Quantifying Distributional Invariance in Causal Subgraph for IRM-Free Graph Generalization

Thank you for your appreciation on our work!

Weakness

• "Intuitive explanation of Section 2.3.2" To better illustrate the connection between cross-environment/domain risk and the norm, we present the examples below:

  1. Consider the weight matrix $W\in\mathbb{R}^{4\times3}$ given by:
  \begin{pmatrix} (1 & 0 & 0), (0 & 1 & 0), (1 & 0 & 0), (0 & 1 & 0) \end{pmatrix}$$ which has rank $\mathrm{rank}(W)=2$. 1. We now pick two 'environmental' inputs $[C1,S1]$ from environment 1 and $[C2,S2]$ from environment 2, each decomposed into a causal part C and a spurious part S. From Theorem 1 that causal subgraph minimizes distribution shift across environments, we suppose $\mathrm{sim}(S_1,S_2)>\mathrm{sim}(C_1,C_2)$. Then we define: - Env 1: $$C_1 = \begin{pmatrix}1,0,0\end{pmatrix},\quad S_1 = \begin{pmatrix}0,1,0\end{pmatrix}.$$ - Env 2: $$C_2 = \begin{pmatrix}0.9,0.1,0\end{pmatrix},\quad S_2 = \begin{pmatrix}0,0.1,0\end{pmatrix}.$$ Here $\mathrm{sim}(C_1,C_2) = 0.9$ while $\mathrm{sim}(S_1,S_2) = 0.1$, which satisfies $\mathrm{sim}(S_1,S_2)>\mathrm{sim}(C_1,C_2)$. 1. Ignoring the bias, we then consider the norm of the causal outputs after the model of weight $W$: $$W\,C_1 = \begin{pmatrix}1,0,1,0\end{pmatrix},\quad W\,C_2 = \begin{pmatrix}0.9,0.1,0.9,0.1\end{pmatrix},\quad$$ $$\|W\,C_1\|_2 = \sqrt2 \approx 1.4142,$$ $$\|W\,C_2\|_2 = \sqrt{0.9^2 + 0.1^2 + 0.9^2 + 0.1^2} = \sqrt{1.64} \approx 1.2806.$$ 1. The norm of the non-causal outputs after the model of weight $W$: $$W\,S_1 = \begin{pmatrix}0,1,0,1\end{pmatrix},\quad \|W\,S_1\|_2 = \sqrt2 \approx 1.4142,$$ $$W\,S_2 = \begin{pmatrix}0,0.1,0,0.1\end{pmatrix},\quad \|W\,S_2\|_2 = \sqrt{0.1^2 + 0.1^2} = \sqrt{0.02} \approx 0.1414.$$ 1. We can get that: $$\mathbb{E}(\|W\,C\|_2) \approx 1.34 > \mathbb{E}(\|W\,C\|_2) \approx 0.78,$$ justifying the relationship between norm and distribution shift. We will refine the description of the relationship in Section 2.3.2 and replace the example in Appendix F with this one to make a better explanation.

• "Compared to environment-inference-based invariant learning methods" To clarify the distinction between the alternating optimization in our IDG method and environment inference approaches (e.g., EIIL, GOODHSE, MoleOOD), the following points are highlighted:

  1. In IDG training, alternating optimization is used mainly to prevent the subgraph extractor and predictor from converging to degenerate solutions or experiencing gradient conflicts when trained jointly under the same objective (classification loss and norm regularization). The extractor and predictor parameters are still updated alternately within each epoch, so they still train in a "collaborative" manner rather than being fully "decoupled" (for example, training the extractor to convergence before the predictor would be completely unworkable). Moreover, no explicit environment splits or pseudo-environment labels are produced.
  2. In environment inference approaches (e.g., EIIL, GOODHSE, MoleOOD), alternating optimization is required because the invariant learning phase demands explicit environment partitions or labels. These methods first infer environment information and then carry out invariant learning or related strategies. This two-stage process—environment inference followed by invariant learning—should be strictlly followed and can be fully decoupled, as in EIIL and MoleOOD.

  We also conduct experiments on DrugOOD to compare performance under alternating versus non-alternating optimization. The results below show that alternating optimization significantly outperforms non-alternating optimization.

  DrugOOD	IC50Scaffold	IC50Size	IC50Assay	EC50Scaffold	EC50Size	EC50Assay
  ERM	68.79	67.50	71.63	64.98	65.10	67.39
  IRM	67.22	61.58	71.15	63.86	59.19	67.77
  MoleOOD	68.02	66.51	71.38	66.69	65.09	73.25
  GOODHSE	69.43	68.64	73.26	69.15	66.87	80.15
  IDG	69.97	69.02	73.34	69.57	68.03	80.54
  IDG-noalter	67.97	67.23	70.84	65.43	65.67	71.24

  Thank you for your kind attention to the training procedure. We will include comparisons with the environment-inference-based invariant learning methods you mentioned—such as GOODHSE and MoleOOD—in both the Related Work and Experimental sections in the future version.

Questions

• "Handle Real-World Shifts" Indeed, on real-world datasets our method can also mitigate distribution shifts to some extent. The results on the DrugOOD benchmark, as you mentioned, are presented in the table above in the response of weakness 2. More importantly, we observe that our assumption—that causal subgraph distribution shifts are relatively small—still holds across multiple real-world datasets. Because real-world datasets lack ground-truth annotations for causal subgraphs, validating this hypothesis on them is extremely challenging. Nevertheless, we devised two distinct experimental setups to validate our assumption under different settings:

  • To further support the validity of the causal subgraphs being more invariant in real applications, we conducte two experiments on real world datasets comparing causal and non-causal subgraphs. Quantifying distribution shifts in real-world graphs is challenging, since such data with ground-truth annotation of causal subgraph are rarely available, so we devise two distinct experimental setups to validate our assumption under different settings. Distribution shift was measured by the maximum mean discrepancy (MMD) between representations produced by the a same GNN model, using a Gaussian kernel on standardized features with bandwidth set by the median heuristic. We denote the cross‐environment distribution shift of causal and non‐causal subgraph by $\Delta{G_c}\approx MMD(G_c^{e1},G_c^{e2}), \Delta{G_s}\approx MMD(G_s^{e1},G_s^{e2})$ and compute the ratio of $\Delta{G_s}/\Delta{G_c}$. Table values above 1 indicate that non‐causal subgraphs undergo larger distribution shifts across environments. The experimental designs and results are described below:

    • Experiment 1. Four real-world molecular graph datasets with ground-truth causal subgraphs—Mutag, Benzene, Fluoride, and Alkane- from the graph explanation literature. Each dataset was split into two environments by molecular size or random. The results are below:

    $\Delta{G_s}/\Delta{G_c}$	Mutag		Benzene		Fluoride		Alkane
    split	size	random	size	random	size	random	size	random
    MMD	3.43	1.56	5.79	1.83	6.67	1.94	6.48	2.03

    The results indicate that, the non-causal subgraph exhibits a larger shift than the causal subgraph across all environment partitions($\Delta{G_s}>\Delta{G_c}$), which aligns with Assumption 2.

    • Experiment 2. We apply the same evaluation to the out-of-distribution (OOD) dataset used in our study. Due to the absence of true ground truth, we treat the subgraphs extracted by our method, GIL, and GSAT as causal subgraphs. The results are below:

    $\Delta{G_s}/\Delta{G_c}$	HIV-scaffold	HIV-size	SST2	Twitter	IC50-size	IC50-scaffold
    IDG	7.28	3.915	1.963	3.983	1.1784	1.5296
    GIL	1.969	1.429	1.044	3.25	1.154	1.74
    GSAT	5.292	4.671	1.868	1.244	1.128	1.156

    The results indicate that, the extracted non-causal subgraph by different methods still exhibits a larger shift than the causal, which also aligns with Assumption 2.

  These two experiments, viewed from different perspectives, clarify the validity of Assumption 2—that causal subgraphs typically exhibit smaller distribution shifts across environments.

• "Continuous IRM" Indeed, our norm can be viewed as a continuous indicator of environments, but unlike continuous IRM, our method does not rely on any hard or soft partitioning of environments. It only assumes that the dataset contains data from different environments. Regarding the continuous IRM you mentioned, such as CORAL and DIR align the full-batch feature distributions—either in a latent or explicitly defined environment space—and enforce a single predictor to perform consistently across all environments. In contrast, our approach does not group or softly assign environments, nor does it explicitly align batch-level environmental distributions. Instead, we maximize a norm-based objective to identify at the sample-level causal subgraphs that are least prone to drift, without any hard or soft environment partitioning.

• "Environment Modeling Help the Method?" Yes our method is fully compatible with IRM. When explicit environment labels are available, shifts in non-causal features across environments can be more evident. Furthermore, we integrat the IRM objective into our framework. As shown in the table below, this integration yields improved performance.

  Method	PIIF	FIIF	SP=0.6	SP=0.7	SP=0.8	SP=0.9
  ERM	62.45	37.22	91.67	91.25	90.63	85.92
  IRM	68.34	44.33	91.85	91.25	90.48	85.38
  IDG	79	83.67	92.67	92.06	91.89	91.33
  IDG+ERM	80.52	84.63	92.67	92.34	92.00	91.89

  In this table, SP denotes the probability of spurious correlation. Lower SP values indicate weakly informative environments. PIIF and FIIF stand for partially informative and fully informative scenarios, respectively. These datasets adhere to the GALA[2], CIGA[3], and LECI[1] settings.

[1] Joint Learning of Label and Environment Causal Independence for Graph Out-of-Distribution Generalization

[2] Does Invariant Graph Learning via Environment Augmentation Learn Invariance?

[3] Learning Causally Invariant Representations for Out-of-Distribution Generalization on Graphs

Thank you for your appreciation on our work!

Weakness

• "Optimization framework comparing to DIR/GIL" The extractor architecture employed here, which maps node features to edge masks, is widely used in Graph OOD research. Besides GIL and DIR, it is also adopted by methods such as LECI[1], GALA[2], CIGA[3], DiSC[4], and GSAT[5]. Its origins trace back to earlier graph explanation methods like PGExplainer[6], as well as similar techniques in the text[7] and vision domains[8]. We would like to point out that the key contribution of this work is a new theoretical framework that, without relying on invariant learning or any environmental labels, guarantees the extractor identifies causal subgraphs rather than the extracting procedure. The main challenge of this framework lies in finding an effective optimization approach that jointly trains the extractor and classifier to improve their coordination on extracting causal features and predicting. Compared to DIR and GIL, our method offers the following key distinctions and advantages:

  • 1, Both DIR and GIL build on invariant risk minimization (IRM) [9] by attempting to find environment-invariant predictors, which requires inferring each sample's environment label. DIR induces different distributions via explicit intervention perturbations, while GIL derives environment labels through unsupervised clustering of k-means. Neither approach has theoretical guarantees that the constructed environments match the real data-generating distributions, which is also noted in related work such as GALA[2] and CIGA[3]. Our experiments below further reveal two points:

    • Removing the invariance (variance) objective from DIR and GIL (denoted DIR-0 and GIL-0) can actually improve performance in some settings.
    • In certain scenarios, both DIR and GIL perform even worse than standard ERM.

    Dataset	PIIF	FIIF	SP=0.6	SP=0.7	SP=0.8	SP=0.9
    ERM	62.45	37.22	91.67	91.25	90.63	85.92
    IRM	68.34	44.33	91.85	91.25	90.48	85.38
    GIL	59.66	64.66	89.78	80.33	72	51.09
    GIL-0	53.67	37.49	90.78	78.45	79	83.46
    DIR	69.33	42	88.7	87.82	86	84.8
    DIR-0	53.67	36	87.67	89.79	87.8	86.74
    IDG	79	83.67	92.67	92.06	91.89	91.33

    These datasets adhere to the GALA[2], CIGA[3], and LECI[1] settings. PIIF and FIIF are two typical distribution-shift scenarios, and SP denotes the spurious-correlation probability. These results indicate that the optimization strategies of DIR and GIL are not universally effective. In contrast, our method is completely IRM-free and requires no environment information: it identifies causal subgraphs simply by finding solutions that satisfy the minimal shift criterion via norm optimization.

  • 2, From an optimization perspective, DIR also demands a large memory bank for non-causal features $S$, which hinders its scalability to large datasets, and GIL must collect representations across the entire batch for clustering, both of which incur substantial resource overhead. By comparison, our approach only requires two forward passes and minimal additional computation of computing the norm, demonstrating huge efficiency advantages.

  • We will include a detailed comparison against DIR, GIL, and other related methods in the Related Work and Discussion sections in the paper.

• "Time complexity analysis" The time complexity of the IDG method is $O(m d + n d^2)$, where $n$, $m$, and $d$ denote the average number of nodes, edges, and feature dimensions per graph. Specifically:

  • Message passing and node updates (per GNN layer) incur a cost of $O(m d + n d^2)$.
  • Edge scoring and top-r selection can be completed in $O(m \log m)$ in the worst case.
  • Norm regularization and compactness term require $O(m)$ time.

  Thus, a single forward–backward pass on one graph remains $O(m d + n d^2)$. In our method, the norm calculation incurs minimal computational overhead (less than that of edge selection). To further demonstrate its efficiency, we measured both training time (per epoch) and inference time on the Motif, HIV, and SST2 datasets. The results are presented in the table below:

  	Motif-basis		HIV-scaffold		SST2-length
  Time(ms)	Training(ms)	Inference(ms)	Training(ms)	Inference(ms)	Training(ms)	Inference(ms)
  GIL	56801	2037	102594	22057	86658	16472
  DIR	14415	794	38477	1098	32392	9377
  IDG	11262	493	35232	995	26326	7659

• "Explanation of 3 assumption" We discuss the rationale of these three core assumptions here:

  • Assumption 1 (Causal Mechanism) The generation of causal labels is fully determined by the causal subgraph $G_c$. This assumption derives from the Independent Causal Mechanism (ICM) hypothesis [9] and underpins Graph OOD methods such as CIGA[3] and LECI[1]. Invariant Risk Minimization (IRM) likewise relies on it by seeking representations that yield the same optimal classifier across environments. For example, if the causal subgraph (e.g., functional groups responsible for key chemical properties) is known, one can infer a molecule's chemical properties regardless of its distribution of origin (environments).

  • Assumption 2 (Environmental Diversity and Interventions) This assumption originates from Peters et al.'s ICP framework and its extensions [10–13], this assumption holds that distributional shifts across environments arise from sparse interventions on the non-causal subgraph $G_s$, while $G_c$ remains stable. This hypothesis has been adopted by Graph OOD methods such as LECI, which impose an even stronger assumption—that the environment $E$ is independent of $G_c$. i.e. $E \perp G_c$. For instance, the types and structures of functional groups in a molecule are limited and stable, whereas non-causal parts (e.g., heteroatoms) vary widely across environments.

  • Assumption 3 (Effective Classifier Support) This assumption is based on the classical domain-adaptation bounds presented first in [14], this assumption requires overlap between the support of source and target distributions to guarantee generalization. Since classification relies on $G_c$, the support of the causal-subgraph distribution in the test domain must lie within that of the training domain. For example, a classifier can correctly and consistently label a functional group in the test environment only if that group appeared during training.

  • To further support the validity of Assumption 2 in real applications, we conducted two experiments on real world datasets comparing causal and non-causal subgraphs. We devised two distinct experimental setups to validate our assumption under different settings. Distribution shift was measured by the maximum mean discrepancy (MMD) between representations produced by the a same GNN model, using a Gaussian kernel on standardized features with bandwidth set by the median heuristic. We denote the cross‐environment distribution shift of causal and non‐causal subgraph by $\Delta{G_c}\approx MMD(G_c^{e1},G_c^{e2}), \Delta{G_s}\approx MMD(G_s^{e1},G_s^{e2})$ and compute the ratio of $\Delta{G_s}/\Delta{G_c}$. The experimental designs and results are described below:

    • Experiment 1. Four real-world molecular graph datasets with ground-truth causal subgraphs—Mutag, Benzene, Fluoride, and Alkane—were selected from the graph explanation literature. Each dataset was split into two environments by molecular size and by random assignment. The results are presented below:

    $\Delta{G_s}/\Delta{G_c}$	Mutag		Benzene		Fluoride		Alkane
    split	size	random	size	random	size	random	size	random
    MMD	3.43	1.56	5.79	1.83	6.67	1.94	6.48	2.03

    The results indicate that, the non-causal subgraph exhibits a larger shift than the causal subgraph across all environment partitions($\Delta{G_s}>\Delta{G_c}$), which aligns with Assumption 2.

    • Experiment 2. We apply the same evaluation to the out-of-distribution (OOD) dataset used in our study. Due to the absence of true ground truth, we treat the subgraphs extracted by our method, GIL, and GSAT as causal subgraphs. The results are presented below:

    $\Delta{G_s}/\Delta{G_c}$	HIV-scaffold	HIV-size	SST2	Twitter	IC50-size	IC50-scaffold
    IDG	7.28	3.915	1.963	3.983	1.1784	1.5296
    GIL	1.969	1.429	1.044	3.25	1.154	1.74
    GSAT	5.292	4.671	1.868	1.244	1.128	1.156

    The results indicate that, the extracted non-causal subgraph by different methods still exhibits a larger shift than the causal subgraph($\Delta{G_s}>\Delta{G_c}$), which also aligns with Assumption 2.

  These two experiments, viewed from different perspectives, clarify the validity of Assumption 2. We will include these explanations and experiments in Section 2.2 (Theoretical Insights) of the paper.

  We will include the explanations of 3 assumptions in Section 2.2 in the paper.

[1] Joint Learning of Label and Environment Causal Independence for Graph Out-of-Distribution Generalization

[2] Does Invariant Graph Learning via Environment Augmentation Learn Invariance?

[3] Learning Causally Invariant Representations for Out-of-Distribution Generalization on Graphs

[4] Debiasing graph neural networks via learning disentangled causal substructure

[5] Interpretable and Generalizable Graph Learning via Stochastic Attention Mechanism

[6] Parameterized Explainer for Graph Neural Network

[7] Rationalizing Neural Predictions

[8] Interpreting Image Classifiers by Generating Discrete Masks

[9] Elements of Causal Inference: Foundations and Learning Algorithms

[10] Causal inference using invariant prediction: identification and confidence intervals

[11] Invariant Causal Prediction for Nonlinear Models

[12] Invariant Causal Prediction for Sequential Data

[13] Invariance, Causality and Robustness

[14] A theory of learning from different domains

Thank you for the feedback and please find below responses to your comments.

Weakness

• "Generalizability to real-world" Because real-world datasets lack ground-truth annotations for causal subgraphs, validating this hypothesis on them is extremely challenging. Nevertheless, we devise two distinct experimental setups to validate our assumption under different settings:

  • To validate that causal subgraphs are more invariant, we ran two experiments on real-world graphs, comparing causal and non-causal subgraphs under different settings. Given the scarcity of ground-truth causal annotations, we design distinct experimental setups and quantify distribution shift via maximum mean discrepancy (MMD) between GNN representations using a Gaussian kernel on standardized features with bandwidth set by the median heuristic. We denote the cross‐environment distribution shift of causal and non‐causal subgraph by $\Delta{G_c}\approx MMD(G_c^{e1},G_c^{e2}), \Delta{G_s}\approx MMD(G_s^{e1},G_s^{e2})$ and compute the ratio of $\Delta{G_s}/\Delta{G_c}$. Table values above 1 indicate that non‐causal subgraphs undergo larger distribution shifts across environments. The experimental designs and results are described below:

    • Experiment 1. Four real-world molecular graph datasets with ground-truth causal subgraphs—Mutag, Benzene, Fluoride, and Alkane- from the graph explanation literature. Each dataset was split into two environments by molecular size or random. The results are below:

    $\Delta{G_s}/\Delta{G_c}$	Mutag		Benzene		Fluoride		Alkane
    split	size	random	size	random	size	random	size	random
    MMD	3.43	1.56	5.79	1.83	6.67	1.94	6.48	2.03

    The results indicate that, the non-causal subgraph exhibits a larger shift than the causal subgraph across all environment partitions($\Delta{G_s}>\Delta{G_c}$), which aligns with Assumption 2.

    • Experiment 2. We apply the same evaluation to the out-of-distribution (OOD) dataset used in our study. Due to the absence of true ground truth, we treat the subgraphs extracted by our method, GIL, and GSAT as causal subgraphs. The results are below:

    $\Delta{G_s}/\Delta{G_c}$	HIV-scaffold	HIV-size	SST2	Twitter	IC50-size	IC50-scaffold
    IDG	7.28	3.915	1.963	3.983	1.1784	1.5296
    GIL	1.969	1.429	1.044	3.25	1.154	1.74
    GSAT	5.292	4.671	1.868	1.244	1.128	1.156

    The results indicate that, the extracted non-causal subgraph by different methods still exhibits a larger shift than the causal, which also aligns with Assumption 2.

  We will include these explanations and experiments in Section 2.2 (Theoretical Insights) of the paper.

• "Limitations about the assumption" We discuss the rationale, limitations, and extreme violation cases here:

  • Assumption 1 (Causal Mechanism) The generation of causal labels is fully determined by the causal subgraph $G_c$. This assumption derives from the Independent Causal Mechanism (ICM) hypothesis [6] and underpins Graph OOD methods such as CIGA[7] and LECI[5]. Invariant Risk Minimization (IRM) likewise relies on it by seeking representations that yield the same optimal classifier across environments. For instance, knowing the causal subgraph (e.g., functional groups) enables inferring a molecule's properties in different environments.

  • Assumption 2 (Environmental Diversity and Interventions) This assumption originates from Peters et al.'s ICP framework and its extensions [1-4], this assumption holds that distributional shifts across environments arise from sparse interventions on the non-causal subgraph $G_s$, while $G_c$ remains stable. This hypothesis has been adopted by Graph OOD methods such as LECI[5], which impose an even stronger assumption—that the environment $E$ is independent of $G_c$. i.e. $E \perp G_c$. For instance, functional groups remain consistent in type and structure, while non-causal elements (e.g., heteroatoms) vary across environments.

  • Assumption 3 (Effective Classifier Support) This assumption is based on the classical domain-adaptation bounds presented first in [8], this assumption requires overlap between the support of source and target distributions to guarantee generalization. Since classification relies on $G_c$, the support of the causal-subgraph distribution in the test domain must lie within that of the training domain. For example, a classifier can only correctly label a functional group at test time seen during training.

  Limitations and Extreme Violations:

  • Assumption 1: If causal labels depend on factors beyond G_c, our model may miss key causal relationships, harming classification. For example, if labels reflect post-reaction metrics rather than intrinsic chemical properties. This scenario is at odds with existing work[1-9]. Like other Graph OOD and causal inference studies, our study focus solely on input–output mapping.
  • Assumption 2: If shifts involve global feature changes rather than sparse interventions on $G_s$, this assumption fails—for instance, if environments are defined by critical functional groups, but these cases are extremely rare and we cannot find supported datasets and evidence from current Graph OOD benchmarks.
  • Assumption 3: If the support of $G_c$ in the test domain does not overlap with that of the training domain—e.g., when test samples include unseen functional groups—no existing method can classify effectively. This scenario also lies outside the scope of Graph OOD tasks in general.

  We will include the discussion about explanations and limitations of 3 assumptions in Section 2.2 in the paper.

• "Incremental contribution of leveraging norm" The work [10] discussed in our paper, to the best of our knowledge, is the only work we have found to be related to us. However, compared to [10], our work differs in three key aspects:

  • Target application. While [10] examines whether inputs are from different distributions, our method focuses on training models to correctly classify inputs from different distribution.
  • Evaluation domain. Work [10] is conducted entirely within the vision domain, whereas our study is dedicated to graph data and graph neural networks.
  • Theoretical foundation. The conclusions in [10] are based on analyses of linear networks, while our framework is grounded in causal inference, domain generalization, and graph learning.

  Moreover, our contributions are threefold:

  • We are the first to empirically establish a quantitative relationship between distribution shift and representation norms in graph data, and to explain its connection to the low-rank property of GNNs. (As recognized by Reviewer Ti4B)
  • We provide a rigorous theoretical proof that causal subgraph minimizes distribution shift across environments. (As recognized by Reviewer Ti4B, rEir)
  • We are the first to introduce norms into graph OOD research and to offer a theoretical justification for using norms to identify causal subgraphs. (As recognized by Reviewer q4r2, Ti4B)

• "Related work on invariant learning and causal reasoning" Due to space constraints, details of existing invariant-learning and causal-reasoning baselines are provided in Appendices G and I. Please refer to our response to reviewer rEir (Weak1: Optimization framework) and Ti4B (Weak2: Compared to environment-inference-based methods). and we plan to add a more in-depth discussion of how our method differs from these approaches in the Related Work section in the paper.

Questions

• "Cases where the causal subgraph does not have a larger norm/distributional shift affects the causal" These settings contradict the assumptions and scenarios of both our work and prior Graph OOD studies like in [5,9,7,11]. In our experiments, we did not encounter any cases in which the causal subgraph fails to exhibit a larger norm than the spurious components, nor cases in which distributional shift affected the causal structure. If possible, could you suggest a dataset or benchmark that exhibits these conditions? We would then use the recommended benchmark to present further experimental results.

• Testing in partial information or weakly informative environments We conduct experiments in these datasets. The results below demonstrate that our method performs strongly across all scenarios (denoting the variant without the norm objective as IDG-0):

  	SP=0.1	SP=0.3	SP=0.6	SP=0.7	SP=0.8	SP=0.9	PIIF	FIIF
  ERM	94.53	93.26	91.67	91.25	90.63	85.92	62.45	37.22
  IRM	93.65	92.87	91.85	91.25	90.48	85.38	68.34	44.33
  GIL	92.14	91.57	89.78	80.33	72.00	51.09	59.66	64.66
  DIR	90.63	90.10	88.70	87.82	86.00	84.80	69.33	42.00
  IDG	95.30	94.67	92.67	92.06	91.89	91.33	79.00	83.67
  IDG-0	94.89	94.29	90.43	90.16	86.74	83.13	66.33	70.66

  SP denotes the probability of spurious correlation, with lower values indicating weakly informative environments. PIIF and FIIF refer to partially and fully informative scenarios. These datasets adhere to the settings in [5,7,9]. We will incorporate these results and their analysis into the comparative experiments in Section 4.2.

[1]Causal inference using invariant prediction: identification and confidence intervals

[2]Invariant Causal Prediction for Nonlinear Models

[3]Invariant Causal Prediction for Sequential Data

[4]Invariance, Causality and Robustness

[5]Joint Learning of Label and Environment Causal Independence for Graph Out-of-Distribution Generalization

[6]Elements of Causal Inference: Foundations and Learning Algorithms

[7]Learning Causally Invariant Representations for Out-of-Distribution Generalization on Graphs

[8]A theory of learning from different domains

[9]Does Invariant Graph Learning via Environment Augmentation Learn Invariance?

[10] Deep Neural Networks Tend To Extrapolate Predictably

[11] GOOD: A Graph Out-of-Distribution Benchmark

Thank you for your appreciation on our work!

Weakness

• "Assumption 2, distribution shifts affect only non-causal parts" This assumption originates from Peters et al.'s ICP framework and its extensions [1–4], this assumption holds that distributional shifts across environments arise from sparse interventions on the non-causal subgraph $G_s$, while $G_c$ remains stable. This hypothesis has been adopted by Graph OOD methods such as LECI[5], which impose an even stronger assumption—that the environment $E$ is independent of $G_c$. i.e. $E \perp G_c$. For instance, the types and structures of functional groups in a molecule are limited and stable, whereas non-causal parts (e.g., heteroatoms) vary widely across environments.

  • To further support the validity of Assumption 2 in real applications, we conducted two experiments on real world datasets comparing causal and non-causal subgraphs. Quantifying distribution shifts in real-world graphs is challenging, since such data with ground-truth annotation of causal subgraph are rarely available, so we devised two distinct experimental setups to validate our assumption under different settings. Distribution shift was measured by the maximum mean discrepancy (MMD) between representations produced by the a same GNN model, using a Gaussian kernel on standardized features with bandwidth set by the median heuristic. We denote the cross‐environment distribution shift of causal and non‐causal subgraph by $\Delta{G_c}\approx MMD(G_c^{e1},G_c^{e2}), \Delta{G_s}\approx MMD(G_s^{e1},G_s^{e2})$ and compute the ratio of $\Delta{G_s}/\Delta{G_c}$. The experimental designs and results are described below:

    • Experiment 1. Four real-world molecular graph datasets with ground-truth causal subgraphs—Mutag, Benzene, Fluoride, and Alkane—were selected from the graph explanation literature. Each dataset was split into two environments by molecular size and by random assignment. The results are presented below:

    $\Delta{G_s}/\Delta{G_c}$	Mutag		Benzene		Fluoride		Alkane
    split	size	random	size	random	size	random	size	random
    MMD	3.43	1.56	5.79	1.83	6.67	1.94	6.48	2.03

    The results indicate that, the non-causal subgraph exhibits a larger shift than the causal subgraph across all environment partitions($\Delta{G_s}>\Delta{G_c}$), which aligns with Assumption 2.

    • Experiment 2. We apply the same evaluation to the out-of-distribution (OOD) dataset used in our study. Due to the absence of true ground truth, we treat the subgraphs extracted by our method, GIL, and GSAT as causal subgraphs. The results are presented below:

    $\Delta{G_s}/\Delta{G_c}$	HIV-scaffold	HIV-size	SST2	Twitter	IC50-size	IC50-scaffold
    IDG	7.28	3.915	1.963	3.983	1.1784	1.5296
    GIL	1.969	1.429	1.044	3.25	1.154	1.74
    GSAT	5.292	4.671	1.868	1.244	1.128	1.156

    The results indicate that, the extracted non-causal subgraph by different methods still exhibits a larger shift than the causal subgraph($\Delta{G_s}>\Delta{G_c}$), which also aligns with Assumption 2.

• "Norm difference to the distributional shift due to spurious correlations" We performe a detailed comparison between IDG and a variant without the norm objective (denoted IDG-0) across varying levels of spurious correlation strength to assess how much the norm objective contributes to mitigating spurious correlations. As shown in the table below, the norm objective's impact on performance grows with increasing spurious correlation strength, with a Pearson correlation coefficient of 0.86 between the spurious correlation probability and the performance gain. This finding demonstrates the effectiveness of our method in addressing spurious correlations.

  	SP=0.5	SP=0.6	SP=0.7	SP=0.8	SP=0.9	SP=0.95
  IDG	93.33	92.67	92.06	91.89	91.33	83.26
  IDG-0	90.66	90.43	90.16	86.74	83.13	70.72
  $\Delta$	2.67	2.24	1.90	5.15	8.20	12.54

  SP denotes the spurious-correlation probability. We will incorporate these results and their analysis into the comparative experiments in Section 4.2.

[1] Causal inference using invariant prediction: identification and confidence intervals

[2] Invariant Causal Prediction for Nonlinear Models

[3] Invariant Causal Prediction for Sequential Data

[4] Invariance, Causality and Robustness

[5] Joint Learning of Label and Environment Causal Independence for Graph Out-of-Distribution Generalization

Dear reviewer q4r2,

Firstly, we would like to express our profound gratitude for your previous suggestions. Your recommendations were meticulous and constructive, prompting deeper reflections on many issues. We truly value the effort and thought you have invested in our work.

We truly appreciate the time and expertise you invested in evaluating our manuscript. In response to your insightful comments, we have prepared detailed replies and implemented the suggested revisions to address each of your concerns.

We sincerely hope to further discuss with you and ensure that we have completely met your expectations. If any aspects of our work remain unclear, or if further concerns arise, please do not hesitate to share them. Our commitment remains steadfast in providing clarity and responding to all feedback.

Best regards,

The Authors.