Original: Debiasing Graph Neural Networks via Learning Disentangled Causal Substructure
Revised: Debiasing Graph Neural Networks via Learning Disentangled Causal Substructure

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Most Graph Neural Networks (GNNs) predict the labels of unseen graphs by learning the correlation between the input graphs and labels.
Revised: Most Graph Neural Networks (GNNs) predict the labels of unseen graphs by learning the correlation between the input graphs and labels.

Original: However, by presentinga graph classification investigation on the training graphs with severe bias, surprisingly, we discover that GNNs always tend to explore the spurious correlationsto make decision, even if the causal correlation always exists.
Revised: However, by presenting a graph classiﬁcation investigation on the training graphs with severe bias, surprisingly, we discover that GNNs always tend to explore the spurious correlations to make decision, even if the causal correlation always exists.

Original: This implies thatexisting GNNs trained on such biased datasets will suffer from poor generalizationcapability.
Revised: This implies that existing GNNs trained on such biased datasets will suffer from poor generalization capability.

Original: By analyzing this problem in a causal view, we find that disentanglingand decorrelating the causal and bias latent variables from the biased graphs areboth crucial for debiasing.
Revised: By analyzing this problem in a causal view, we ﬁnd that disentangling and decorrelating the causal and bias latent variables from the biased graphs are both crucial for debiasing.

Original: Inspiring by this, we propose a general disentangled
Revised: Inspiring by this, we propose a general disentangled

Original: GNN framework to learn the causal substructure and bias substructure, respectively.
Revised: GNN framework to learn the causal substructure and bias substructure, respectively.

Original: Particularly, we design a parameterized edge mask generator to explicitly split theinput graph into causal and bias subgraphs.
Revised: Particularly, we design a parameterized edge mask generator to explicitly split the input graph into causal and bias subgraphs.

Original: Then two GNN modules supervisedby causal/bias-aware loss functions respectively are trained to encode causal andbias subgraphs into their corresponding representations.
Revised: Then two GNN modules supervised by causal/bias-aware loss functions respectively are trained to encode causal and bias subgraphs into their corresponding representations.

Original: With the disentangledrepresentations, we synthesize the counterfactual unbiased training samples tofurther decorrelate causal and bias variables.
Revised: With the disentangled representations, we synthesize the counterfactual unbiased training samples to further decorrelate causal and bias variables.

Original: Moreover, to better benchmark thesevere bias problem, we construct three new graph datasets, which have controllable bias degrees and are easier to visualize and explain.
Revised: Moreover, to better benchmark the severe bias problem, we construct three new graph datasets, which have controllable bias degrees and are easier to visualize and explain.

Original: Experimental results welldemonstrate that our approach achieves superior generalization performance overexisting baselines.
Revised: Experimental results well demonstrate that our approach achieves superior generalization performance over existing baselines.

Original: Furthermore, owing to the learned edge mask, the proposedmodel has appealing interpretability and transferability.
Revised: Furthermore, owing to the learned edge mask, the proposed model has appealing interpretability and transferability.

Original:  Introduction
Revised:  Introduction

Original: Graph Neural Networks (GNNs) have exhibited powerful performance on graph data with variousapplications [15, 33, 11].
Revised: Graph Neural Networks (GNNs) have exhibited powerful performance on graph data with various applications [19, 38, 14, 10, 9].

Original: One major category of applications are the graph classification task,such as molecular graph property prediction [13, 18, 42], superpixel graph classification [12] andsocial network category classification [44, 42].
Revised: One major category of applications are the graph classiﬁcation task, such as molecular graph property prediction [16, 22, 48], superpixel graph classiﬁcation [15], and social network category classiﬁcation [50, 48].

Original: It is well known that graph classification is usuallydetermined by a relevant substructure but not the whole graph structure [41, 24, 43].
Revised: It is well known that graph classiﬁcation is usually determined by a relevant substructure, but not the whole graph structure [47, 28, 49].

Original: For example,for MNIST superpixel graph classification task, the digit subgraphs are causal ( i.e., deterministic) forlabels [34].
Revised: For example, for MNIST superpixel graph classiﬁcation task, the digit subgraphs are causal ( i.e., deterministic) for labels [39].

Original: The mutagenic property of a molecular graph depends on the functional groups ( i.e.,nitrogen dioxide (NO 2 )), rather than the irrelevant patterns ( i.e., carbon rings) [25].
Revised: The mutagenic property of a molecular graph depends on the functional groups ( i.e., nitrogen dioxide (NO 2 )), rather than the irrelevant patterns ( i.e., carbon rings) [29].

Original: Therefore, it is afundamental requirement for GNNs to identify causal substructures, so as to make correct prediction.
Revised: Therefore, it is a fundamental requirement for GNNs to identify causal substructures, so as to make correct prediction.

Original: Ideally, when the graphs are unbiased, i.e. , only the causal substructures are related with the graphlabels, the GNNs are able to utilize such substructure to predict the labels.
Revised: Ideally, when the graphs are unbiased, i.e. , only the causal substructures are related with the graph labels, the GNNs are able to utilize such substructure to predict the labels.

Original: However, due to theuncontrollable data collection process, the graphs are inevitably biased, i.e. , existing meaninglesssubstructures spuriously correlates with labels.
Revised: However, due to the uncontrollable data collection process, the graphs are inevitably biased, i.e. , existing meaningless substructures spuriously correlates with labels.

Original: Taking a colored MNIST superpixel graph dataset in
Revised: Taking a colored MNIST superpixel graph dataset in

Original: Sec.
Revised: Sec.

Original: 3.1 as an example (illustrated in Fig. 1(a)), each category of digit subgraphs mainly correspond to one kind of color background subgraphs, e.g. , digit 0 subgraph is related with red backgroundsubgraph.
Revised: 3.1 as an example (illustrated in Fig. 1(a)), each category of digit subgraphs mainly correspond to one kind of color background subgraphs, e.g. , digit 0 subgraph is related with red background subgraph.

Original: Therefore, the color background subgraph will be treated as bias information, which highlycorrelates with labels but does not determines them in the training set.
Revised: Therefore, the color background subgraph will be treated as bias information, which highly correlates with labels but does not determines them in the training set.

Original: Under this situation, will
Revised: Under this situation, will

Original: GNNs still stably utilize the causal substructure to make decision?
Revised: GNNs still stably utilize the causal substructure to make decision?

Original: To investigate the impact of bias on GNNs, we conduct an experimental investigation to demonstratethe impact of bias (especially in the severe bias scenarios) on the generalization capability of GNNs(Sec. 3.1).
Revised: To investigate the impact of bias on GNNs, we conduct an experimental investigation to demonstrate the impact of bias (especially in the severe bias scenarios) on the generalization capability of GNNs (Sec. 3.1).

Original: We find that GNNs actually utilize both bias and causal substructures to make prediction.
Revised: We ﬁnd that GNNs actually utilize both bias and causal substructures to make prediction.

Original: However, with severer bias correlation, even bias substructure still could not exactly determine labelslike causal substructure, GNNs majorly utilize bias substructure as shortcuts to make prediction,causing a large generalization performance degradation.
Revised: However, with severer bias correlation, even bias substructure still could not exactly determine labels like causal substructure, GNNs majorly utilize bias substructure as shortcuts to make prediction, causing a large generalization performance degradation.

Original: Why this happens?
Revised: Why this happens?

Original: We analyze the datagenerating process and model prediction mechanism behind the graph classification using a causalgraph (Sec. 3.2).
Revised: We analyze the datagenerating process and model prediction mechanism behind the graph classiﬁcation using a causal graph (Sec. 3.2).

Original: The casual graph illustrates that the observed graphs are generated by the causal andbias latent variables and existing GNNs could not distinguish the causal substructure from entangledgraphs.
Revised: The casual graph illustrates that the observed graphs are generated by the causal and bias latent variables and existing GNNs could not distinguish the causal substructure from entangled graphs.

Original: How can we disentangle the causal and bias substructures from observed graphs, so that
Revised: How can we disentangle the causal and bias substructures from observed graphs, so that

Original: GNNs can only utilize the causal substructures to make stable prediction when severe bias appears?
Revised: GNNs can only utilize the causal substructures to make stable prediction when severe bias appears?

Original: To address the question, two challenges need to be faced.
Revised: To address the question, two challenges need to be faced.

Original: 1) How to identify the causal substructureand bias substructure in the severe biased graphs?
Revised: 1) How to identify the causal substructure and bias substructure in the severe biased graphs?

Original: In the severe bias scenarios, bias substructure willbe “easier to learn” for GNNs and finally dominate the prediction.
Revised: In the severe bias scenarios, bias substructure will be “easier to learn” for GNNs and ﬁnally dominate the prediction.

Original: Using the normal cross-entropyloss, like DIR [37], could not fully capture such aggressive property of bias.
Revised: Using the normal cross-entropy loss, like DIR [43], could not fully capture such aggressive property of bias.

Original: 2) How to extractthe causal substructure from an entangled graph?
Revised: 2) How to extract the causal substructure from an entangled graph?

Original: The statistically causal substructure is usuallydetermined by the global property of the entire graph population, rather than a single graph.
Revised: The statistically causal substructure is usually determined by the global property of the entire graph population, rather than a single graph.

Original: Whenextracting causal substructure from a graph, we need to establish the relations among all the graphs.
Revised: When extracting causal substructure from a graph, we need to establish the relations among all the graphs.

Original: In this paper, we propose a novel debiasing framework for GNNs via learning Dis entangled C ausalsubstructure, called DisC .
Revised: In this paper, we propose a novel debiasing framework for GNNs via learning Dis entangled C ausal substructure, called DisC .

Original: Given an input biased graph, we propose to explicitly filter edges intocausal and bias subgraphs by a parameterized edge mask generator, whose parameters are sharedacross entire graph population.
Revised: Given an input biased graph, we propose to explicitly ﬁlter edges into causal and bias subgraphs by a parameterized edge mask generator, whose parameters are shared across entire graph population.

Original: As a result, the edge masker is naturally capable to specify theimportance for each edge and extract causal and bias subgraphs from a global view of the entireobservations.
Revised: As a result, the edge masker is naturally capable to specify the importance for each edge and extract causal and bias subgraphs from a global view of the entire observations.

Original: Then, a “casual”-aware (weighted cross-entropy) loss and a “bias”-aware (generalizedcross-entropy) loss are respectively utilized to supervise two functional GNN modules.
Revised: Then, a “casual”-aware (weighted cross-entropy) loss and a “bias”-aware (generalized cross-entropy) loss are respectively utilized to supervise two functional GNN modules.

Original: Based onthe supervision, the edge mask generator could generate corresponding subgraphs and the GNNscould encode corresponding subgraphs into their disentangled embeddings.
Revised: Based on the supervision, the edge mask generator could generate corresponding subgraphs and the GNNs could encode corresponding subgraphs into their disentangled embeddings.

Original: With the disentangledembeddings, we randomly permute the latent vectors extracted from different graphs to generatemore unbiased counterfactual samples in embedding space.
Revised: With the disentangled embeddings, we randomly permute the latent vectors extracted from different graphs to generate more unbiased counterfactual samples in embedding space.

Original: The new generated samples still containboth causal and bias information, while their correlation has been decorrelated.
Revised: The new generated samples still contain both causal and bias information, while their correlation has been decorrelated.

Original: In this time, there isonly correlation between causal variables with labels, so that the model could concentrate on the truecorrelation between the causal subgraphs and labels.
Revised: In this time, there is only correlation between causal variables with labels, so that the model could concentrate on the true correlation between the causal subgraphs and labels.

Original: Our major contributions are as follows:
Revised: Our major contributions are as follows:

Original: • To our knowledge, we first study the generalization problem of GNNs in a more challenging yetpractical scenario, i.e., the graphs are with severe bias.
Revised: • To our knowledge, we ﬁrst study the generalization problem of GNNs in a more challenging yet practical scenario, i.e., the graphs are with severe bias.

Original: We systematically analyze the bias impacton GNNs from both experimental study and causal analysis.
Revised: We systematically analyze the bias impact on GNNs from both experimental study and causal analysis.

Original: We find that the bias substructure,compared with causal substructure, is much easier to dominate the training of GNNs.
Revised: We ﬁnd that the bias substructure, compared with causal substructure, is much easier to dominate the training of GNNs.

Original: • To debias GNNs, we develop a novel GNN framework for disentangling causal substructure,which is flexible to build upon various GNNs for improving generalization ability while enjoying inherent interpretability, robustness and transferability.
Revised: • To debias GNNs, we develop a novel GNN framework for disentangling causal substructure, which is ﬂexible to build upon various GNNs for improving generalization ability while enjoying inherent interpretability, robustness and transferability.

Original: • We construct three new datasets with various properties and controllable bias degrees, which can better benchmark the new problem.
Revised: • We construct three new datasets with various properties and controllable bias degrees, which can better benchmark the new problem.

Original: Our model outperforms the corresponding base models witha large margin (from 4.47% to 169. % average improvements).
Revised: Our model outperforms the corresponding base models with a large margin (from 4.47% to 169.17% average improvements).

Original: Various investigation studiesdemonstrate that our model could discover and leverage causal substructure for prediction.
Revised: Various investigation studies demonstrate that our model could discover and leverage causal substructure for prediction.

Original:  Related Works
Revised:  Related Works

Original: Generalization of GNNs in wild environments.
Revised: Generalization of GNNs in wild environments.

Original: Most existing GNN methods are proposed underthe IID hypothesis, i.e., training and testing set are independently sampled from the identicaldistribution [32, 15, 33, 11, 22].
Revised: Most existing GNN methods are proposed under the IID hypothesis, i.e., training and testing set are independently sampled from the identical distribution [36, 19, 38, 14, 26].

Original: However, in reality, thus ideal assumption is hard to be satisfied.
Revised: However, in reality, thus ideal assumption is hard to be satisﬁed.

Original: Recently, several methods have been proposed to improve the generalization ability of GNNs inwild OOD environments.
Revised: Recently, several methods have been proposed to improve the generalization ability of GNNs in wild OOD environments.

Original: Several works [27, 7, 36] study the OOD problem of node classification.
Revised: Several works [31, 8, 42] study the OOD problem of node classiﬁcation.

Original: For OOD graph classification task, StableGNN [6] propose to learn the stable causal relationship ingraphs.
Revised: For OOD graph classiﬁcation task, StableGNN [7] propose to learn the stable causal relationship in graphs.

Original: OOD-GNN [20] propose to constrain each dimension of learned embedding independent.
Revised: OOD-GNN [24] propose to constrain each dimension of learned embedding independent.

Original: DIR [37] discovers the invariant rationales for generalizing GNNs.
Revised: DIR [43] discovers the invariant rationales for generalizing GNNs.

Original: Although they have achievedbetter OOD performance, they are not designed for the datasets with severe bias, which is morechallenging for guaranteeing the generalization ability of GNNs.
Revised: Although they have achieved better OOD performance, they are not designed for the datasets with severe bias, which is more challenging for guaranteeing the generalization ability of GNNs.

Original: Disentangled graph neural networks.
Revised: Disentangled graph neural networks.

Original: Recently, there are a couple of methods that study thedisentangled GNNs.
Revised: Recently, there are a couple of methods that study the disentangled GNNs.

Original: DisenGCN [26] utilizes neighbourhood routing mechanism to divide theneighbours of the node into several mutually exclusive parts.
Revised: DisenGCN [30] utilizes neighbourhood routing mechanism to divide the neighbours of the node into several mutually exclusive parts.

Original: IPGDN [23] promotes DisenGCNby constraining the different parts of the embedding feature independent.
Revised: IPGDN [27] promotes DisenGCN by constraining the different parts of the embedding feature independent.

Original: DisenGCN and IPGDNare node-level disentanglement, thus FactorGCN [40] considers the whole graph information anddisentangles the target graph into several factorized graphs.
Revised: DisenGCN and IPGDN are node-level disentanglement, thus FactorGCN [46] considers the whole graph information and disentangles the target graph into several factorized graphs.

Original: Despite results of the previous works,they do not consider disentangling the causal and bias information for graphs.
Revised: Despite results of the previous works, they do not consider disentangling the causal and bias information for graphs.

Original: General debiasing methods.
Revised: General debiasing methods.

Original: Recently, debiasing problem has drawn much attention in machinelearning community [14, 21, 31, 1, 2, 9].
Revised: Recently, debiasing problem has drawn much attention in machine learning community [17, 25, 35, 1, 2, 12].

Original: One category of these methods is pre-defining a certainbias type explicitly to mitigate [14, 21, 31, 1, 35].
Revised: One category of these methods is pre-deﬁning a certain bias type explicitly to mitigate [17, 25, 35, 1, 40].

Original: For example, Wang et al.
Revised: For example, Wang et al.

Original: [35] and Bahng etal.
Revised: [40] and Bahng et al.

Original: [1] design a texture- and color-guided model to adversarially train a debiased neural networkagainst the biased one.
Revised: [1] design a texture- and color-guided model to adversarially train a debiased neural network against the biased one.

Original: Instead of defining certain types of bias, recent approaches [28, 5, 19] relyon the straightforward assumption that models are prone to exploit the bias as shortcuts to makeprediction [8].
Revised: Instead of deﬁning certain types of bias, recent approaches [32, 6, 23] rely on the straightforward assumption that models are prone to exploit the bias as shortcuts to make prediction [11].

Original: In the line with the recent studies, our study belongs to the second category.
Revised: In the line with the recent studies, our study belongs to the second category.

Original: However,most of existing methods are designed for image datasets and could not effectively extract causalsubstructure from graph data.
Revised: However, most of existing methods are designed for image datasets and could not effectively extract causal substructure from graph data.

Original: Distinctly, we first study the severe bias problem on graph data, andour method could effectively extract causal substructure from graph data.(a) Examples
Revised: Distinctly, we ﬁrst study the severe bias problem on graph data, and our method could effectively extract causal substructure from graph data.

Original: [Figure]
Revised: [Figure]

Original:  Preliminary Study and Analysis
Revised:  Preliminary Study and Analysis

Original: In this section, we first illustrate the existing GNNs tend to exploit the bias substructure as shortcutsfor prediction through a motivating experiment.
Revised: In this section, we ﬁrst illustrate the existing GNNs tend to exploit the bias substructure as shortcuts for prediction through a motivating experiment.

Original: Then we analyze the prediction process of GNNs incausal view.
Revised: Then we analyze the prediction process of GNNs in causal view.

Original: Based on this causal view, it motivates our solution to relieve the impact of bias.
Revised: Based on this causal view, it motivates our solution to relieve the impact of bias.

Original: 3.1 Motivating Example
Revised: 3.1 Motivating Example

Original: To measure the generalization ability of GNNs with the effect of bias, we construct a graph classification dataset with controllable bias degrees, called Colored MNIST-75sp.
Revised: To measure the generalization ability of GNNs with the effect of bias, we construct a graph classiﬁcation dataset with controllable bias degrees, called CMNIST-75sp.

Original: We first construct a biased MNIST image dataset like [1], where each category of digit highly correlates with a pre-defined colorin their background.
Revised: We ﬁrst construct a biased MNIST image dataset like [1], where each category of digit highly correlates with a pre-deﬁned color in their background.

Original: For example, in the training set, 90% of 0 digits are with red background ( i.e., biased samples), and remaining 10% images are with random background color ( i.e., unbiased samples),whose the bias degree is 0.9 in this situation.
Revised: For example, in the training set, 90% of 0 digits are with red background ( i.e., biased samples), and remaining 10% images are with random background color ( i.e., unbiased samples), whose the bias degree is 0.9 in this situation.

Original: We consider four bias degrees { 0 . 8 , 0 . 85 , 0 . 9 , 0 . 95 } .
Revised: We consider four bias degrees { 0 . 8 , 0 . 85 , 0 . 9 , 0 . 95 } .

Original: For the testing set, we construct both biased testing set and unbiased testing set.
Revised: For the testing set, we construct both biased testing set and unbiased testing set.

Original: The biased testingset has the same bias degree with training set, aiming to measure the extent of models relying onbias.
Revised: The biased testing set has the same bias degree with training set, aiming to measure the extent of models relying on bias.

Original: The unbiased testing set, where the digit labels uncorrelate with the background colors, aims totest whether the model could utilize the inherent digit signals for prediction.
Revised: The unbiased testing set, where the digit labels uncorrelate with the background colors, aims to test whether the model could utilize the inherent digit signals for prediction.

Original: Note that training setand testing set have the same pre-defined color set.
Revised: Note that training set and testing set have the same pre-deﬁned color set.

Original: Then, we convert the biased MNIST images intosuperpixel graphs with at most 75 nodes each graph using [16], where the edges are constructed bythe KNN method based on the 2D coordinates of superpixels and node features are the concatenation of coordinates and average color of superpixels.
Revised: Then, we convert the biased MNIST images into superpixel graphs with at most 75 nodes each graph using [20], where the edges are constructed by the KNN method based on the 2D coordinates of superpixels and node features are the concatenation of coordinates and average color of superpixels.

Original: Each graph is labeled by its digit class, so that itsdigital subgraph is deterministic for label and background subgraph is spuriously correlated withlabels but not deterministic.
Revised: Each graph is labeled by its digit class, so that its digital subgraph is deterministic for label and background subgraph is spuriously correlated with labels but not deterministic.

Original: The examples of graphs are illustrated in Fig. 1(a).
Revised: The examples of graphs are illustrated in Fig. 1(a).

Original: We perform three popular GNN methods: GCN [15], GIN [39], and GCNII [3] on Colored MNIST75sp and the results are shown in Fig. 1(b).
Revised: We perform three popular GNN methods: GCN [19], GIN [45], and GCNII [3] on CMNIST-75sp and the results are shown in Fig. 1(b).

Original: The same color of dashed line and solid line represent theresults of the corresponding methods on the biased testing set and the unbiased testing set respectively.
Revised: The same color of dashed line and solid line represent the results of the corresponding methods on the biased testing set and the unbiased testing set respectively.

Original: Overall, the GNNs achieve much better performance on biased testing set than unbiased testing set.
Revised: Overall, the GNNs achieve much better performance on biased testing set than unbiased testing set.

Original: The phenomenon indicates that although GNNs could still learn some causal signals for prediction,the unexpected bias information is also being utilized for prediction.
Revised: The phenomenon indicates that although GNNs could still learn some causal signals for prediction, the unexpected bias information is also being utilized for prediction.

Original: More specifically, with biasdegree becoming larger, the performance of GNNs on biased testing set is increased and the valueof accuracy is nearly in line with the bias degree, while the performance on unbiased testing dropsdramatically.
Revised: More speciﬁcally, with bias degree becoming larger, the performance of GNNs on biased testing set is increased and the value of accuracy is nearly in line with the bias degree, while the performance on unbiased testing drops dramatically.

Original: Hence, although causal substructure could determine labels perfectly, in severe biasscenarios, the GNNs lean to utilize the easier to learn bias information to make prediction rather thanthe inherent causal signals, and bias substructure will finally dominate the prediction. 
Revised: Hence, although causal substructure could determine labels perfectly, in severe bias scenarios, the GNNs lean to utilize the easier to learn bias information to make prediction rather than the inherent causal signals, and bias substructure will ﬁnally dominate the prediction. BB [Equation] [Figure]

Revised: [Figure]

Original:  3.2 Problem Analysis
Revised: [Figure] [Equation] [Figure] 3.2 Problem Analysis

Original: Debiasing GNNs for unbiased prediction requires understanding the natural mechanisms of graphclassification task.
Revised: Debiasing GNNs for unbiased prediction requires understanding the natural mechanisms of graph classiﬁcation task.

Original: We present a causal view of the union of the data-generating process and the modelprediction process behind the task.
Revised: We present a causal view of the union of the data-generating process and the model prediction process behind the task.

Original: Here we formalize the causal view as a Structure Causal Model(SCM) or causal graph [10, 29] by inspecting on the causalities among five variables: unobservedcausal variable C , unobserved bias variable B , observed graph G , graph embedding E , and groundtruth label / prediction Y  .
Revised: Here we formalize the causal view as a Structure Causal Model (SCM) or causal graph [13, 33] by inspecting on the causalities among ﬁve variables: unobserved causal variable C , unobserved bias variable B , observed graph G , graph embedding E , and ground truth label / prediction Y 4 .

Original: Fig.
Revised: Fig.

Original: 2(a) illustrates the SCM, where each link denotes a causal relationship.
Revised: 2(a) illustrates the SCM, where each link denotes a causal relationship.

Original: • C → G ← B 
Revised: • C → G ← B .

Original: The observed graph data is generated by two unobserved latent variables: thecausal variable C and the bias variable B , such as digit subgraphs and background subgraphs in the Colored MNIST-75sp dataset.
Revised: The observed graph data is generated by two unobserved latent variables: the causal variable C and the bias variable B , such as digit subgraphs and background subgraphs in the CMNIST-75sp dataset.

Original: And all bellow relations are illustrated by Colored MNIST-75sp.
Revised: And all bellow relations are illustrated by CMNIST-75sp.

Original: • C → Y .
Revised: • C → Y .

Original: This link means that the causal variable C is the only endogenous parent to determine the generation of ground-truth label Y .
Revised: This link means that the causal variable C is the only endogenous parent to determine the generation of ground-truth label Y .

Original: For example, C is the oracle digit subgraph, which exactlyexplains why the label is labeled as Y .
Revised: For example, C is the oracle digit subgraph, which exactly explains why the label is labeled as Y .

Original: • C ⇠⇢ B .
Revised: • C ⇠⇢ B .

Original: This link indicates the spurious correlation between C and B .
Revised: This link indicates the spurious correlation between C and B .

Original: Such probabilistic dependencies is usually caused by the direct cause or unobserved confounder [30].
Revised: Such probabilistic dependencies is usually caused by the direct cause or unobserved confounder [34].

Original: Here we do not distinguish these scenarios and only observe the spurious correlation between B and C , such as the spurious correlation between the color background subgraphs and digit subgraphs.
Revised: Here we do not distinguish these scenarios and only observe the spurious correlation between B and C , such as the spurious correlation between the color background subgraphs and digit subgraphs.

Original: • G
Revised: • G

Original: → E → Y .
Revised: → E → Y .

Original: Existing GNNs usually learn the graph embedding E based on the observed graph G and make the prediction Y based on the learned embedding E .
Revised: Existing GNNs usually learn the graph embedding E based on the observed graph G and make the prediction Y based on the learned embedding E .

Original: According to the SCM, GNNs will utilize both information to make prediction.
Revised: According to the SCM, GNNs will utilize both information to make prediction.

Original: As bias substructure( e.g., background subgraph) usually has simpler structure than meaningful causal substructure ( e.g.,digit subgraph), if GNN utilizes such simple substructure, it could achieve low loss very fast.
Revised: As bias substructure ( e.g., background subgraph) usually has simpler structure than meaningful causal substructure ( e.g., digit subgraph), if GNN utilizes such simple substructure, it could achieve low loss very fast.

Original: Hence,
Revised: Hence,

Original: GNN inclines to utilizes bias information when most graphs are biased.
Revised: GNN inclines to utilizes bias information when most graphs are biased.

Original: Based on the SCM in
Revised: Based on the SCM in

Original: Fig. 2(a), according to d -connection theory [29] (see App.
Revised: Fig. 2(a), according to d -connection theory [33] (see App.

Original: A): two variables are dependent if they areconnected by at least one unblocked path, we could find two paths that would induce the spuriouscorrelation between the bias variable B and label Y : (1) B → G → E → Y and (2) B 
Revised: A): two variables are dependent if they are connected by at least one unblocked path, we could ﬁnd two paths that would induce the spurious correlation between the bias variable B and label Y : (1) B → G → E → Y and (2) B ↔ C

Original: ↔ C → Y .
Revised: → Y .

Original: To make the prediction Y being uncorrelated with the bias B , we need to intercept the two unblockedpaths.
Revised: To make the prediction Y being uncorrelated with the bias B , we need to intercept the two unblocked paths.

Original: For this purpose, we propose to debias GNNs in causal view, as in Fig. 2(b).
Revised: For this purpose, we propose to debias GNNs in causal view, as in Fig. 2(b).

Original: • C ← G → B and C → Y .
Revised: • C ← G → B and C → Y .

Original: To intercept the path (1), we should disentangle the latent variables C and B from the observed graph G and make prediction only based on the causal variable C 
Revised: To intercept the path (1), we should disentangle the latent variables C and B from the observed graph G and make prediction only based on the causal variable C .

Original: • C ⇠ (cid:8)(cid:8)(cid:72)(cid:72) −− ⇢ B .
Revised: • C ⇠ (cid:8)(cid:8)(cid:72)(cid:72) −− ⇢ B .

Original: To intercept the path (2), as we cannot change the link between C and Y , one possible solution is to make C and B uncorrelated
Revised: To intercept the path (2), as we cannot change the link between C and Y , one possible solution is to make C and B uncorrelated.

Original:  Methodology
Revised:  Methodology

Original: Motivated by the above causal analysis, in this section, we present our proposed debiasing GNNframework DisC , to remove the spurious correlation.
Revised: Motivated by the above causal analysis, in this section, we present our proposed debiasing GNN framework DisC , to remove the spurious correlation.

Original: The overall framework is shown in Fig.
Revised: The overall framework is shown in Fig.

Original: First,an edge mask generator is learnt to mask the edges of original input graphs into causal subgraphs 
Revised: First, an edge mask generator is learnt to mask the edges of original input graphs into causal subgraphs and bias subgraphs.

Original: [Figure] C C B and bias subgraphs. Second, two separate GNN modules with their corresponding masked subgraphsare trained to encode corresponding causal substructure and bias substructure into disentangledrepresentations, respectively.
Revised: Second, two separate GNN modules with their corresponding masked subgraphs are trained to encode corresponding causal substructure and bias substructure into disentangled representations, respectively.

Original: Last, after the disentangled representations are well-trained, we permutethe bias representations among the training graphs to generate counterfactual unbiased samples, sothat the correlation between causal representations and bias representations is removed.
Revised: Last, after the disentangled representations are well-trained, we permute the bias representations among the training graphs to generate counterfactual unbiased samples, so that the correlation between causal representations and bias representations is removed.

Original: [Figure]
Revised: [Figure]

Original: 4.1 Causal and Bias Substructure Generator
Revised: 4.1 Causal and Bias Substructure Generator

Original: [Figure] Given a mini-batch of biased graphs G = { G 1 , ⋯ , G n } , our idea is that: we take a collection of graphinstances and design a generative probabilistic model to learn to mask the edges into causal subgraphor bias subgraph.
Revised: Given a mini-batch of biased graphs G = { G 1 , ⋯ , G n } , our idea is that: we take a collection of graph instances and design a generative probabilistic model to learn to mask the edges into causal subgraph or bias subgraph.

Original: Particularly, given a graph G = { A , X } , where A is the adjacency matrix and X isthe node feature matrix, we utilize a multi-layer perceptron (MLP) upon the concatenation of nodefeatures x i of node i and x j of node j to measure the importance of edge ( i, j ) for causal subgraph:
Revised: Particularly, given a graph G = { A , X } , where A is the adjacency matrix and X is the node feature matrix, we utilize a multi-layer perceptron (MLP) upon the concatenation of node features x i of node i and x j of node j to measure the importance of edge ( i, j ) for causal subgraph:

Original: [Equation]
Revised: [Equation]

Original: Then a sigmoid function σ ( ⋅ ) is employed to project α ij into the range of (0,1), which indicates theprobability of edge ( i, j ) being the edge in the causal subgraph as follows:
Revised: Then a sigmoid function σ ( ⋅ ) is employed to project α ij into the range of (0,1), which indicates the probability of edge ( i, j ) being the edge in the causal subgraph as follows:

Original: [Equation]
Revised: [Equation]

Original: Naturally, we could get the probability of edge ( i, j ) being the edge in the bias subgraph by:b ij = 1 − c ij .
Revised: Naturally, we could get the probability of edge ( i, j ) being the edge in the bias subgraph by: b ij = 1 − c ij .

Original: Now we could construct the causal edge mask M c =
Revised: Now we could construct the causal edge mask M c =

Original: [ c ij ] and bias edge mask
Revised: [ c ij ] and bias edge mask

Original: M b =
Revised: M b =

Original: [ b ij ] .
Revised: [ b ij ] .

Original: Finally, we decompose the original graph G into causal subgraph G c
Revised: Finally, we decompose the original graph G into causal subgraph G c

Original: =
Revised: =

Original: { M c ⊙ A , X }and bias subgraph G b = { M b ⊙ A , X } .
Revised: { M c ⊙ A , X } and bias subgraph G b = { M b ⊙ A , X } .

Original: Intuitively, the edge mask could highlight different part ofstructure information of original graphs, thus GNNs built on the different subgraphs could encodedifferent parts of graph information.
Revised: Intuitively, the edge mask could highlight different part of structure information of original graphs, thus GNNs built on the different subgraphs could encode different parts of graph information.

Original: Moreover, the mask generator has two advantages.
Revised: Moreover, the mask generator has two advantages.

Original: (1) Globalview: In individual graph level, the mask generator ( i.e. , MLP), whose parameters are shared byall the edges in a graph, take a global view of all the edges in a graph, which enables us to identifycommunity in graph.
Revised: (1) Global view: In individual graph level, the mask generator ( i.e. , MLP), whose parameters are shared by all the edges in a graph, take a global view of all the edges in a graph, which enables us to identify community in graph.

Original: It is well known that the effect of an edge cannot be judged independently,because edges usually collaborate with each other, forming a community, to make prediction.
Revised: It is well known that the effect of an edge cannot be judged independently, because edges usually collaborate with each other, forming a community, to make prediction.

Original: Thus,it is critical to evaluate an edge in a global view.
Revised: Thus, it is critical to evaluate an edge in a global view.

Original: In whole graph population level, the mask generatortakes a global view of all the graphs in the training set, which enables us to identify causal/biassubgraph.
Revised: In whole graph population level, the mask generator takes a global view of all the graphs in the training set, which enables us to identify causal/bias subgraph.

Original: Particularly, as the causal/bias is the statistical information in the population level, itis necessary to view all the graphs to identify the causal/bias substructure.
Revised: Particularly, as the causal/bias is the statistical information in the population level, it is necessary to view all the graphs to identify the causal/bias substructure.

Original: Considering both suchcoalition effects and population-level statistical information, the generator is able to measure the importance of edges more accurately.
Revised: Considering both such coalition effects and population-level statistical information, the generator is able to measure the importance of edges more accurately.

Original: (2) Generalization : The mask generator can generalize themechanism of mask generation to new graphs without retraining, so it is capable and efficient toprune unseen graphs.
Revised: (2) Generalization : The mask generator can generalize the mechanism of mask generation to new graphs without retraining, so it is capable and efﬁcient to prune unseen graphs.

Original: 4.2 Learning Disentangled Graph Representations
Revised: 4.2 Learning Disentangled Graph Representations

Original: Given G c and G b , how to ensure they are causal subgraph and bias subgraph, respectively?
Revised: Given G c and G b , how to ensure they are causal subgraph and bias subgraph, respectively?

Original: Inspiredby [19], our approach simultaneously trains a pair of GNNs ( g b , g c ) with linear classifiers ( C b , C c )as follows: (1) Motivated by the observation in Sec.
Revised: Inspired by [23], our approach simultaneously trains a pair of GNNs ( g b , g c ) with linear classiﬁers ( C b , C c ) as follows: (1) Motivated by the observation in Sec.

Original: 3.1 that bias substructure is easier to learn, weutilize a bias-aware loss to train a bias GNN g b and a bias classifier C b and (2) in contrast, we train acausal GNN g c and a causal classifier C c on the training graphs that the bias GNN struggles to learn.
Revised: 3.1 that bias substructure is easier to learn, we utilize a bias-aware loss to train a bias GNN g b and a bias classiﬁer C b and (2) in contrast, we train a causal GNN g c and a causal classiﬁer C c on the training graphs that the bias GNN struggles to learn.

Original: Next, we would present each component in detail.
Revised: Next, we would present each component in detail.

Original: As shown in Fig.
Revised: As shown in Fig.

Original: 3, GNN g c and g b embed the corresponding subgraphs into causal embeddingz c
Revised: 3, GNN g c and g b embed the corresponding subgraphs into causal embedding z c = g c

Original: = g c ( G c ; γ c ) and bias embedding z b = g b ( G b ; γ b ) , respectively, where γ is the parameters of
Revised: ( G c ; γ c ) and bias embedding z b = g b ( G b ; γ b ) , respectively, where γ is the parameters of

Original: GNNs.
Revised: GNNs.

Original: Subsequently, concatenated vector z =
Revised: Subsequently, concatenated vector z =

Original: [ z c ; z b ] is fed into linear classifiers C c and C b topredict the target label y .
Revised: [ z c ; z b ] is fed into linear classiﬁers C c and C b to predict the target label y .

Original: To train g b and C b as bias extractor, we utilize the generalized cross entropy(GCE) [45] loss to amplify the bias of the bias GNN and classifier:
Revised: To train g b and C b as bias extractor, we utilize the generalized cross entropy (GCE) [51] loss to amplify the bias of the bias GNN and classiﬁer:

Original: [Equation]
Revised: [Equation]

Original: where C b ( z ; α b ) and C yb ( z ; α b ) are softmax output of the bias classifier and its probability belongingto the target category y , respectively, and α is the parameters of classifier.
Revised: where C b ( z ; α b ) and C yb ( z ; α b ) are softmax output of the bias classiﬁer and its probability belonging to the target category y , respectively, and α is the parameters of classiﬁer.

Original: Here q ∈ ( 0 , 1 ] is ahyperparameter that controls the degree of amplifying bias.
Revised: Here q ∈ ( 0 , 1 ] is a hyperparameter that controls the degree of amplifying bias.

Original: Given θ b =
Revised: Given θ b =

Original: [ γ b , α b ] , the gradient of the
Revised: [ γ b , α b ] , the gradient of the

Original: GCE loss up-weights the gradient of the standard cross entropy (CE) loss for the samples with a highconfidence C yb of predicting the correct target category as follows:
Revised: GCE loss up-weights the gradient of the standard cross entropy (CE) loss for the samples with a high conﬁdence C yb of predicting the correct target category as follows:

Original: [Equation]
Revised: [Equation]

Original: Therefore, compared with CE loss, GCE loss will amplify the gradients of θ b on samples by theconfidence score ( C yb ) q .
Revised: Therefore, compared with CE loss, GCE loss will amplify the gradients of θ b on samples by the conﬁdence score ( C yb ) q .

Original: Based on our observation that the bias information is usually easier to belearned, so the biased graphs will have higher ( C yb )
Revised: Based on our observation that the bias information is usually easier to be learned, so the biased graphs will have higher ( C yb )

Original: q than unbiased graphs.
Revised: q than unbiased graphs.

Original: Therefore, the modelg b and C b trained by GCE loss will focus on bias information and finally get the bias subgraph.
Revised: Therefore, the model g b and C b trained by GCE loss will focus on bias information and ﬁnally get the bias subgraph.

Original: Note that, to ensure that C b predicts target labels mainly based on this z b , the loss from C b is notbackpropagated to g c , i.e., only update θ b in Eq.
Revised: Note that, to ensure that C b predicts target labels mainly based on this z b , the loss from C b is not backpropagated to g c , i.e., only update θ b in Eq.

Original: (4), and vice versa.
Revised: (4), and vice versa.

Original: Meanwhile, we also train a causal GNN simultaneously with the weighted CE loss.
Revised: Meanwhile, we also train a causal GNN simultaneously with the weighted CE loss.

Original: The graphs withhigh CE loss from C b can be regarded as the unbiased samples compared with the samples with low
Revised: The graphs with high CE loss from C b can be regarded as the unbiased samples compared with the samples with low

Original: CE loss.
Revised: CE loss.

Original: In this regard, we could obtain the unbias score of each graph as
Revised: In this regard, we could obtain the unbias score of each graph as

Original: [Equation]
Revised: [Equation]

Original: Large value of W implies the graph is an unbiased sample, hence we could use these weights toreweight the loss of these graphs to train g c and C c , enforcing them to learn the unbiased information.
Revised: Large value of W implies the graph is an unbiased sample, hence we could use these weights to reweight the loss of these graphs to train g c and C c , enforcing them to learn the unbiased information.

Original: Thus, the objective function for learning disentangled representation is:
Revised: Thus, the objective function for learning disentangled representation is:

Original: [Equation]
Revised: [Equation]

Original: 4.3 Counterfactual Unbiased Sample Generation
Revised: 4.3 Counterfactual Unbiased Sample Generation

Original: Until now, we have achieved the first goal analyzed in Sec. 3.2 that is the disentanglement ofcausal and bias substructures.
Revised: Until now, we have achieved the ﬁrst goal analyzed in Sec. 3.2 that is the disentanglement of causal and bias substructures.

Original: Next, we will show how to achieve the second goal that makes thecausal variable z c and bias variable z b uncorrelated.
Revised: Next, we will show how to achieve the second goal that makes the causal variable z c and bias variable z b uncorrelated.

Original: Although we have disentangled causal and biasinformation, they are disentangled from the biased observed graphs.
Revised: Although we have disentangled causal and bias information, they are disentangled from the biased observed graphs.

Original: Hence, there will exist statisticalcorrelation between causal and bias variables inheriting from the biased observed graphs.
Revised: Hence, there will exist statistical correlation between causal and bias variables inheriting from the biased observed graphs.

Original: To furtherdecorrelate z c and z b , according to the causal relation of data-generating process: C → G ← B , wepropose to generate the counterfactual unbiased samples in embedding space by swapping z b .
Revised: To further decorrelate z c and z b , according to the causal relation of data-generating process: C → G ← B , we propose to generate the counterfactual unbiased samples in embedding space by swapping z b .

Original: Morespecifically, we randomly permute bias vectors in each mini-batch and obtain z unbiased =
Revised: More speciﬁcally, we randomly permute bias vectors in each mini-batch and obtain z unbiased =

Original: [ z c ; ˆ z b ] ,where ˆ z b represents the randomly permuted bias vectors of z b .
Revised: [ z c ; ˆ z b ] , where ˆ z b represents the randomly permuted bias vectors of z b .

Original: As z c and ˆ z b in z unbiased are randomly combined from different graphs, they will have much less correlation than z =
Revised: As z c and ˆ z b in z unbiased are randomly combined from different graphs, they will have much less correlation than z =

Original: [ z c ; z b ] where bothare from the same graph.
Revised: [ z c ; z b ] where both are from the same graph.

Original: To make g b and C b still focus on the bias information, we also swap label yas ˆ y along with ˆ z b , so that the spurious correlation between ˆ z b and ˆ y still exists.
Revised: To make g b and C b still focus on the bias information, we also swap label y as ˆ y along with ˆ z b , so that the spurious correlation between ˆ z b and ˆ y still exists.

Original: With the generatedunbiased samples, we utilize the following loss function to train two GNN modules:
Revised: With the generated unbiased samples, we utilize the following loss function to train two GNN modules:

Original: [Equation]
Revised: [Equation]

Original: Together with the disentanglement loss, total loss function is defined as:
Revised: Together with the disentanglement loss, total loss function is deﬁned as:

Original: [Equation]
Revised: [Equation]

Original: where λ G is a hyperparameter for weighting the importance of generation component.
Revised: where λ G is a hyperparameter for weighting the importance of generation component.

Original: Moreover,training with more diverse samples would also benefit with better generalization on unseen testingscenarios.
Revised: Moreover, training with more diverse samples would also beneﬁt with better generalization on unseen testing scenarios.

Original: Our approach is summarized in App.
Revised: Our approach is summarized in App.

Original: B. Note that, as we need well-disentangled representations to generate the high-quality unbiased samples, in the early stage of training, we only train themodel with L D .
Revised: B. Note that, as we need well-disentangled representations to generate the high-quality unbiased samples, in the early stage of training, we only train the model with L D .

Original: After certain epochs, we train the model with L .
Revised: After certain epochs, we train the model with L .

Original:  Experiment
Revised:  Experiment

Original: Datasets.
Revised: Datasets.

Original: We construct three datasets with various properties and bias ratios to benchmark thisnew problem, where the datasets have clear causal subgraphs making the results explainable.
Revised: We construct three datasets with various properties and bias ratios to benchmark this new problem, where the datasets have clear causal subgraphs making the results explainable.

Original: Following Colored MNIST-75sp introduced in Sec.
Revised: Following CMNIST-75sp introduced in Sec.

Original: 3.1, we use the similar way to construct Colored
Revised: 3.1, we use the similar way to construct CFashion-75sp and

Original: Fashion-75sp and Colored Kuzushiji-75sp datasets based on the Fashion-MNIST [38] and KuzushijiMNIST [4] datasets.
Revised: CKuzushiji-75sp datasets based on the Fashion-MNIST [44] and Kuzushiji-MNIST [4] datasets.

Original: As the causal subgraphs of these two datasets are more complicated (fashionproduct and hiragana characters), they are more challenging.
Revised: As the causal subgraphs of these two datasets are more complicated (fashion product and hiragana characters), they are more challenging.

Original: Due to the page limits, here we set biasdegrees as { 0 . 8 , 0 . 9 , 0 . 95 } .
Revised: Due to the page limits, here we set bias degrees as { 0 . 8 , 0 . 9 , 0 . 95 } .

Original: We report the main results on unbiased test sets.
Revised: We report the main results on unbiased test sets.

Original: Details are in App.
Revised: Details are in App.

Original: C.1.
Revised: C.1.

Original: Baselines and experimental setup.
Revised: Baselines and experimental setup.

Original: As DisC is a general framework which could be built on variousbase GNN models, we select three popular GNNs: GCN [15], GIN [39], and GCNII [3].
Revised: As DisC is a general framework which could be built on various base GNN models, we select three popular GNNs: GCN [19], GIN [45], and GCNII [3].

Original: Thecorresponding models are termed as DisC GCN , DisC GIN and DisC GCNII , respectively.
Revised: The corresponding models are termed as DisC GCN , DisC GIN and DisC GCNII , respectively.

Original: Hence, basemodels are the most straight baselines.
Revised: Hence, base models are the most straight baselines.

Original: Another baseline is the causal-inspired GNN method DIR [37].
Revised: Another kind fo baselines are the causal-inspired GNN method DIR [43] and StableGNN [7].

Original: We also compare against a general debiasing method LDD [19] by replacing its encoder with GNNs.
Revised: We also compare against a general debiasing method LDD [23] by replacing its encoder with GNNs.

Original: Graph Pooling method DiffPool [42] and graph disentangling method FactorGCN [40] are alsocompared.
Revised: Graph Pooling method DiffPool [48] and graph disentangling method FactorGCN [46] are also compared.

Original: To keep fair comparison, our model uses the same GNN architecture and hyperparameterswith the corresponding base model.
Revised: To keep fair comparison, our model uses the same GNN architecture and hyperparameters with the corresponding base model.

Original: All the experiments are run 4 times with different random seedsand we report the accuracy and the standard error.
Revised: All the experiments are runtimes with different random seeds and we report the accuracy and the standard error.

Original: More details are in App.
Revised: More details are in App.

Original: C.2.
Revised: C.2.

Original: 5.1 Quantitative Evaluation
Revised: 5.1 Quantitative Evaluation

Original: Main results.
Revised: Main results.

Original: The overall results are summarized in Table 1, and we have following observations:
Revised: The overall results are summarized in Table 1, and we have following observations:

Original: (1) DisC has much better generalization ability than base models.
Revised: (1) DisC has much better generalization ability than base models.

Original: DisC outperforms the correspond- ing base model consistently with a large margin.
Revised: DisC outperforms the corresponding base model consistently with a large margin.

Original: With heavier biases, our model achieves larger 
Revised: With heavier biases, our model achieves larger improvements over

Original: improvements over base models.
Revised: base models.

Original: Specifically, for Colored MNIST-75sp, Colored Fashion-75sp
Revised: Speciﬁcally, for CMNIST-75sp, CFashion-75sp and

Original: and Colored Kuzushiji-75sp with smaller bias degree ( i e., 0.8), our model achieves 40.02%, 4.47% and 29.82% average improvements over corresponding base models, respectively.
Revised: CKuzushiji-75sp with smaller bias degree ( i.e., 0.8), our model achieves 40.02%, 4.47% and 29.82% average improvements over corresponding base models, respectively.

Original: Surprisingly, with severer biases (0.9 and 0.95), DisC achieves 169.17%, 14.67% and 49.35% averageimprovements over base models on three datasets, respectively.
Revised: Surprisingly, with severer biases (0.9 and 0.95), DisC achieves 169.17%, 14.67% and 49.35% average improvements over base models on three datasets, respectively.

Original: It indicates that the proposed method is a general framework helping existing GNNs against the negative impact of bias.
Revised: It indicates that the proposed method is a general framework helping existing GNNs against the negative impact of bias.

Original: (2) DisC significantly outperforms existing debiasing methods.
Revised: (2) DisC signiﬁcantly outperforms existing debiasing methods.

Original: We notice that DIR could not achieve satisfying results.
Revised: We notice that DIR could not achieve satisfying results.

Original: The reason is that DIR utilizes CE loss to extract bias information, which could not fully capture the property of bias in severe bias scenarios. 
Revised: The reason is that DIR utilizes CE loss to extract bias information, which could not fully capture the property of bias in severe bias scenarios. And DIR sets one ﬁxed threshold to spilt subgraphs, which is suboptimal.

Original: Moreover, LDD is a general
Revised: StableGNN outperforms their base model DiffPool and achieve competitive results, indicating the effectiveness of their proposed causal variable distinguishing regularizer. However, their framework adjusts data distribution based on the original dataset, it is hard to generate unbiased distribution when the unbiased samples are scarce. DisC could generate more unbiased samples based on the disentangled representations.

Original:  debiasing method which is not designed for graph data
Revised: Moreover, LDD is a general debiasing method which is not designed for graph data.

Original: DisC outperforms corresponding LDDvariants with average 23.15%, indicating that the seamless joint of global-population-aware edgemasker with debiasing disentangle framework is very effective. 
Revised: DisC outperforms corresponding LDD variants with average 23.15%, indicating that the seamless joint of global-population-aware edge masker with debiasing disentangle framework is very effective. [Table]

Original: Ablation studies.
Revised: Ablation studies.

Original: To validate the importance of each module in our method, in Fig.
Revised: To validate the importance of each module in our method, in Fig.

Original: 4, we conductablation studies on our variants (w.o. G means without the sample generation module) and therelated variants of LDD.
Revised: 4, we conduct ablation studies on our variants (w.o. G means without the sample generation module) and the related variants of LDD.

Original: The major difference between DisC/w.o. G with LDD /w.o.
Revised: The major difference between DisC/w.o. G with LDD /w.o.

Original: G is the edge[Table] mask module.
Revised: G is the edge mask module.

Original: In most cases, DisC/w.o. G significantly outperforms LDD /w.o.
Revised: In most cases, DisC/w.o. G signiﬁcantly outperforms LDD /w.o.

Original: G, indicating thenecessity of learning edge mask for graph data.
Revised: G, indicating the necessity of learning edge mask for graph data.

Original: And DisC which has counterfactual sample generationmodule could further boost the performances based on the disentangled embeddings of DisC/w.o. G.
Revised: And DisC which has counterfactual sample generation module could further boost the performances based on the disentangled embeddings of DisC/w.o. G.

Original: However, LDD seldomly outperforms LDD /w.o.
Revised: However, LDD seldomly outperforms LDD /w.o.

Original: G or even achieves worse performances.
Revised: G or even achieves worse performances.

Original: That is,generating high-quality counterfactual samples needs well-disentangled causal and bias embeddings.
Revised: That is, generating high-quality counterfactual samples needs well-disentangled causal and bias embeddings.

Original: If embeddings are not well-disentangled, counterfactual samples may act as noisy samples, whichwould prevent models from achieving further improvement.
Revised: If embeddings are not well-disentangled, counterfactual samples may act as noisy samples, which would prevent models from achieving further improvement.

Original: The edge masker could help the modelgenerate well-disentangled embeddings, which is crucial for overall performance.
Revised: The edge masker could help the model generate well-disentangled embeddings, which is crucial for overall performance.

Original: [Figure]
Revised: [Figure]

Original: Robustness on unseen bias.
Revised: Robustness on unseen bias.

Original: Table 2 reports the results of DisC compared with its correspondingbase models on testing set with unseen bias, i.e., the pre-defined color (bias) sets of training set and testing set are disjoint.
Revised: Table 2 reports the results of DisC compared with its corresponding base models on testing set with unseen bias, i.e., the pre-deﬁned color (bias) sets of training set and testing set are disjoint.

Original: The performances of base models further drop compared with the resultson seen bias scenario in Table 1.
Revised: The performances of base models further drop compared with the results on seen bias scenario in Table 1.

Original: However, our model still achieves very stable performances, fullydemonstrating the generalization ability of our model on agnostic bias scenario. 
Revised: However, our model still achieves very stable performances, fully demonstrating the generalization ability of our model on agnostic bias scenario. Hyperparameter experiments Fig.

Original: [Table]
Revised: 5 is the hyperparameter experiments of the degree of amplifying bias q in GCE loss and the importance of generation component λ G . For q , we ﬁx λ G = 10 and vary q from { 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 } .

Original: 5.2 Qualitative Evaluation
Revised: For λ G , we ﬁx q = 0 . 7 and vary λ G from { 1 , 5 , 10 , 15 } . From the results, we can see that our model achieves stable performance across different values of q and [Table]

Original: Visualization of edge mask.
Revised: [Table] [Figure] λ G . When q = 0 .

Original: To better illustrate the significant causal and bias subgraphs extractedby DisC GCN , we visualize the original images, original graph
Revised: 1 , it means the GCE loss will nearly reduce to normal CE loss. We can see the performance of DisC GCN is worse than other scenarios, demonstrating the effectiveness of utilizing GCE loss.

Original: [Table]
Revised: [Figure]

Original:  and corresponding causal subgraphand bias subgraph of Colored MNIST-75sp with 0.9 bias degree in Fig.[Table]
Revised: Visualization of edge mask. To better illustrate the signiﬁcant causal and bias subgraphs extracted by DisC GCN , we visualize the original images, original graph, and corresponding causal subgraph and bias subgraph of CMNIST-75sp with 0.9 bias degree in Fig.

Original:  where the width of edgerepresents the value of learned weight c ij or b ij .
Revised: 6, where the width of edge represents the value of learned weight c ij or b ij .

Original: Fig.
Revised: Fig.

Original: 5(a) shows the visualization results of testinggraphs with the bias (color) that has been seen in the training set.
Revised: 6(a) shows the visualization results of testing graphs with the bias (color) that has been seen in the training set.

Original: As we can see, our model coulddiscover the causal subgraphs where the most salient edges are in the digital subgraphs.
Revised: As we can see, our model could discover the causal subgraphs where the most salient edges are in the digital subgraphs.

Original: With thesecausal subgraphs that highlight the structure information of digital, the GNNs will more easily extractthis causal information.
Revised: With these causal subgraphs that highlight the structure information of digital, the GNNs will more easily extract this causal information.

Original: Fig.
Revised: Fig.

Original: 5(b) shows the visualization results of testing graphs with unseen bias.
Revised: 6(b) shows the visualization results of testing graphs with unseen bias.

Original: According to the visualization, our model could still discover the causal subgraph outline, indicating our model could recognize causal subgraphs, whether the bias is seen or unseen.
Revised: According to the visualization, our model could still discover the causal subgraph outline, indicating our model could recognize causal subgraphs, whether the bias is seen or unseen.

Original: The visualizationresults of Colored Fashion-75sp and Colored Kuzushiji-75sp are shown in App.
Revised: The visualization results of CFashion-75sp and CKuzushiji-75sp are shown in App.

Original: D.
Revised: D.

Original: [Figure]

Original: Projection of disentangled representation.
Revised: Projection of disentangled representation.

Original: Fig.
Revised: Fig.

Original: 6 shows the projection of latent vectors z c and z bextracted from the causal GNN g c and bias GNN g b of DisC GCN , respectively, using t-SNE [17] on
Revised: 7 shows the projection of latent vectors z c and z b extracted from the causal GNN g c and bias GNN g b of DisC GCN , respectively, using t-SNE [21] on

Original: Colored MNIST-75sp.
Revised: [Figure]

Original: Fig.
Revised: Fig.

Original: 6 (a-b) are the projections of z c labeled by the target labels (digit) andbias labels (color), respectively.
Revised: color)

Original: Fig.
Revised: Fig.

Original: 6 (c-d) are the projectionsof z b labeled by the target labels andbias labels, respectively.
Revised: ) are the projections labeled by the target labels and labels respectively.

Original: We observe that z c are clustered according to the target labels while z b areclustered with the bias labels.

Original: And z c are  mixed with bias labelsand z b are mixed with target labels.
Revised: z c are clustered according to the target labels b c are mixed with bias labels b are mixed with target labels

Original: The results indicate that DisC successfully learns the disentangled causal and bias representations.

Original: Transferability of the learned mask.
Revised: Transferability of the learned mask.

Original: As our model could extract
Revised: As our model could extract

Original: GNN-independent subgraphs, the learning edge weights can be usedto purify original biased graphs.
Revised: GNN-independent subgraphs, the learning edge weights can be used to purify original biased graphs.

Original: These sparse subgraphs represent significant semantic information and can be universally transferred to any
Revised: These sparse subgraphs represent signiﬁcant semantic information and can be universally transferred to any

Original: GNNs.
Revised: GNNs.

Original: To validate this point, we learn the edge mask by DisC GCNand prune the edges with least { 0% , 20% , 40% , 60% } weights whilekeeping the remaining edge weights.
Revised: To validate this point, we learn the edge mask by DisC GCN and prune the edges with least { 0% , 20% , 40% , 60% } weights while keeping the remaining edge weights.

Original: Then we train vanilla GIN and
Revised: Then we train vanilla GIN and

Original: GCNII on these weighted pruned datasets.
Revised: GCNII on these weighted pruned datasets.

Original: Fig.
Revised: Fig.

Original: 7 is the comparison ofthe results, where the dashed lines represent the results of base modelon original biased graphs and the solid lines represent the performance of GNNs on weighted pruned datasets.
Revised: 8 is the comparison of the results, where the dashed lines represent the results of base model on original biased graphs and the solid lines represent the performance of GNNs on weighted pruned datasets.

Original: The results show thatthe GNNs trained on the pruned datasets achieve better performances,indicating our learned edge mask has considerable transferability.
Revised: The results show that the GNNs trained on the pruned datasets achieve better performances, indicating our learned edge mask has considerable transferability.

Original:  Conclusion
Revised:  Conclusion

Original: In this paper, we are first to study the generalization problem of GNNs on severe bias datasets, whichis crucial to study the transparently knowledge learning mechanism of GNNs.
Revised: In this paper, we are ﬁrst to study the generalization problem of GNNs on severe bias datasets, which is crucial to study the transparently knowledge learning mechanism of GNNs.

Original: We analyze the problemin a causal view that the generalization of GNNs will be hindered by entangled representations aswell as the correlation between causal and bias variables.
Revised: We analyze the problem in a causal view that the generalization of GNNs will be hindered by entangled representations as well as the correlation between causal and bias variables.

Original: To remove the impact from these twoaspects, we propose a general disentangling framework, DisC, which extracts causal substructureand bias substructure by two different functional GNNs, respectively.
Revised: To remove the impact from these two aspects, we propose a general disentangling framework, DisC, which extracts causal substructure and bias substructure by two different functional GNNs, respectively.

Original: After the representations arewell-disentangled, we proliferate the counterfactual unbiased samples by randomly swapping thedisentangled vectors.
Revised: After the representations are well-disentangled, we proliferate the counterfactual unbiased samples by randomly swapping the disentangled vectors.

Original: With the new constructed benchmarks, we clearly validate the effectiveness,robustness, interpretability, and transferability of our method. 
Revised: With the new constructed benchmarks, we clearly validate the effectiveness, robustness, interpretability, and transferability of our method. Acknowledgments and Disclosure of Funding

Original: Checklist
Revised: This work is supported in part by the National Natural Science Foundation of China (No. U20B2045, 62192784, 62172052, 62002029, 62172052, U1936014). This work is also partially supported by the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant, the Canada CIFAR AI Chair Program, collaboration grants between Microsoft Research and Mila, Samsung Electronics Co., Ltd., Amazon Faculty Research Award, Tencent AI Lab Rhino-Bird Gift Fund and a NRC Collaborative R&D Project (AI4D-CORE-06).

Original: For all authors
Revised: This project was also partially funded by IVADO Fundamental Research Project grant PRF-2019- 3583139727. The work of Shaohua Fan is supported by the China Scholarship Council (No.202006470078). The computation resource of this project is supported by Compute Canada 5 .

Original:  (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
Revised: For all authors... (a) Do the main claims made in the abstract and introduction accurately reﬂect the paper’s contributions and scope?

Original: [Yes] (b) Did you describe the limitations of your work?
Revised: [Yes] (b) Did you describe the limitations of your work?

Original: [Yes] See App.
Revised: [Yes] See App.

Original: E. (c) Did you discuss any potential negative societal impacts of your work?
Revised: E. (c) Did you discuss any potential negative societal impacts of your work?

Original: No] We could not forsee any potential negative societal impacts of our work.
Revised: [No] We could not foresee any potential negative societal impacts of our work.

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to them?

Original: [Yes] 
Revised: [Yes] 2.

Original: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?
Revised: If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes] See Sec. 3.2.
Revised: [Yes] See Sec. 3.2.

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] 
Revised: [Yes] 3.

Original: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either in the supplemental material or as a URL)?
Revised: If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?

Original: [Yes] We provide the URL of code and data for reproducing the main results.
Revised: [Yes] We provide the URL of code and data for reproducing the main results.

Original: See Sec. 5.

Original: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?
Revised: (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?

Original: [Yes] See Sec.
Revised: [Yes] See Sec.

Original: 5 and App.
Revised: 5 and App.

Original: C.
Revised: C.2.

Original: (c) Did you report error bars (e.g., with respect to the random seed after running experi- ments multiple times)?
Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?

Original: [Yes] See Sec.5.
Revised: [Yes] See Sec.

Original: (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?

Original: [Yes] See Sec.5.
Revised: [Yes] See Sec.

Original: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?
Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators?

Original: [Yes] We construct new assets based on existing assets.
Revised: [Yes] We construct new assets based on existing assets.

Original: We have cited them in Sec. 5.
Revised: We have cited them in Sec. 5.

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [Yes] See App.
Revised: [Yes] See App.

Original: C. 
Revised: C. (c)

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: Did you include any new assets either in the supplemental material or as a URL?

Original: [Yes] We release new assets through a URL in Sec.5.
Revised: [Yes] We release new assets through a URL in Sec.

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?

Original: [No] The source data for generating our data is publicly available.
Revised: [No] The source data for generating our data is publicly available.

Original: (e) Did you discuss whether the data you are using/curating contains personally identifi- able information or offensive content?
Revised: (e) Did you discuss whether the data you are using/curating contains personally identiﬁable information or offensive content?

Original: [No] We do not have personally identifiable information or offensive content.
Revised: [No] We do not have personally identiﬁable information or offensive content.

Original: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?
Revised: If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable?

Original: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?
Revised: [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?

Revised: [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you include the estimated hourly wage paid toparticipants and the total amount
Revised: In order to rigorously formalize our causal assumption behind the dataset, we resort to the Structural Causal Models, or SCM. SCM is a way of describing the relevant features (variables) of a particular problem and how they interact with each other.

Original: spent on participant compensation?
Revised: In particular, an SCM describes how the system assigns values to variables of interest. Formally, an SCM consists of a set of exogenous variables U and a set of endogenous variables V , and a set of functions f that determines the values of variables in V based on the other variables in the model. Casually, a variable X is a direct cause of a variables Y if X exists in the function that determines the value of Y .

Revised: If X is a direct cause of Y or of any cause of Y , X is a cause of Y .

Revised: Exogenous variables U roughly means that they are external to the model, hence, in most scenarios, we choose not to explain how they are caused.

Revised: Every endogenous variable is a descendant of at least one exogenous variable.

Revised: Exogenous variables can only be the root variables.

Revised: If we know the value of every exogenous variable, with the functions in f , we can perfectly determine the value of every endogenous variable.

Revised: In many cases, we usually assume that all exogenous variables are unobserved variables like noise and are independently distributed with an expected value zero, so we only interest with the interaction with endogenous variables.

Revised: Every SCM is associated with a graphical causal model or simply referred to “casual graph”.

Revised: Causal graph consists of nodes representing the variables in U and V , and the direct edges between the nodes representing the functions in f .

Revised: Note the in our

Revised: SCM in Section 3.2, we only show the endogenous variables we are interested in.

Revised: A.2 d -separation/connection

Revised: Given an SCM, we are particularly interested in (conditional) dependence information that is embedded in the model.

Revised: There are three basic relationships of variables in an SCM, i.e., chains, forks and colliders, as shown in Fig. 9.

Revised: For chains and forks, X and Y would be dependent if Z is not in the conditional set, i.e., the path is unblocked, and vice versa.

Revised: And for colliders, X and Y would be independent if Z is not in the conditional set, i.e., the path is blocked.

Revised: Built upon these rules, d -separation is a criterion that can be applied in causal graphs of any complexity in order to predict dependencies that are shared by all datasets generated by the graph [13].

Revised: Two nodes X

Revised: and Y are d -separated if every path between them is blocked.

Revised: If even one path between X and Y is unblocked,

Revised: X and Y are d -connected.

Revised: Formally, we have following deﬁnition of d -separation:

Revised: Deﬁnition 1 ( d -separation [13])

Revised: A path p is blocked by a set of nodes Z if and only if 1.

Revised: p contains a chain of nodes A → B → C or a fork A ← B → C such that the middle node B is in

Revised: Z ( i.e., B is conditioned on), or 2.

Revised: p contains a collider A

Revised: → B ← C such that the collision node B is not in Z , and no descendant of B is in Z .

Revised: With this principle, we could ﬁnd that the paths (1) B → G → E → Y and (2) B

Revised: ↔ C

Revised: → Y in

Revised: Section 3.2 are unblocked paths, which would induce unexpected correlation between bias variable

Revised: B and prediction Y .

Revised: [Figure]

Revised: More knowledge of causal inference, please refer to [13, 33].

Revised: Input : graph G , label y , max iteration T , generation iteration t gen

Revised: Output : Learned edge mask generator MLP, two GNN networks ( g c , g b ) , and two classiﬁers ( C c , C b )

Revised: Initialization : iteration t = 0 ; Initialize MLP, ( g c , g b ) and ( C c , C b ) 1 while not converged or t < T do

Revised: Extract subgraphs G c and G b based on edge mask generator;

Revised: Encode subgraphs G c and G b into z c and z b via g c ( G c ) and g b ( G b ) ;

Revised: Concatenate z

Revised: =

Revised: [ z c ; z b ] ;

Revised: Update ( g c , g b ) and ( C c , C b ) via L D in Eq.

Revised: (6); if t > t gen :

Revised: Randomly swap z =

Revised: [ z c ; z b ] into z unbiased =

Revised: [ z c ; ˆ z b ] ;

Revised: Update ( g c , g b ) and ( C c , C b ) via L in Eq.

Revised: (8); t = t + 1 ; 10 end

Revised: C Experimental Details C.1 Datasets details

Original: [N/A]
Revised: [Table]

Revised: We summarize statistics of datasets constructed in this paper in Table 3.

Revised: Note that the bias degree of validation set is 0.5, we use it to adjust the learning rate during training process.

Revised: Without loss of any generality, here we subsample original 60K training samples into 10K training samples to make the training process more efﬁcient.

Revised: One could easily construct full dataset with our method.

Revised: Each graph of CFashion-75sp is labeled by the category of fashion product it belongs to and each graph of

Revised: CKuzushiji-75sp is labeled by one of 10 Hiragana characters.

Revised: Moreover, we would like to list the map between label and predeﬁned correlated color for all datasets in Table 4.

Revised: The links for source image datasets are as follows:

Revised: MNIST: http://yann.lecun.com/exdb/mnist/. 2.

Revised: Fashion-MNIST: https://github.com/zalandoresearch/fashion-mnist.

Revised: MIT License.

Revised: Kuzushiji-MNIST: https://github.com/rois-codh/kmnist.

Revised: CC BY-SA 4.0 License.

Revised: [Table]

Revised: For unbiased testing dataset with unseen bias used in Table 2, the RGB value of predeﬁned color set is {(199, 21, 133), (255, 140, 105), (255, 127, 36), (139, 71, 38), (107, 142, 35), (173, 255, 47), (60, 179, 113), (0, 255,

Revised: C.2 Experimental setup

Revised: For GCN and GIN, we use the same model architectures as [15] 6 , which have 4 layers, andhidden dimension for GCN and 110 hidden dimension for GIN.

Revised: And GIN utilizes its GIN0 variant.

Revised: For GCNII, it has 4 layers and 146 hidden dimension.

Revised: DIR 7 utilizes the default parameters in original paper for MNIST-75sp dataset.

Revised: For causal GNN or bias GNN in our model, it has the same architecture with base model.

Revised: We optimize all models with the Adam [18] optimizer and 0.01 learning rate with for all experiments.

Revised: The batch-size for all the methods is 256.

Revised: We train all the models with 200 epochs and set the generation iteration of our method t gen as 100.

Revised: For our model, we set q of

Revised: GCE loss as 0 .

Revised: 7 and λ G as 10 for all experiments.

Revised: Our substructure generator is a two-layer MLP, whose activation function is sigmoid function.

Revised: For StableGNN, we use their GraphSAGE variant.

Revised: For other baselines, we use their default hyperparameters.

Revised: LDD 8 has same hyparameters with our model.

Revised: To better reﬂect the performance of unbiased sample generation, we take the performances of last step as ﬁnal results.

Revised: All the experiments are conducted on the single NVIDIA V100 GPU.

Revised: D Visualization of CFashion-75sp and CKuzushiji-75sp

Revised: Figure 10 and Figure 11 are visualization results of CFashion-75sp and CKuzushiji-75sp dataset.

Revised: As we can see, our model could also discover reasonable causal subgraphs for these challenging datasets.

Revised: E Limitations and societal impacts

Revised: Our method assumes that the graph consists of causal subgraph and bias subgraph.

Revised: In reality, it may also exist non-informative subgraphs, which is neither causal nor biased for label.

Revised: We would like to consider more ﬁne-grained splitting of graphs in the future.

Revised: When deploying GNNs to realworld applications, especially safety-critical ﬁelds, whether the results of GNNs are stable is an important factor.

Revised: The demands for a stable model are universal and extensive such as in the ﬁeld of disease prediction [37], trafﬁc states prediction [5], and ﬁnancial applications [41], where utilizing human-understandable causal knowledge for prediction is necessary.