ_i Σ_k σ((s(f(G̃,X̃,θ,x_i),k)−η̂)/τ)."

The headline efficiency improvement ('up to a 31% reduction in prediction set size on the WIKI dataset') is the same quantity that Algorithm 1 optimizes: Ω is trained by backward propagation on Eq. 13, a smooth proxy for prediction-set size, and λ1,λ2 are grid-searched. The reported test-set efficiency is therefore a selected outcome of the fitting procedure, not a prediction or derived consequence of the framework. The ablation comparisons are largely statements about how well the optimization modules improve their own objective.
ansatz smuggled in via citation [Section 3, after Lemma 3.4; Appendix B Eq. 20; Section 4.1 Eq. 12]
"In our M2 (Section 4.1), we optimize weights for efficiency without violating theoretical assumptions, as Lemma 3.4 permits efficiency-aware weights and arbitrary non-conformity scores."

The paper asserts that Lemma 3.4 permits arbitrary efficiency-aware weights, but the lemma's own proof in Appendix B (Eq. 20) requires (ω_{n+1}−ω_k)(s_{n+1}−s_k) ≥ 0 for every calibration point k. Algorithm 1 optimizes Ω by unconstrained gradient descent with no monotonicity or ordering constraint relative to the scores, and Eq. 12 replaces the lemma's cumulative-weight quantile with η̂ = S_π B, a softmax-weighted average. Thus the coverage guarantee is not derived for NCPNet's actual prediction rule; it is imported from Barber et al.'s weighted-quantile setting under conditions that are neither stated as assumptions nor enforced by the algorithm.

1 more flagged steps

renaming known result [Section 3, Lemma 3.4]
"Our theory is inspired by [4], but we extend it to temporal graphs, with unique challenges like non-Euclidean structure and evolving dependencies."

Lemma 3.4, Eq. (6), is a restatement of Barber et al.'s non-exchangeable conformal coverage bound; the 'temporal graph extension' consists of reinterpreting the total-variation terms for graph-derived non-conformity scores, not of deriving a new coverage guarantee. The paper presents this as its own theoretical proof ('we provide theoretical proofs demonstrating that a predefined coverage level can still be guaranteed'), renaming a known external result as a novel contribution while the algorithm that is actually deployed does not implement the weighted quantile used in that result.

full rationale

The most serious issue is not self-citation but a mismatch between the imported theory and the implemented algorithm. Lemma 3.4 and its Appendix B proof require a fixed weighted quantile with weights ordered relative to the scores (Eq. 20); NCPNet instead learns unconstrained Ω by gradient descent and uses a softmax-weighted average η̂ = S_π B from Eq. 12. The claim that 'Lemma 3.4 permits efficiency-aware weights and arbitrary non-conformity scores' is therefore an unsupported ansatz smuggled in via citation to [4]. Separately, the headline 31% efficiency reduction is the objective being optimized, making it a fitted outcome rather than an independent prediction. There is no load-bearing self-citation chain here: the authors' own prior works are cited only incidentally. However, because the central coverage guarantee is not actually established for the deployed algorithm, and because the efficiency claim reduces to the fitted objective, the paper warrants a partial circularity score of 6.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the unproven applicability of an existing coverage bound to adaptively fitted weights, several hyperparameters selected by grid search or left unspecified, and the assumption that the soft quantile matches the exact weighted quantile. No new physical entities are introduced.

free parameters (5)

λ1, λ2 (diffusion weights) = λ1=λ2=0.01 chosen by grid search
Eq. 11 mixes each node's non-conformity score with its topological neighbors (λ1) and temporal neighbors (λ2). Section 5.5 reports a limited grid search and states the small-scale setting works well, making these fitted hyperparameters.
Quantile weights Ω = learned via backprop on Eq. 13
The efficiency loss L_Efficiency in Eq. 13 optimizes the n weights Ω to minimize prediction set size on the calibration-training split; these are free parameters fit to data.
Temperature T (Eq. 12) = not reported
Controls soft assignment of the weighted quantile; a hyperparameter not specified in the paper, yet it affects the learned Ω.
Sigmoid temperature τ (Eq. 13) = not reported
Hyperparameter in the surrogate efficiency loss; not specified.
d_st, t_st (neighbor thresholds) = not reported
User-defined distance and time thresholds in Definition 4.1 for determining temporal neighbors; not discussed further, unclear how set.

assumptions (4)

ad hoc to paper The weighted quantile in Barber et al. (2023) remains valid for weights Ω learned from the calibration-training split.
Algorithm 1 fits Ω to minimize efficiency loss; the paper gives no theorem that this preserves the coverage bound of Lemma 3.4.
ad hoc to paper The soft quantile η in Eq. 12 coincides with the exact weighted quantile for the stated miscoverage level.
The softmax-with-temperature construction is a differentiable approximation; its finite-T version does not exactly reproduce the quantile used in the theory.
domain assumption The weights must be monotone non-increasing when sorted against non-conformity scores for the coverage bound to hold.
Barber et al.'s bound requires ordered weights; the learned Ω are unrestricted, and the paper neither states nor enforces this.
domain assumption The temporal shift between calibration-validation and test is small enough that the total variation distances in Eq. 6 are bounded.
Without a bound on d_TV, the coverage gap can be arbitrarily large; the paper neither estimates these distances nor discusses their scale.

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Cite this review

Pith. "Pith review of Non-exchangeable Conformal Prediction for Temporal Graph Neural Networks." pith.science (2026). https://pith.science/paper/IDHYHMCM

@misc{pith2026250702151,
  author       = {Pith},
  title        = {Pith review of: Non-exchangeable Conformal Prediction for Temporal Graph Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IDHYHMCM}},
  note         = {Machine review of arXiv:2507.02151}
}

read the original abstract

Conformal prediction for graph neural networks (GNNs) offers a promising framework for quantifying uncertainty, enhancing GNN reliability in high-stakes applications. However, existing methods predominantly focus on static graphs, neglecting the evolving nature of real-world graphs. Temporal dependencies in graph structure, node attributes, and ground truth labels violate the fundamental exchangeability assumption of standard conformal prediction methods, limiting their applicability. To address these challenges, in this paper, we introduce NCPNET, a novel end-to-end conformal prediction framework tailored for temporal graphs. Our approach extends conformal prediction to dynamic settings, mitigating statistical coverage violations induced by temporal dependencies. To achieve this, we propose a diffusion-based non-conformity score that captures both topological and temporal uncertainties within evolving networks. Additionally, we develop an efficiency-aware optimization algorithm that improves the conformal prediction process, enhancing computational efficiency and reducing coverage violations. Extensive experiments on diverse real-world temporal graphs, including WIKI, REDDIT, DBLP, and IBM Anti-Money Laundering dataset, demonstrate NCPNET's capability to ensure guaranteed coverage in temporal graphs, achieving up to a 31% reduction in prediction set size on the WIKI dataset, significantly improving efficiency compared to state-of-the-art methods. Our data and code are available at https://github.com/ODYSSEYWT/NCPNET.

Figures

Figures reproduced from arXiv: 2507.02151 by the authors.

**Figure 1.** Illustration of non-exchangeability in temporal graphs, where the shapes of nodes indicate the class memberships [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** The Overview of our proposed framework NCPNet, which is composed of two modules: (M1) topological and temporal diffusion-based non-conformity scores and (M2) efficiency-aware optimization. optimized quantile calculation through a combination of topological and temporal diffusion non-conformity scores and learnable weighted parameters that are backward efficient aware. The overarching framework is illustrated in [… view at source ↗

**Figure 3.** Efficiency and coverage on various training data sizes and mis-coverage level. Fig. 3a and Fig. 3b: [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Case study on IBM anti-money laundering dataset [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Scalability test on the node number and edge density [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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