SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Kexin Huang $^{1}$  Ying Jin $^{2}$  Emmanuel Candès $^{2,3}$  Jure Leskovec $^{1}$
$^{1}$  Department of Computer Science, Stanford University
$^{2}$  Department of Statistics, Stanford University
$^{3}$  Department of Mathematics, Stanford University
kexinh@cs.stanford.edu,ying531@stanford.edu,
candes@stanford.edu,jure@cs.stanford.edu
Graph Neural Networks (GNNs) are powerful machine learning prediction models on graph-structured data. However, GNNs lack rigorous uncertainty estimates, limiting their reliable deployment in settings where the cost of errors is significant. We propose conformalized GNN (CF-GNN), extending conformal prediction (CP) to graph-based models for guaranteed uncertainty estimates. Given an entity in the graph, CF-GNN produces a prediction set/interval that provably contains the true label with pre-defined coverage probability (e.g.  $90\%$ ). We establish a permutation invariance condition that enables the validity of CP on graph data and provide an exact characterization of the test-time coverage. Besides valid coverage, it is crucial to reduce the prediction set size/interval length for practical use. We observe a key connection between non-conformity scores and network structures, which motivates us to develop a topology-aware output correction model that learns to update the prediction and produces more efficient prediction sets/intervals. Extensive experiments show that CF-GNN achieves any pre-defined target marginal coverage while significantly reducing the prediction set/interval size by up to  $74\%$  over the baselines. It also empirically achieves satisfactory conditional coverage over various raw and network features.
Graph Neural Networks (GNNs) have shown great potential in learning representations for graph-structured data, which has led to their widespread adoption in weather forecasting [29], drug discovery [31], and recommender systems [46], etc. As GNNs are increasingly deployed in high-stakes settings, it is important to understand the uncertainty in the predictions they produce. One prominent approach to uncertainty quantification is to construct a prediction set/interval that informs a plausible range of values the true outcome may take. A large number of methods have been proposed to achieve this goal [17, 49, 28, 44]. However, these methods often lack theoretical and empirical guarantees regarding their validity, i.e. the probability that the prediction set/interval covers the outcome [2]. This lack of rigor hinders their reliable deployment in situations where errors can be consequential.
Conformal prediction [43] (CP) is a framework for producing statistically guaranteed uncertainty estimates. Given a user-specified miscoverage level  $\alpha \in (0,1)$ , it leverages a set of "calibration" data to output prediction sets/intervals for new test points that provably include the true outcome with probability at least  $1 - \alpha$ . Put another way, the conformal prediction sets provably only miss the test outcomes at most  $\alpha$  fraction of the time. With its simple formulation, clear guarantee and
distribution-free nature, it has been successfully applied to various problems in computer vision [2, 4], causal inference [30, 22, 47], time series forecasting [11, 48], and drug discovery [21].
Despite its success in numerous domains, conformal prediction has remained largely unexplored in the context of graph-structured data. One primary challenge is that it is unclear if the only, yet crucial, assumption for CP-exchangeability between the test and calibration samples-holds for graph data. When applying conformal prediction, exchangeability is usually ensured by independence among the trained model, the calibration data, and test samples (see Appendix B for more discussion). However, in the transductive setting, GNN training employs all nodes within the same graph-including test points-for message passing, creating intricate dependencies among them. Thus, to deploy conformal prediction for graph data, the first challenge is to identify situations where valid conformal prediction is possible given a fitted GNN model that already involves test information.
Efficiency is another crucial aspect of conformal prediction for practical use: a prediction set/interval with an enormous set size/interval length might not be practically desirable even though it achieves valid coverage. Therefore, the second major challenge is to develop a graph-specific approach to reduce the size of the prediction set or the length of the prediction interval (dubbed as inefficiency hereafter for brevity) while retaining the attractive coverage property of conformal prediction.
Present work. We propose conformalized GNN (CF-GNN),<sup>1</sup> extending conformal prediction to GNN for rigorous uncertainty quantification over graphs. We begin by establishing the validity of conformal prediction for graphs. We show that in the transductive setting, regardless of the dependence among calibration, test, and training nodes, standard conformal prediction [43] is valid as long as the score function (whose definition will be made clear in Section 2) is invariant to the ordering of calibration and test samples. This condition is easily satisfied by popular GNN models. Furthermore, we provide an exact characterization of the empirical test-time coverage.
Subsequently, we present a new approach that learns to optimize the inefficiencies of conformal prediction. We conduct an empirical analysis which reveals that inefficiencies are highly correlated along the network edges. Based on this observation, we add a topology-aware correction model that updates the node predictions based on their neighbors. This model is trained by minimizing a differentiable efficiency loss that simulates the CP set sizes/interval lengths. In this way, unlike the raw prediction that is often optimized for prediction accuracy, the corrected GNN prediction is optimized to yield smaller/shorter conformal prediction sets/intervals. Crucially, our approach aligns with the developed theory of graph exchangeability, ensuring valid coverage guarantees while simultaneously enhancing efficiency.
We conduct extensive experiments across 15 diverse datasets for both node classification and regression with 8 uncertainty quantification (UQ) baselines, covering a wide range of application domains. While all previous UQ methods fail to reach pre-defined target coverage, CF-GNN achieves the pre-defined empirical marginal coverage. It also significantly reduces the prediction set sizes/interval lengths by up to  $74\%$  compared with a direct application of conformal prediction to GNN. Such improvement in efficiency does not appear to sacrifice adaptivity: we show that CF-GNN achieves strong empirical conditional coverage over various network features.
Let  $G = (\mathcal{V}, \mathcal{E}, \mathbf{X})$  be a graph, where  $\mathcal{V}$  is a set of nodes,  $\mathcal{E}$  is a set of edges, and  $\mathbf{X} = \{\mathbf{x}_v\}_{v \in \mathcal{V}}$  is the attributes, where  $\mathbf{x}_v \in \mathbb{R}^d$  is a  $d$ -dimensional feature vector for node  $v \in \mathcal{V}$ . The label of node  $v$  is  $y_v \in \mathcal{V}$ . For classification,  $\mathcal{V}$  is the discrete set of possible label classes. For regression,  $\mathcal{V} = \mathbb{R}$ .
Transductive setting. We focus on transductive node classification/regression problems with random data split. In this setting, the graph  $G$  is fixed. At the beginning, we have access to  $\{(\mathbf{x}_v, y_v)\}_{v \in \mathcal{D}}$  as the "training" data, as well as test data  $\mathcal{D}_{\text{test}}$  with unknown labels  $\{y_v\}_{v \in \mathcal{D}_{\text{test}}}$ . Here  $\mathcal{D}$  and  $\mathcal{D}_{\text{test}}$  are disjoint subsets of  $\mathcal{V}$ . We work on the prevalent random split setting where nodes in  $\mathcal{D}$  and  $\mathcal{D}_{\text{test}}$  are randomly allocated from the entire graph, and the test sample size is  $m = |\mathcal{D}_{\text{test}}|$ . The training node set  $\mathcal{D}$  is then randomly split into  $\mathcal{D}_{\text{train}} / \mathcal{D}_{\text{valid}} / \mathcal{D}_{\text{calib}}$  of fixed sizes, the training/validation/calibration set, correspondingly. A perhaps nonstandard point here is that we withhold a subset  $\mathcal{D}_{\text{calib}}$  as "calibration" data in order to apply conformal prediction later on. During the training step, the data
<sup>1</sup>See Figure 1 for an overview. The code is available at https://github.com/snap-stanford/conformalized-gnn.
Figure 1: Conformal prediction for graph-structured data. (1) GNN training. We first use standard GNN training to obtain a base GNN model  $(\mathrm{GNN}_{\theta})$  that produces prediction scores  $\widehat{\mu}(X_i)$  for node  $i$ . It is fixed once trained. (2) Conformal correction. Since the training step is not aware of the conformal calibration step, the size/length of prediction sets/intervals (i.e. efficiency) are not optimized. We propose a novel correction step that learns to correct the prediction to achieve desirable properties such as efficiency. We use a topology-aware correction model  $\mathrm{GNN}_{\vartheta}$  that takes  $\widehat{\mu}(X_i)$  as the input node feature and aggregates information from its local subgraph to produce an updated prediction  $\widehat{\mu}(X_i)$ .  $\vartheta$  is trained by simulating the conformal prediction step and optimizing a differentiable inefficiency loss. (3) Conformal prediction. We prove that in a transductive random split setting, graph exchangeability holds (Section 3) given permutation invariance. Thus, standard CP can be used to produce a prediction set/interval based on  $\widehat{\mu}$  that includes true label with pre-specified coverage rate  $1 - \alpha$ .
$\{(\mathbf{x}_v, y_v)\}_{v \in \mathcal{D}_{\mathrm{train}} \cup \mathcal{D}_{\mathrm{valid}}}$ , the attribute information in  $\{\mathbf{x}_v\}_{v \in \mathcal{D}_{\mathrm{calib}} \cup \mathcal{D}_{\mathrm{test}}}$  and the entire graph structure  $(\mathcal{V}, \mathcal{E})$  are available to the GNN to compute training nodes representations, while  $\{y_v\}_{v \in \mathcal{D}_{\mathrm{calib}} \cup \mathcal{D}_{\mathrm{test}}}$  are not seen.
Graph Neural Networks (GNNs). GNNs learn compact representations that capture network structure and node features. A GNN generates outputs through a series of propagation layers [12], where propagation at layer  $l$  consists of the following three steps: (1) Neural message passing. GNN computes a message  $\mathbf{m}_{uv}^{(l)} = \mathrm{MSG}(\mathbf{h}_u^{(l-1)}, \mathbf{h}_v^{(l-1)})$  for every linked nodes  $u, v$  based on their embeddings from the previous layer  $\mathbf{h}_u^{(l-1)}$  and  $\mathbf{h}_v^{(l-1)}$ . (2) Neighborhood aggregation. The messages between node  $u$  and its neighbors  $\mathcal{N}_u$  are aggregated as  $\widehat{\mathbf{m}}_u^{(l)} = \mathrm{AGG}(\widehat{\mathbf{m}}_{uv}^{(l)}|v \in \mathcal{N}_u)$ . (3) Update. Finally, GNN uses a non-linear function to update node embeddings as  $\mathbf{h}_u^{(l)} = \mathrm{UPD}(\widehat{\mathbf{m}}_u^{(l)}, \widehat{\mathbf{h}}_u^{(l-1)})$  using the aggregated message and the embedding from the previous layer. The obtained final node representation is then fed to a classifier or regressor to obtain a prediction  $\widehat{\mu}(X)$ .
Conformal prediction. In this work, we focus on the computationally efficient split conformal prediction method [43]. Given a predefined miscoverage rate  $\alpha \in [0,1]$ , it proceeds in three steps: (1) non-conformity scores. CP first obtains any heuristic notion of uncertainty called non-conformity score  $V\colon \mathcal{X}\times \mathcal{Y}\to \mathbb{R}$ . Intuitively,  $V(x,y)$  measures how  $y$  "conforms" to the prediction at  $x$ . An example is the predicted probability of a class  $y$  in classification or the residual value  $V(x,y) = |y - \hat{\mu} (x)|$  in regression for a predictor  $\hat{\mu}\colon \mathcal{X}\rightarrow \mathcal{Y}$ . (2) Quantile computation. CP then takes the  $1 - \alpha$  quantile of the non-conformity scores computed on the calibration set. Let  $\{(X_i,Y_i)\}_{i = 1}^n$  be the calibration data, where  $n = |\mathcal{D}_{\mathrm{calib}}|$ , and compute  $\hat{\eta} = \mathrm{quantile}(\{V(X_1,Y_1),\dots ,V(X_n,Y_n)\},(1-\alpha)(1 + \frac{1}{n}))$ . (3) Prediction set/interval construction. Given a new test point  $X_{n + 1}$ , CP constructs a prediction set/interval  $C(X_{n + 1}) = \{y\in \mathcal{Y}:V(X_{n + 1},y)\leq \hat{\eta}\}$ . If  $\{Z_i\}_{i = 1}^{n + 1}\coloneqq \{(X_i,Y_i)\}_{i = 1}^{n + 1}$  are exchangeable, then  $V_{n + 1}\coloneqq V(X_{n + 1},Y_{n + 1})$  is exchangeable with  $\{V_i\}_{i = 1}^n$  since  $\hat{\mu}$  is given. Thus,  $\widehat{C} (X_{n + 1})$  contains the true label with predefined coverage rate [43]:  $P\{Y_{n + 1}\in C(X_{n + 1})\} = \mathbb{P}\{V_{n + 1}\geq \mathrm{Quantile}(\{V_1,\ldots ,V_{n + 1}\} ,1 - \alpha)\geq 1 - \alpha$  due to exchangeability of  $\{V_i\}_{i = 1}^{n + 1}$ . This framework works for any non-conformity score. CF-GNN is similarly non-conformity score-agnostic. However, for demonstration, we focus on two popular scores, described in detail below.
Adaptive Prediction Set (APS). For the classification task, we use the non-conformity score in APS proposed by [37]. It takes the cumulative sum of ordered class probabilities till the true class. Formally, given any estimator  $\widehat{\mu}_j(x)$  for the conditional probability of  $Y$  being class  $j$  at  $X = x$ ,  $j = 1, \ldots, |\mathcal{Y}|$ , we denote the cumulative probability till the  $k$ -th most promising class as  $V(x, k) = \sum_{j=1}^{k} \widehat{\mu}_{\pi_{(j)}}(x)$ ,
See Appendix B for discussion on full and split conformal prediction. We refer to the split conformal prediction when we describe conformal prediction throughout the paper.
3Exchangeability definition: for any  $z_{1},\ldots ,z_{n + 1}$  and any permutation  $\pi$  of  $\{1,\dots ,n + 1\}$  , it holds that  $\mathbb{P}((Z_{\pi (1)},\ldots ,Z_{\pi (n + 1)}) = (z_1,\ldots ,z_{n + 1})) = \mathbb{P}((Z_1,\ldots ,Z_{n + 1}) = (z_1,\ldots ,z_{n + 1})).$
where  $\pi$  is a permutation of  $\mathcal{V}$  so that  $\widehat{\mu}_{\pi(1)}(x) \geq \widehat{\mu}_{\pi(2)}(x) \geq \dots \geq \widehat{\mu}_{\pi(|\mathcal{V}|)}(x)$ . Then, the prediction set is constructed as  $C(x) = \{\pi(1), \dots, \pi(k^*)\}$ , where  $k^* = \inf \{k : \sum_{j=1}^{k} \widehat{\mu}_{\pi(j)}(x) \geq \widehat{\eta}\}$ .
Conformalized Quantile Regression (CQR). For the regression task, we use CQR in [36]. CQR is based on quantile regression (QR). QR obtains heuristic estimates  $\widehat{\mu}_{\alpha /2}(x)$  and  $\widehat{\mu}_{1 - \alpha /2}(x)$  for the  $\alpha /2$  th and  $1 - \alpha /2$  -th conditional quantile functions of  $Y$  given  $X = x$ . The non-conformity score is  $V(x,y) = \max \{\widehat{\mu}_{\alpha /2}(x) - y,y - \widehat{\mu}_{1 - \alpha /2}(x)\}$ , interpreted as the residual of true label projected to the closest quantile. The prediction interval is then  $C(x) = [\widehat{\mu}_{\alpha /2}(x) - \widehat{\eta},\widehat{\mu}_{1 - \alpha /2}(x) + \widehat{\eta} ]$
In its vanilla form, the non-conformity score (including APS and CQR) in CP does not depend on the calibration and test data. That means,  $\{\mathbf{x}_v\}_{v\in \mathcal{D}_{\mathrm{calib}}\cup \mathcal{D}_{\mathrm{test}}}$  are not revealed in the training process of  $V$  which is the key to exchangeability. In contrast, GNN training typically leverages the entire graph, and hence the learned model depends on the calibration and test attributes in a complicated way. In the following, for clarity, we denote any non-conformity score built on a GNN-trained model as
$$
V (x, y; \{z _ {v} \} _ {v \in D _ {\text {t r a i n}} \cup D _ {\text {v a l i d}}}, \{\mathbf {x} _ {v} \} _ {v \in D _ {\text {c a l i b}} \cup D _ {\text {t e s t}}}, \mathcal {V}, \mathcal {E})
$$
to emphasize its dependence on the entire graph, where  $z_v = (\mathbf{x}_v, Y_v)$  for  $v \in \mathcal{V}$ .
Evaluation metrics. The goal is to ensure valid marginal coverage while decreasing the inefficiency as much as possible. Given the test set  $\mathcal{D}_{\mathrm{test}}$ , the empirical marginal coverage is defined as Coverage :=  $\frac{1}{|\mathcal{D}_{\mathrm{test}}|}\sum_{i\in \mathcal{D}_{\mathrm{test}}}\mathbb{I}(Y_i\in C(X_i))$ . For the regression task, inefficiency is measured as the interval length while for the classification task, the inefficiency is the size of the prediction set: Ineff :=  $\frac{1}{|\mathcal{D}_{\mathrm{test}}|}\sum_{i\in \mathcal{D}_{\mathrm{test}}} |C(X_i)|$ . The larger the length/size, the more inefficient. Note that inefficiency of conformal prediction is different from accuracy of the original predictions. Our method does not change the trained prediction but modifies the prediction sets from conformal prediction.
3 Exchangeability and Validity of Conformal Prediction on Graph
To deploy CP for graph-structured data, we first study the exchangeability of node information under the transductive setting. We show that under a general permutation invariant condition (Assumption 1), exchangeability of the non-conformity scores is still valid even though GNN training uses the calibration and test information; this paves the way for applying conformal prediction to GNN models. We develop an exact characterization of the test-time coverage of conformal prediction in such settings. Proofs of these results are in Appendix A.1.
Assumption 1. For any permutation  $\pi$  of  $\mathcal{D}_{\mathrm{calib}} \cup \mathcal{D}_{\mathrm{test}}$ , the non-conformity score  $V$  obeys
$$
\begin{array}{l} V (x, y; \{z _ {v} \} _ {v \in \mathcal {D} _ {\text {t r a i n}} \cup \mathcal {D} _ {\text {v a l i d}}}, \{\mathbf {x} _ {v} \} _ {v \in \mathcal {D} _ {\text {c a l i b}} \cup \mathcal {D} _ {\text {t e s t}}}, \mathcal {V}, \mathcal {E}) \\ = V (x, y; \{z _ {v} \} _ {v \in \mathcal {D} _ {\text {t r a i n}} \cup \mathcal {D} _ {\text {v i d i l}}}, \{\mathbf {x} _ {\pi (v)} \} _ {v \in \mathcal {D} _ {\text {c a l i b}} \cup \mathcal {D} _ {\text {t e s t}}}, \mathcal {V} _ {\pi}, \mathcal {E} _ {\pi}), \\ \end{array}
$$
where  $(\mathcal{V}_{\pi},\mathcal{E}_{\pi})$  represents a graph where  $\mathcal{D}_{\mathrm{calib}}\cup \mathcal{D}_{\mathrm{test}}$  nodes (indices) are permuted according to  $\pi$ .
Assumption 1 imposes a permutation invariance condition for the GNN training, i.e., model output/non-conformity score is invariant to permuting the ordering of the calibration and test nodes (with their edges permuted accordingly) on the graph. To put it differently, different selections of calibration sets do not modify the non-conformity scores for any node in the graph. GNN models (including those evaluated in our experiments) typically obey Assumption 1, because they only use the structures and attributes in the graph without information on the ordering of the nodes [24, 16, 12].
For clarity, we write the calibration data as  $\{(X_i,Y_i)\}_{i = 1}^n$ , where  $X_{i} = X_{v_{i}}$ , and  $v_{i}\in \mathcal{D}_{\mathrm{calib}}$  is the  $i$ -th node in the calibration data under some pre-defined ordering. Similarly, the test data are  $\{(X_{n + j},Y_{n + j})\}_{j = 1}^{m}$ , where  $X_{n + j} = X_{v_j}$ , and  $v_{j}\in \mathcal{D}_{\mathrm{test}}$  is the  $j$ -th node in the test data. We write
$$
V _ {i} = V \left(X _ {i}, Y _ {i}; \{z _ {i} \} _ {i \in \mathcal {D} _ {\text {t r a i n}} \cup \mathcal {D} _ {\text {v a l i d}}}, \{\mathbf {x} _ {v} \} _ {v \in \mathcal {D} _ {\text {c a l i b}} \cup \mathcal {D} _ {\text {t e s t}}}, \mathcal {V}, \mathcal {E}\right), \quad i = 1, \dots , n, n + 1, \dots , n + m.
$$
$V_{i}$  is a random variable that depends on the training process and the split of calibration and test data. The next lemma shows that under Assumption 1, the non-conformity scores are still exchangeable.
Lemma 2. In the transductive setting described in Section 2, conditional on the entire unordered graph  $(\mathcal{V},\mathcal{E})$ , all the attribute and label information  $\{(\mathbf{x}_v,y_v)\}_{v\in \mathcal{V}}$ , and the index sets  $\mathcal{D}_{train}$  and  $\mathcal{D}_{ct} := \mathcal{D}_{calib} \cup \mathcal{D}_{test}$ , the unordered set of the scores  $[V_i]_{i=1}^{n+m}$  are fixed. Also, the calibration scores  $\{V_i\}_{i=1}^n$  are a simple random sample from  $\{V_i\}_{i=1}^{n+m}$ . That is, for any subset  $\{v_1,\ldots,v_n\} \subseteq \{V_i\}_{i=1}^{n+m}$  of size  $n$ ,  $P(\{V_i\}_{i=1}^n = \{v_1,\ldots,v_n\} \mid \{V_i\}_{i=1}^{n+m}) = 1 / \binom{n}{|\mathcal{D}_{ct}|}$ .
Based on Lemma 2, we next show that any permutation-invariant non-conformity score leads to valid prediction sets, and provide an exact characterization of the distribution of test-time coverage.
Theorem 3. Given any score  $V$  obeying Assumption 1 and any confidence level  $\alpha \in (0,1)$ , we define the split conformal prediction set as  $\widehat{C}(x) = \{y: V(x,y) \leq \widehat{\eta}\}$ , where
$$
\widehat {\eta} = \inf  \left\{\eta : \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbb {1} \{V (X _ {i}, Y _ {i}) \leq \eta \} \geq (1 - \alpha) (1 + 1 / n) \right\}.
$$
Then  $\mathbb{P}(Y_{n + j}\in \widehat{C} (X_{n + j}))\geq 1 - \alpha ,\forall j = 1,\ldots ,m.$  Moreover, define  $\widehat{\mathrm{Cover}} = \frac{1}{m}\sum_{j = 1}^{m}\mathbb{1}\{Y_{n + j}\in$ $\widehat{C} (X_{n + j})\}$  . If the  $V_{i}$  s,  $i\in \mathcal{D}_{calib}\cup \mathcal{D}_{test},$  have no ties almost surely, then for any  $t\in (0,1)$
$$
\mathbb {P} \big (\widehat {\text {C o v e r}} \leq t \big) = 1 - \Phi_ {\mathrm {H G}} \big (\lceil (n + 1) (1 - q) \rceil - 1; m + n, n, \lceil (1 - q) (n + 1) \rceil + \lceil m t \rceil \big),
$$
where  $\Phi_{\mathrm{HG}}(\cdot ;N,n,k)$  denotes the cumulative distribution function of a hyper-geometric distribution with parameters  $N,n,k$  (drawing  $k$  balls from an urn wherein  $n$  out of  $N$  balls are white balls).
Figure 2 plots the probability density functions (p.d.f.) of  $\widehat{\text{Cover}}$  at a sequence of  $t \in [0,1]$  fixing  $n = 1000$  while varying  $m$ . The exact distribution described in Theorem 3 is useful in determining the size of the calibration data in order for the test-time coverage to concentrate sufficiently tightly around  $1 - \alpha$ . More discussion and visualization of  $\widehat{\text{Cover}}$  under different values of  $(n,m)$  are in Appendix A.2. Note that similar exchangeability and validity results are obtained in several concurrent works [15, 33], yet without exact characterization of the test-time coverage.
4 CF-GNN: Conformalized Graph Neural Networks
Figure 2: P.d.f. of Cover for  $n = 1000$  and  $\alpha = 0.05$ ; curves represent different values of test sample size  $m$ .
We now propose a new method called CF-GNN to reduce inefficiency while maintaining valid coverage. The key idea of CF-GNN is to boost any given non-conformity score with graphical information. The method illustration is in Figure 1 and pseudo-code is in Appendix C.
Efficiency-aware boosting. Standard CP takes any pre-trained predictor to construct the prediction set/interval (see Section 2). A key observation is that the training stage is not aware of the post-hoc stage of conformal prediction set/interval construction. Thus, it is not optimized for efficiency. Our high-level idea is to include an additional correction step that boosts the non-conformity score, which happens after the model training and before conformal prediction. To ensure flexibility and practicality, our framework is designed as a generic wrapper that works for any pre-trained GNN model, without changing the base model training process.