MixupExplainer: Generalizing Explanations for Graph Neural Networks with
  Data Augmentation
Graph Neural Networks (GNNs) have received increasing attention due to their
ability to learn from graph-structured data. However, their predictions are
often not interpretable. Post-hoc instance-level explanation methods have been
proposed to understand GNN predictions. These methods seek to discover
substructures that explain the prediction behavior of a trained GNN. In this
paper, we shed light on the existence of the distribution shifting issue in
existing methods, which affects explanation quality, particularly in
applications on real-life datasets with tight decision boundaries. To address
this issue, we introduce a generalized Graph Information Bottleneck (GIB) form
that includes a label-independent graph variable, which is equivalent to the
vanilla GIB. Driven by the generalized GIB, we propose a graph mixup method,
MixupExplainer, with a theoretical guarantee to resolve the distribution
shifting issue. We conduct extensive experiments on both synthetic and
real-world datasets to validate the effectiveness of our proposed mixup
approach over existing approaches. We also provide a detailed analysis of how
our proposed approach alleviates the distribution shifting issue.
Jiaxing Zhang, Dongsheng Luo, Hua Wei
# MixupExplainer: Generalizing Explanations for Graph Neural Networks with Data Augmentation

###### Abstract.

Graph Neural Networks (GNNs) have received increasing attention due to their ability to learn from graph-structured data. However, their predictions are often not interpretable. Post-hoc instance-level explanation methods have been proposed to understand GNN predictions. These methods seek to discover substructures that explain the prediction behavior of a trained GNN. In this paper, we shed light on the existence of the distribution shifting issue in existing methods, which affects explanation quality, particularly in applications on real-life datasets with tight decision boundaries. To address this issue, we introduce a generalized Graph Information Bottleneck (GIB) form that includes a label-independent graph variable, which is equivalent to the vanilla GIB. Driven by the generalized GIB, we propose a graph mixup method, MixupExplainer, with a theoretical guarantee to resolve the distribution shifting issue. We conduct extensive experiments on both synthetic and real-world datasets to validate the effectiveness of our proposed mixup approach over existing approaches. We also provide a detailed analysis of how our proposed approach alleviates the distribution shifting issue.

graph neural network, explainability, data augmentation +
Footnote †: dagger}\)Corresponding author

+
Footnote †: dagger}\)Corresponding author
and its explanation shows that the explanation embeddings are out of distribution with respect to the original graphs, which leads to impaired safe usage of the approximation because of the inductive bias in \(f\). The negative impact of the distribution shifting problem on explanation quality is especially pronounced when applied to complex real-world datasets with tight decision boundaries.

While the distribution shifting issue in post-hoc explanations has gained growing attention in computer vision (Beng et al., 2017), this issue is less explored in the graph domain. In computer vision, (Beng et al., 2017) optimizes image classifier explanations to highlight contextual information relevant to the prediction and consistent with the training distribution. (Kang et al., 2018) addresses the distribution shifting issue in image explanation via a module that quantifies affinity between perturbed data and original dataset distribution. In the graph domain, while a recent work (Kang et al., 2019) attempts to address distribution shifting by annealing the size constraint coefficient at the start of the explanation process, the distribution shifting issue still persists throughout the explanation process.

To address the distribution shifting issue in post-hoc graph explanation, we introduce a general form of Graph Information Bottleneck (GIB) that includes another label-independent graph variable \(G^{\Delta}\). This new form of GIB is proven equivalent to vanilla GIB. By having \(G^{\Delta}\) in the objective, we can alleviate the distribution shifting problem with theoretical guarantees. To further improve the explanation method, we propose MixupExplainer using an improved Mixup approach. The MixupExplainer assumes that a non-explainable part of a graph is label-independent and mixes the explanation with a non-explainable structure from another randomly sampled graph. The explanation substructure is obtained by minimizing the difference between the predicted labels of the original graph and the mixup graph.

To the end, we summarize our contributions as follows.

* For the first time, we point out that the distribution shifting problem is prevalent in the most popular post-hoc explanation framework for graph neural networks.
* We derive a generalized framework with a solid theoretical foundation to alleviate the problem and propose a straightforward yet effective instantiation based on mixing up the explanation with a randomly sampled base structure by aligning the graph and mixing the graph masks.
* Comprehensive empirical studies on both synthetic and real-life datasets demonstrate that our method can dramatically and consistently improve the quality of the explanations, with up to 35.5% in AUC scores.

## 2. Related Work

### Graph Neural Networks

The use of graph neural networks (GNNs) is on the rise for analyzing graph structure data, as seen in recent research studies (Beng et al., 2017; Chen et al., 2018; Chen et al., 2018). There are two main types of GNNs: spectral-based approaches (Chen et al., 2018; Chen et al., 2018; Chen et al., 2018) and spatial-based approaches (Chen et al., 2018; Chen et al., 2018; Chen et al., 2018). Despite the differences, message passing is a common framework for both, using pattern extraction and message interaction between layers to update node embeddings. However, GNNs are still considered a black box model with a hard-to-understand mechanism, particularly for graph data, which is harder to interpret compared to image data. To fully utilize GNNs, especially in high-risk applications, it is crucial to develop methods for understanding how they work.

### GNN Explanation

Many attempts have been made to interpret GNN models and explain their predictions (Kang et al., 2018; Chen et al., 2018; Chen et al., 2018; Chen et al., 2018; Chen et al., 2018). These methods can be grouped into two categories based on granularity: (1) instance-level explanation, which explains the prediction for each instance by identifying significant substructures (Chen et al., 2018; Chen et al., 2018; Chen et al., 2018), and (2) model-level explanation, which seeks to understand the global decision rules captured by the GNN (Kang et al., 2018; Chen et al., 2018; Chen et al., 2018). From a methodological perspective, existing methods can be classified as (1) self-explainable GNNs (Kang et al., 2018; Chen et al., 2018), where the GNN can provide both predictions and explanations, and (2) post-hoc explanations (Kang et al., 2018; Chen et al., 2018; Chen et al., 2018), which use another model or strategy to explain the target GNN. In this work, we focus on post-hoc instance-level explanations, which involve identifying instance-wise critical substructures to explain the prediction. Various strategies have been explored, including gradient signals, perturbed predictions, and decomposition.

Perturbed prediction-based methods are the most widely used in post-hoc instance-level explanations. The idea is to learn a perturbation mask that filters out non-important connections and identifies dominant substructures while preserving the original predictions. For example, GNNExplainer (Chen et al., 2018) uses end-to-end learned soft masks on node attributes and graph structure, while PGExplainer (Kang et al., 2018) incorporates a graph generator to incorporate global information. RG-Explainer (Chen et al., 2018) uses reinforcement learning technology with starting point selection to find important substructures for the explanation.

However, most of these methods fail to consider the distribution shifting issue. The explanation should contain the same information that contributes to the prediction, but the GNN is trained on a data pattern that consists of an explanation subgraph relevant to labels, and a label-independent structure, leading to a distribution shifting problem when feeding the explanation directly into the GNN. Our

Figure 1. Visualization of original graphs \(G\), explanation subgraphs \(G^{*}\), and our generated graphs \(G^{(\text{mix})}\). There is a large distributional divergence between explanation subgraphs \(G^{*}\) and original graphs \(G\). \(G_{a}\) and \(G_{b}\) are two graphs in the original dataset. More experimental results on the existence of the distributional divergence can be found in Section 5.3.
method aims to capture the distribution information of the graph and build the explanation with a label-independent structure to help the explainer better minimize the objective function and retrieve a higher-quality explanation.

### Graph Data Augmentation with Mixup

Data augmentation addresses issues such as noise, scarcity, and out-of-distribution problems. One popular data augmentation approach is using Mixup (Wang et al., 2017) strategy to generate synthetic training examples based on feature mixing and label mixing. Specifically, (Wang et al., 2017; Wang et al., 2018) mix the graph representation learned from GNNs to avoid dealing with the arbitrary structure in the input space for mixing a node or graph pair. ifMixup (Wang et al., 2018) interpolates both the node features and the edges of the input pair based on feature mixing and graph generation. (Wang et al., 2018) and (Wang et al., 2018) generate interpolated graphs with the estimation of the properties in the graph data, like the graphon of each class or nearest neighbors of target nodes. All the previous methods (Wang et al., 2018; Wang et al., 2018; Wang et al., 2018; Wang et al., 2018) aim to generalize the mixup approach to improve the performance of classification models like GNNs. Unlike existing graph mixup approaches, this paper solves a different task, which is to generalize the explanations for GNN.

## 3. Preliminary

### Notations and Problem Definition

We denote a graph as \(G=(\mathcal{V},\mathcal{E};X,\mathbf{A})\), where \(\mathcal{V}=\{v_{1},v_{2},...,v_{n}\}\) represents a set of \(n\) nodes and \(\mathcal{E}\in\mathcal{V}\times\mathcal{V}\) represents the edge set. Each graph has a feature matrix \(\mathbf{X}\in\mathbb{R}^{n\times d}\) for the nodes, where in \(\mathbf{X},\mathbf{x}_{i}\in\mathbb{R}^{1\times d}\) is the \(d\)-dimensional node feature of node \(v_{i}\). \(\mathcal{E}\) is described by an adjacency matrix \(\mathbf{A}\in\{0,1\}^{n\times n}\). \(A_{ij}=1\) means that there is an edge between node \(v_{i}\) and \(v_{j}\); otherwise, \(A_{ij}=0\).

For graph classification task, each graph \(G_{i}\) has a label \(Y_{i}\in\mathcal{C}\), with a GNN model \(f\) trained to classify \(G_{i}\) into its class, i.e., \(f:(\mathbf{X},\mathbf{A})\mapsto\{1,2,...,C\}\). For the node classification task, each graph \(G_{i}\) denotes a \(K\)-hop sub-graph centered around node \(v_{i}\), with a GNN model \(f\) trained to predict the label for node \(v_{i}\) based on the node representation of \(v_{i}\) learned from \(G_{i}\).

Problem 1 (Post-hoc Instance-level GNN Explanation).: _Given a trained GNN model \(f\), for an arbitrary input graph \(G=(\mathcal{V},\mathcal{E};X,\mathbf{A})\), the goal of post-hoc instance-level GNN explanation is to find a subgraph \(G^{*}\) that can explain the prediction of \(f\) on \(G\)._

Informative feature selection has been well studied in non-graph structured data (Wang et al., 2017), and traditional methods, such as concrete autoencoder (Bengio et al., 2017), can be directly extended to explain features in GNNs. In this paper, we focus on discovering important typologies. Formally, the obtained explanation \(G^{*}\) is depicted by a binary mask \(\mathbf{M}\in\{0,1\}^{n\times n}\) on the adjacency matrix, e.g., \(G^{*}=(\mathcal{V},\mathcal{E},\mathbf{A}\odot\mathbf{M};\mathbf{X}),\odot\) means elements-wise multiplication. The mask highlights components of \(G\) which are essential for \(f\) to make the prediction.

### Graph Information Bottleneck

The Information Bottleneck (IB) (Wang et al., 2018; Wang et al., 2018) provides an intuitive principle for learning dense representations that an optimal representation should contain _minimal_ and _sufficient_ information for the downstream prediction task. Based on IB, a recent work unifies the most existing post-hoc explanation methods for GNN, such as GNNExplainer (Wang et al., 2018), PGExplainer (Wang et al., 2018), with the graph information bottleneck (GIB) principle (Wang et al., 2018; Wang et al., 2018; Wang et al., 2018). Formally, the objective of explaining the prediction of \(f\) on \(G\) can be represented by

\[\operatorname*{arg\,min}_{G^{*}}I(G,G^{*})-\alpha I(G^{*},Y), \tag{1}\]

where \(G^{*}\) is the explanation subgraph, \(Y\) is the original or ground truth label, and \(\alpha\) is a hyper-parameter to get the trade-off between minimal and sufficient constraints. GIB uses the mutual information \(I(G,G^{*})\) to select the minimal explanation that inherits only the most indicative information from \(G\) to predict the label \(Y\) by maximizing \(I(G^{*},Y)\), where \(I(G,G^{*})\) avoids imposing potentially biased constraints, such as the size of the connectivity of the selected subgraphs (Wang et al., 2018). Through the optimization of the subgraph, \(G^{*}\) provides model interpretation. Further, from the definition of mutual information, we have \(I(G^{*},Y)=H(Y)-H(Y|G^{*})\), where the entropy \(H(Y)\) is static and independent of the explanation process. Thus, minimizing the mutual information between the explanation subgraph \(G^{*}\) and \(Y\) can be reformulated as maximizing the conditional entropy of \(Y\) given \(G^{*}\). Formally, we rewrite the GIB objective as follows:

\[\operatorname*{arg\,min}_{G^{*}}I(G,G^{*})+\alpha H(Y|G^{*}), \tag{2}\]

As is shown in Figure 2(a), the objective function in Eq. (2) optimizes \(G^{*}\) to have the minimal mutual information with the original graph \(G\), which could be expressed as a subgraph from \(G\) with a smaller size, or scattered components in \(G\), while at the same time provides maximum mutual information for \(Y\), which is equivalent to have minimum entropy \(H(Y|G^{*})\).

Due to the intractability of entropy of the label conditioned on explanation, a widely-adopted approximation in previous methods (Wang et al., 2018; Wang et al., 2018; Wang et al., 2018) is:

\[\operatorname*{arg\,min}_{G^{*}}I(G,G^{*})+\alpha H(Y|G^{*})\approx \operatorname*{arg\,min}_{G^{*}}I(G,G^{*})+\alpha CE(Y,Y^{*}), \tag{3}\]

where \(Y^{*}=f(G^{*})\) is the predicted label of \(G^{*}\) made by the model to be explained, \(f\) and the cross-entropy \(\operatorname{CE}(Y,Y^{*})\) between the ground truth label \(Y\) and \(Y^{*}\) is used to approximate \(H(Y|G^{*})\).

## 4. Methodology

In this section, we first introduce an overlooked problem in the GIB objective. Then we propose a generalized GIB objective to address the problem, which directly inspires our method through a mixup approach.

### Generalized GIB

#### 4.1.1. Diverging Distributions in Eq. (3).

Although prevalent, the approximation with \(Y^{*}=f(G^{*})\) in Eq. (3) overlooks the distributional divergence between the original graph \(G\) and the dense subgraph \(G^{*}\) after the processing of the prediction model \(f\). An intuitive example from the MUTAG dataset (Chen et al., 2017) is shown in Figure 1. The prediction model \(f\), represented by a hypothesis line, performs well in classifying the positive and negative samples. Due to the distribution shifting problem naturally inherent in \(f(G)\) and \(f(G^{*})\) on explanation subgraphs, \(f\) maps some explanation subgraphs across the decision boundary to the negative region. As a result, the explanation subgraph achieved by Eq. (3) may be suboptimal and even
far away from the ground truth explanation due to the significant divergence between \(f(G^{*})\) and \(f(G)\). The existing GIB framework could work for simple synthetic datasets by relying on the implicit knowledge associated with the class and assuming a large decision margin between the two or more classes. However, in more practical scenarios like MUTAG, the existing approximation may be heavily affected by the distribution shifting problem (Kraus et al., 2019; Kraus et al., 2019).

#### 4.1.2. Addressing with Label-independent Subgraph

To address the above challenge in the previous GIB methods, we first generalize the existing GIB framework by taking a label-independent subgraph \(G^{\Delta}\) into consideration. The intuition is that for an original graph \(G_{a}\) with label \(Y_{a}\), the label-independent subgraph \(G^{\Delta}_{a}\) also contains useful information. For example, \(G^{\Delta}_{a}\) makes sure that connecting it with the label-preserving subgraph \(G^{*}_{a}\) will not lead to another label. Formally, given a graph variable \(G^{\Delta}\) that satisfies \(I(G^{\Delta},Y|G^{*})=0\), the GIB objective can be generalized as follows.

\[\operatorname*{arg\,min}_{G^{*}}I(G,G^{*})+\alpha H(Y|G^{*},G^{\Delta}), \quad\text{s.t.}\quad I(G^{\Delta},Y|G^{*})=0. \tag{4}\]

As shown below, our generalized GIB has the following property.

Property 1 ().: _The generalized GIB objective, Eq. (4) is equivalent to vanilla GIB, Eq. (2)._

This can be proved by the definition of conditional entropy. With the condition that \(I(G^{\Delta},Y|G^{*})=0\), we have \(H(Y|G^{*})=H(Y|G^{*})+I(G^{\Delta},Y|G^{*})=H(Y|G^{*},G^{\Delta})\). Thus, the optimal solutions of GIB and our generalized version are equivalent. In addition, the advantage of our objective is that by choosing a suitable \(G^{\Delta}\) that minimizes the distribution distance, \(D(G^{*}+G^{\Delta},G)\), we can approximate the GIB without including the distribution shifting problem. An intuitive illustration is given in Figure 2(b).

Following exiting work (Kraus et al., 2019; Kraus et al., 2019), we can further approximate \(H(Y|G^{*},G^{\Delta})\) with \(\operatorname{CE}(Y,Y^{m})\), where \(Y^{m}=f(G^{*}+G^{\Delta})\) is the predicted label of \(G^{*}+G^{\Delta}\) made by the model \(f\) to be explained. Especially when \(G^{\Delta}\) is an empty graph, our objective degenerates to the vanilla approximation. Formally, we derive our new objective for GNN explanation as follows:

\[\operatorname*{arg\,min}_{G^{\Delta},G^{*}} I(G,G^{*})+\alpha\operatorname{CE}(Y,Y^{m})\] \[\text{s.t.}\operatorname{D}(G^{*}+G^{\Delta},G)=0,I(G^{\Delta},Y| G^{*})=0. \tag{5}\]

### MixupExplainer

Inspired by Eq. 5, in this section, we introduce a straightforward yet theoretically guaranteed instantiation, MixupExplainer, to resolve the distribution shifting issue. Figure 3 demonstrates the overall framework of the proposed MixupExplainer and the differences between MixupExplainer and previous GIB methods. MixupExplainer includes a graph generation phase after extracting the explanation of the graph with the explainer. Specifically, we instantiate the \(G^{\Delta}\) from the distribution of label-independent subgraphs from the graph dataset, denoted as \(\mathbb{P}_{\mathcal{G}^{(i)}}\), and connect \(G^{*}\) and \(G^{\Delta}\) to generate a new graph \(G^{(\text{mix})}\). Formally,

\[G^{\Delta}\sim\mathbb{P}_{\mathcal{G}^{(i)}},\quad G^{(\text{mix})}=G^{*}+G^{ \Delta}. \tag{6}\]

To avoid the trivial case that \(G=G^{(\text{mix})}\), when sampling \(G^{\Delta}\), we dismiss the original graph itself. In addition, since \(G^{\Delta}\) is sampled without considering the label information, we can make a safe assumption that \(I(G^{\Delta},Y|G^{*})=0\).

As stated in Problem 1, given a graph \(G_{a}=(\mathbf{A}_{a},\mathbf{X}_{a})\)1 and a to-be-explained model \(f\), an explanation model \(g\) aims to learns a subgraph \(G^{*}_{a}\), represented with the edge mask \(\mathbf{M}_{a}=g(G_{a})\) on the adjacency matrix \(\mathbf{A}_{a}\). To generate a graph distributed similarly to \(G_{a}\), we need to generate a label-independent subgraph, where we randomly sample another graph instance from the dataset, denoted by \(G_{b}\), without considering the label information. With the explanation model \(g\), we obtain the corresponding edge mask \(\mathbf{M}_{b}\) for \(G_{b}\). Then, we mix these two graphs by connecting the informative part in \(G_{a}\) and the label-independent part in \(G_{b}\). We first assume that \(G_{a}\) and \(G_{b}\) share the same set of nodes, and more general cases are discussed in the next section. Formally, the mask of the mixed graph, \(\mathbf{M}_{a}^{(\text{mix})}\), is calculated as follows.

Footnote 1: We dismiss \(\mathcal{V}\) and \(\mathcal{E}\) to simplify the notations.

\[\mathbf{M}_{a}^{(\text{mix})}=\lambda\mathbf{M}_{a}+(\mathbf{A}_{b}-\lambda\mathbf{M}_{b}), \tag{7}\]

where \(\mathbf{A}_{b}\) is the adjacency matrix of graph \(G_{b}\) and \(\lambda\) is a hyper-parameter to support flexible usage of mixup operation. Then, we have \(G_{a}^{(\text{mix})}=(\mathbf{X}_{a},\mathbf{M}_{a}^{(\text{mix})})\). The mask matrix \(\mathbf{M}_{a}\) and \(\mathbf{M}_{b}\) denote the weight of the edges in \(\mathbf{A}_{a}\) and \(\mathbf{A}_{b}\), respectively, with the same size of the matrix. By default, we mix up \(G^{*}_{a}\) with the rest part of the \(G_{b}\) by setting \(\lambda=1\) and above formula could be further simplified as:

\[\mathbf{M}_{a}^{(\text{mix})}=\mathbf{M}_{a}+(\mathbf{A}_{b}-\mathbf{M}_{b}). \tag{8}\]

Note that our proposed mixup approach is different from traditional mixup approaches (Kraus et al., 2019; Kraus et al., 2019; Kraus et al., 2019) in data augmentation, where they usually follow a form similar to \(\mathbf{M}^{(\text{mix})}=\lambda\mathbf{M}_{a}+(1-\lambda)\mathbf{M}_{b}\). This form of mixup does not differentiate label-dependent from label-independent parts. On the contrary, our proposed mixup approach in Eq. (7) includes the label-dependent part in \(G_{a}\) with \(\lambda\mathbf{M}_{a}\) and excludes the label-dependent part in \(G_{b}\) by subtracting the same \(\lambda\) on \(\mathbf{M}_{b}\) from \(\mathbf{A}_{b}\).

Figure 2. Illustration of GIB and our proposed new objective. (a) Previous vanilla GIB objective aims to minimize \(I(G^{*},Y)\) and \(H(Y|G^{*})\), with a smaller overlap between \(G^{*}\) and \(G\). (b) Our generalized GIB objective has the same objective as vanilla GIB, with a larger lap between \(G\) and \(G^{*}+G^{\Delta}\), resulting in less distribution shifting issue.
#### 4.2.1. Implementation

In this section, we introduce the implementation details of the mixup function and provide the pseudo-code of graph mixup in Algorithm 1.

Given a graph \(G_{a}\) with \(n_{a}\) nodes and another graph \(G_{b}\) with \(n_{b}\) nodes, the addition in Eq. (7) between two matrices requires \(\mathbf{M}_{a}\) and \(\mathbf{M}_{b}\) have the same dimensions, i.e., \(G_{a}\) and \(G_{b}\) have the same number of nodes. However, in real-world graph datasets, this assumption may not hold, leading to a mismatch between the dimensions of \(\mathbf{M}_{a}\) and \(\mathbf{M}_{b}\). In order to merge two graphs with different sets of nodes, we first extend node sets in \(G_{a}\) and \(G_{b}\) to a single node set \(\mathcal{V}_{a}\cup\mathcal{V}_{b}\), and their adjacency matrices are calculated with the following functions:

\[\mathbf{A}_{a}^{\text{ext}}=\left[\begin{array}{cc}\mathbf{A}_{a}&0\\ 0&0_{b}\end{array}\right],\mathbf{A}_{b}^{\text{ext}}=\left[\begin{array}{cc} \mathbb{0}_{a}&0\\ 0&\mathbf{A}_{b}\end{array}\right], \tag{9}\]

where \(\mathbb{0}_{a}\) and \(\mathbb{0}_{b}\) are zero matrices with shapes \(n_{a}\times n_{a}\) and \(n_{b}\times n_{b}\), respectively.

After extending \(G_{a}\) and \(G_{b}\), we then merge them into \(G^{(\text{mix})}=(\mathbf{X}^{(\text{mix})},\mathbf{M}_{a}^{(\text{mix})}\odot\mathbf{A} ^{(\text{mix})})\), where \(\mathbf{X}^{(\text{mix})}=[\mathbf{X}_{a};\mathbf{X}_{b}]\) is the concatenation of node features \(\mathbf{X}_{a}\) and \(\mathbf{X}_{b}\); \(\mathbf{A}^{(\text{mix})}\) is the merged adjacency matrix; \(\mathbf{M}_{a}^{(mix)}\) is the edge mask indicating the edge weights for the explanation.

Specifically, the adjacency matrix of \(G^{(\text{mix})}\) is:

\[\mathbf{A}^{(\text{mix})}=\left[\begin{array}{cc}\mathbf{A}_{a}&\mathbf{A}_{c}\\ \mathbf{A}_{c}^{T}&\mathbf{A}_{b}\end{array}\right], \tag{10}\]

where \(\mathbf{A}_{c}\) is a matrix indicating the cross-graph connectivity between the nodes in \(G_{a}\) and \(G_{b}\). In practice, we randomly sample \(\eta\) cross-graph edges to connect \(G_{a}\) and \(G_{b}\) at each mixup step to ensure the mixed graph is a connected graph to be optimized together on both label-dependent and label-independent subgraphs.

Similarly, the edge mask matrix is obtained from extended \(\mathbf{M}_{a}\) and \(\mathbf{M}_{b}\) and calculated with Eq. (7). Formally, we have

\[\mathbf{M}_{a}^{(\text{mix})}=\left[\begin{array}{cc}\lambda\mathbf{M}_{a}&\mathbf{M}_ {c}\\ \mathbf{M}_{c}^{T}&\mathbf{A}_{b}-\lambda\mathbf{M}_{b}\end{array}\right] \tag{11}\]

where the explainer \(g\) gives the \(\mathbf{M}_{a}\) and \(\mathbf{M}_{b}\), \(\mathbf{M}_{c}\) is the weight matrix on the randomly-sampled cross-graph edges corresponding with \(\mathbf{A}_{c}\), where the values are randomly sampled on connected edges in \(\mathbf{A}_{c}\) at each mixup step and thus will not be optimized by \(g\).

Finally, we can mixup the edge weight matrices \(\mathbf{M}_{a}^{\text{ext}}\) and \(\mathbf{M}_{b}^{\text{ext}}\) together with Eq. (7). The mixed graph \(G_{a}^{(\text{mix})}\) is then fed into the GNN model \(f\) to calculate the predicted result \(Y^{(\text{mix})}\). The detailed implementation is shown in Algorithm 1.

#### 4.2.2. Computational Complexity Analysis

Here, we analyze the computational complexity of our mixup approach. Given a graph \(G_{a}\) and a randomly sampled graph \(G_{b}\), the complexity of graph extension on adjacency matrices and edge masks is \(\mathcal{O}(|\mathcal{E}_{a}|+|\mathcal{E}_{b}|)\), where \(|\mathcal{E}_{a}|\) and \(|\mathcal{E}_{b}|\) denote the number of edges in \(G_{a}\) and \(G_{b}\), respectively. To generate \(\eta\) cross-graph edges, the computational complexity is \(\mathcal{O}(\eta)\). For mixup, the complexity is \(\mathcal{O}(|\mathcal{E}_{a}|+|\mathcal{E}_{b}|)\). By considering \(\eta\) as a small constant, the overall complexity of our mixup approach is \(\mathcal{O}(|\mathcal{E}_{a}|+|\mathcal{E}_{b}|)\).

#### 4.2.3. Theoretical Justification

In the following, we theoretically prove that: _the proposed mixup approach could reduce the distance between the explanation and original graphs_. Formally, we have the following theorem:

**Theorem 1**.: _Given an original graph \(G\), graph explanation \(G^{*}\) and \(G^{(\text{mix})}\) generated by Eq. (7), we have \(KL(G,G^{*})\geq KL(G,G^{(\text{mix})})\)._

Proof Sketch.: According to the previous work (Zhu et al., 2019; Zhang et al., 2020), a graph \(G\) can be treated as \(G=G^{(e)}+G^{(i)}\), where \(G^{(e)}\) presents the underlying subgraph that makes important contributions to GNN's predictions, which is the expected explanatory graph, and \(G^{(i)}\) consists of

Figure 3. Illustration of the GIB-based explanation and our proposed MixupExplainer. (a) Vanilla GIB directly minimizes \(\text{CE}(Y,Y^{*})\), which is the cross entropy between the original prediction \(Y\) and the prediction of explanation subgraph \(G^{*}\) made by the to-be-explained model \(f\). (b) Our MixupExplainer first generates an augmented graph \(G^{(\text{mix})}\) by mixing up the explanation subgraph \(G^{*}\) with the label-independent part from another randomly sampled graph. Then we minimize the cross entropy between \(Y\) and \(Y^{(\text{mix})}\), the prediction made by \(f\) on \(G^{(\text{mix})}\).
the remaining label-independent edges for predictions made by the GNN. Assuming the graph \(G^{(\mathbf{e})}\) and \(G^{(i)}\) independently follow the distribution \(\mathbb{P}_{\mathbf{\mathcal{G}}^{(e)}}\) and \(\mathbb{P}_{\mathbf{\mathcal{G}}^{(i)}}\) respectively, denoted as \(G^{(\mathbf{e})}\sim\mathbb{P}_{\mathbf{\mathcal{G}}^{(e)}}\) and \(G^{(i)}\sim\mathbb{P}_{\mathbf{\mathcal{G}}^{(i)}}\), we randomly sample \(G_{b}=G_{b}^{(\mathbf{e})}+G_{b}^{(i)}\) from the data set. Both \(G\) and \(G_{b}\) follow the distribution \(\mathbb{P}_{\mathbf{\mathcal{G}}}=\mathbb{P}_{\mathbf{\mathcal{G}}^{(e)}}\).6 We could get our Mixup explanation:

Footnote 6: All the dataset and codes can be found in [https://github.com/jr48/MixupExplainer](https://github.com/jr48/MixupExplainer)

\[G^{(\text{mix})}\coloneqq G^{(\mathbf{e})}+(G_{b}-G_{b}^{(\mathbf{e})})=G^{(\mathbf{e})}+G _{b}^{(i)}, \tag{12}\]

Then, we have \(\mathbb{P}_{\mathbf{\mathcal{G}}^{(\text{mix})}}=\mathbb{P}_{\mathbf{\mathcal{G}}^{(e )}}\ast\mathbb{P}_{\mathbf{\mathcal{G}}^{(i)}}=\mathbb{P}_{\mathbf{\mathcal{G}}}\). It is easy to show that \(KL(G,G^{(\text{mix})})=0\). Thus, we have

\[KL(G,G^{*})\geq\text{KL}(G,G^{(\text{mix})}) \tag{13}\]

The theoretical justification shows that our objective function could better estimate the explanation distribution and resolve the distribution shifting issue than the previous approach. In addition, with a safe assumption that \(I(G^{\Delta},Y|G^{*})=0\), as discussed in Eq. (6), we have MixupExplainer satisfy the s.t. condition in Eq. (5). Thus, we can simplify the objective for MixupExplainer as:

\[\operatorname*{arg\,min}_{G^{*}}\quad I(G,G^{*})+\alpha\text{CE}(Y,Y^{(\text{ mix})}) \tag{14}\]

## 5. Experimental Study

We conduct comprehensive experimental studies on benchmark datasets to empirically verify the effectiveness of the proposed MixupExplainer. Specifically, we aim to answer the following research questions:

* RQ1: Can the proposed framework outperform the GIB in identifying explanatory substructures for GNNs?
* RQ2: Is the distribution shifting issue severe in the existing GNN explanation methods? Could the proposed Mixup approach alleviate this issue?
* RQ3: How does the proposed approach perform under different hyperparameters?

### Experiment Settings

#### 5.1.1. Datasets

We focus on analyzing the effects of the distribution shifting problem between the ground truth explanation and the original graphs. Thus, we select six publicly available benchmark datasets with ground truth explanations in our empirical studies 2.

Footnote 2: All the dataset and codes can be found in [https://github.com/jr48/MixupExplainer](https://github.com/jr48/MixupExplainer)

* **BA-Shapes**(Zhang et al., 2019): This is a node classification dataset based on a 300-node Barabasi-Albert (BA) graph, to which 80 "house" motifs have been randomly attached. The nodes are labeled for use by GNN classifiers, while the edges within the corresponding motif serve as ground truth for explainers. There are four classes in the classification task, with one class indicating nodes in the base graph and the others indicating the relative location of nodes in the motif.
* **BA-Community**(Zhang et al., 2019): This extends the BA-Shapes dataset to more complex scenarios with eight classes. Two types of motifs are attached to the base graph, with nodes in different motifs having different labels.
* **Tree-Circles**(Zhang et al., 2019): This is a node classification dataset with two classes, with a binary tree serving as the base graph and a 6-node cycle structure as the motif. The labels only indicate if the nodes are in the motifs.
* **Tree-Grid**(Zhang et al., 2019): This is a node classification dataset created by attaching 80 grid motifs to a single 8-layer balanced binary tree. The labels only indicate if the nodes are in the motifs, and edges within the relative motif are used as ground-truth explanations.
* **BA-2motifs**(Zhang et al., 2019): This is a graph classification dataset where the label of the graph depends on the type of motif attached to the base graph, which is a BA random graph. The two types of motifs are a 5-node house structure and a 5-node circle structure.
* **MUTAG**(Chen et al., 2019): Unlike other synthetic datasets, MUTAG is a real-world molecular dataset commonly used for graph classification explanations. Each graph in MUTAG represents a molecule, with nodes representing atoms and edges representing bonds between atoms. The labels for the graphs are based on the chemical functionalities of the corresponding molecules.

#### 5.1.2. Baselines

To assess the effectiveness of the proposed framework, we use representative GIB-based explanation methods, GNNExplainer (Zhang et al., 2019) and PGExplainer (Zhang et al., 2019) as baselines. We include these two backbone explainers in our framework MixupExplainer and replace the GIB objective with the new proposed mixup objective. The methods are denoted by MixUp-GNNExplainer and MixUp-PGExplainer, respectively. We also include other types of post-hoc explanation methods for comparison, including GRAD (Zhang et al., 2019), ATT (Zhang et al., 2019), SubgraphX (Zhang et al., 2019), MetaGNN (Zhang et al., 2019), and RG-Explainer (Zhang et al., 2019).

* **GRAD**(Zhang et al., 2019): GRAD learns weight vectors of edges by computing gradients of GNN's objective function.
* **ATT**(Zhang et al., 2019): ATT distinguishes the edge attention weights in the input graph with the self-attention layers. Each edge's importance is obtained by averaging its attention weights across all attention layers.
* **SubgraphX**(Zhang et al., 2019): SubgraphX uses Monte Carlo Tree Search (MCTS) to find out the connected sub-graphs, which could preserve the predictions as explanations.
* **MetaGNN**(Zhang et al., 2019) MetaGNN proposes a meta-explainer for improving the level of explainability of a GNN directly at training time by training the GNNs and the explainer in turn.
* **RG-Explainer**(Zhang et al., 2019): RG-Explainer is an RL-enhanced explainer for GNN, which constructs the explanation subgraph by starting from a seed and sequentially adding nodes with an RL agent.
* **GNNExplainer**(Zhang et al., 2019): GNNExplainer is a post-hoc method, which provides explanations for every single instance by learning an edge mask for the edges in the graph. The weight of the edge could be treated as important.
* **PGExplainer**(Zhang et al., 2019): PGExplainer extends GNNExplainer by adopting a deep neural network to parameterize the generation process of explanations, which enables PGExplainer to explain the graphs in a global view. It also generates the substructure graph explanation with the edge importance mask.
#### 5.1.3. Configurations

The experiment configurations are set following prior research (Kumar et al., 2017). A three-layer GCN model was trained on 80% of each dataset's instances as the target model. All explanation methods used the Adam optimizer with a weight decay of \(5e\)-4 (Kumar et al., 2017). The learning rate for GNNExplainer was initialized to 0.01, with 100 training epochs. For PGExplainer, the learning rate was set to 0.003, and the training epoch was 30. The weight of mix-up processing, controlled by \(\lambda\), was determined through grid search. Explanations are tested in all instances. While running our approach MixUp-GNNExplainer and MixUp-PGExplainer and comparing them to the original GNNExplainer and PGExplainer, we set them with the same configurations, respectively. Hyperparameters are kept as the default values in other baselines.

#### 5.1.4. Evaluation Metrics

Due to the existence of gold standard explanations, we follow existing works (Kumar et al., 2017; Wang et al., 2018; Wang et al., 2019) and adopt AUC-ROC score on edge importance to evaluate the faithfulness of different methods. Other metrics, such as fidelity (Zhao et al., 2018), are not included because the metrics themselves are affected by the distribution shifting problem, making them unsuitable in our setting.

To quantitatively measure the distribution shifting between the original graph and the explanation graph, we use _Cosine score_ and _Euclidean distance_ to measure the distances between the graph embeddings learned by the GNN model. For the Cosine score, the range is \([-1,1]\), with 1 being the most similar and -1 being the least similar. For the Euclidean distance, the smaller, the better.

### Quantitative Evaluation (RQ1)

To answer RQ1, we compare MixupExplainer with other baseline methods in terms of the AUC-ROC score. Our approach is evaluated using the weighted vector of the graph generated by the explainers, which serves as the explanation and is compared against the ground truth to calculate the AUC-ROC score. Each experiment is conducted 10 times with random seeds. We summarize the average performances in Table 1.

As shown in Table 1, across all six datasets, with both GNNExplainer or PGExplainer as the backbone methods, MixupExplainer can consistently and significantly improve the quality of obtained explanations. Specifically, Mixup-GNNExplainer improves the AUC scores by 12.3%, on average, on the node classification datasets, and 22.6% on graph classification tasks. Similarly, MixUp-PGExplainer achieves average improvements of 5.41% and 19.4% for node/graph classification tasks, respectively. The comparisons between our MixupExplainer and the original counterparts indicate the advantage of the proposed explanation framework. In addition, MixUp-PGExplainer achieves competitive and even state-of-the-art performances compared with other sophisticated baselines, such as reinforcement learning-based RG-Explainer.

### Alleviating Distribution Shifts (RQ2)

In the previous section, we showed that our MixUp approach outperforms existing explanation methods in terms of AUC-ROC. In this section, we show the existence of the distribution shifting issue and show our proposed mixup approach alleviates this issue and improves the performance in explanation w.r.t. AUC.

**Visualizing Distributing Shifting.** In this section, we show the existence of the distribution shifting issue by visualizing the distribution vector (the output of the last layer in a well-trained GNN model \(f\)) for the original graph \(G\), the explanation from MixupExplainer \(G^{\text{(mix)}}\), and the ground truth explanation \(G^{*}\) with t-Distributed Stochastic Neighbor Embedding(t-SNE) (Srivastava et al., 2017). To calculate distribution vectors, we use the output of the last GNN layer in \(f\) as the representation vector \(\mathbf{h}\) for the original graph. Ground truth explanations \(G^{*}\) and the mixup graph from MixupExplainer, \(G^{\text{(mix)}}\) are also fed into the model to achieve corresponding representations, denoted by \(\mathbf{h}^{*}\), \(\mathbf{h}^{\text{(mix)}}\), respectively. The visualization results can be found in Figure 4. The red points represent the vectors from original graphs \(\mathbf{G}\); the blue points represent vectors from substructure explanations \(\mathbf{G}^{\prime}\), and the green points represent the vectors from the mixup explanations \(\mathbf{G}^{\prime}_{mix}\). Note that for BA-2motifs, while there are multiple graphs in the dataset, with only two kinds of motifs as explanations, t-SNE only shows two blue points, which are actually multiple overlapping blue points. From Figure 4, we have the following observations:

\(\bullet\) The blue points shift away from the red points in most datasets, including both synthetic and real-world datasets. It means that the distribution shifting issue exists in most cases, where most existing work overlooked this issue.

\(\bullet\) The green points are inseparable from the red points in most

\begin{table}
\begin{tabular}{c|c c c c c c} \hline \hline  & BA-Shapes & BA-Community & Tree-Circles & Tree-Grid & BA-2motifs & MUTAG \\ \hline GRAD & 0.882 & 0.750 & 0.905 & 0.612 & 0.717 & 0.783 \\ ATT & 0.815 & 0.739 & 0.824 & 0.667 & 0.667 & 0.765 \\ SubgraphX & 0.548 & 0.473 & 0.617 & 0.516 & 0.610 & 0.529 \\ MetaGNN & 0.851 & 0.688 & 0.523 & 0.628 & 0.500 & 0.680 \\ RG-Explainer & 0.985 & 0.919 & 0.787 & 0.927 & 0.657 & 0.873 \\ \hline GNNExplainer & 0.884\({}_{\pm 0.002}\) & 0.682\({}_{\pm 0.004}\) & 0.683\({}_{\pm 0.009}\) & 0.379\({}_{\pm 0.001}\) & 0.660\({}_{\pm 0.006}\) & 0.539\({}_{\pm 0.002}\) \\ + MixUp & 0.890\({}_{\pm 0.004}\) & 0.788\({}_{\pm 0.006}\) & 0.690\({}_{\pm 0.014}\) & 0.501\({}_{\pm 0.003}\) & 0.869\({}_{\pm 0.004}\) & 0.612\({}_{\pm 0.043}\) \\ (improvement) & 0.60\% & 15.5\% & 1.02\% & 32.2\% & 31.7\% & 13.5\% \\ \hline PGExplainer & 0.999\({}_{\pm 0.001}\) & 0.829\({}_{\pm 0.040}\) & 0.762\({}_{\pm 0.014}\) & 0.679\({}_{\pm 0.008}\) & 0.679\({}_{\pm 0.043}\) & 0.843\({}_{\pm 0.084}\) \\ + MixUp & 0.999\({}_{\pm 0.001}\) & 0.955\({}_{\pm 0.017}\) & 0.774\({}_{\pm 0.004}\) & 0.712\({}_{\pm 0.000}\) & 0.920\({}_{\pm 0.031}\) & 0.871\({}_{\pm 0.079}\) \\ (improvement) & 0.00\% & 15.2\% & 1.57\% & 4.86\% & 35.5\% & 3.32\% \\ \hline \hline \end{tabular}
\end{table}
Table 1. Explanation faithfulness in terms of AUC-ROC on edges under six datasets. The higher, the better. Our mixup approach achieves consistent improvements over backbone GIB-based explanation methods.
datasets. It means that the explanation from MixupExplainer aligns well with the original graph's distribution, which indicates our mixup approach's effectiveness in alleviating the distribution shifting issue.

\(\bullet\) The shifting between blue points and red points is more obvious in the MUTAG dataset, where the green points generated by MixupExplainer still align well with the red points. This shows our method with mixup works well not only in synthetic datasets but also in the real-world dataset.

**Measuring Distances.** In this section, we quantitatively assess the distribution shifting issue by measuring the distances between the distribution vector \(\mathbf{h}\) from the original graphs and the explanation subgraphs \(\mathbf{h}^{*}\) and \(\mathbf{h}^{(\text{mix})}\). We report the averaged Cosine score and the Euclidean distance between different types of representation vectors in Table 2. From the results, we can see that, on average, \(\mathbf{h}^{(\text{mix})}\) has a higher Cosine score and a smaller Euclidean distance with \(\mathbf{h}\) than \(\mathbf{h}^{*}\), indicating more similarity of distribution between \(G^{(\text{mix})}\) and \(G\) than that between \(G^{*}\) and \(G\). The smaller distances between representation vectors demonstrate that our Mixup approach can effectively alleviate the distribution shifting problem caused by the inductive bias in the prediction model \(f\). As \(G^{(\text{mix})}\) better estimates the distribution of the original graphs, MixupExplainer can consistently improve the performance of existing explainers.

**Correlation with Performance Improvements.** We quantitatively evaluate the correlation between the improvements of AUC-ROC scores of MixupExplainer over basic counterparts and the improvements in distances with our mixup approach. We calculate the improvements of AUC-ROC scores from GNNExplainer and PGExplainer over GNNExplainer and PGExplainer without mixup (denoted as \(\Delta_{\text{AUC}}^{\text{GNNExplainer}}\) and \(\Delta_{\text{AUC}}^{\text{PGExplainer}}\), respectively). The improvements of average \(\text{Cosine}(\mathbf{h},\mathbf{h}^{(\text{mix})})\) over average \(\text{Cosine}(\mathbf{h},\mathbf{h}^{*})\) is denoted by \(\Delta_{\text{Cosine}}\), and the improvements on Euclidean distance is

\begin{table}
\begin{tabular}{c|c c c c c c} \hline  & BA-Shapes & BA-Community & Tree-Circles & Tree-Grid & BA-2motifs & MUTAG \\ \hline Avg. Cosine\((\mathbf{h},\mathbf{h}^{*})\) & 0.574 & 0.483 & 0.962 & 0.629 & 0.579 & 0.775 \\ Avg. Cosine\((\mathbf{h},\mathbf{h}^{(\text{mix})})\) & 0.940\(\pm\)0.005 & 0.644\(\pm\)0.006 & 0.953\(\pm\)0.006 & 0.810\(\pm\)0.004 & 0.901\(\pm\)0.000 & 0.852\(\pm\)0.006 \\ \hline Avg. Euclidean\((\mathbf{h},\mathbf{h}^{*})\) & 1.30 & 1.31 & 0.213 & 0.921 & 1.32 & 1.07 \\ Avg. Euclidean\((\mathbf{h},\mathbf{h}^{(\text{mix})})\) & 0.440\(\pm\)0.014 & 1.10\(\pm\)0.010 & 0.211\(\pm\)0.011 & 0.582\(\pm\)0.006 & 0.587\(\pm\)0.001 & 0.816\(\pm\)0.011 \\ \hline \end{tabular}
\end{table}
Table 2. The Cosine score and Euclidean distance between the distribution vectors of the original graph \(\mathbf{h}\), explanation subgraph \(\mathbf{h}^{*}\), and our mixup graph \(\mathbf{h}^{(\text{mix})}\) on different datasets. Large Cosine scores and small Euclidean distances indicate high similarities between representations. The standard deviations of Avg. \(\text{Cosine}(\mathbf{h},\mathbf{h}^{*})\) and Avg. \(\text{Euclidean}(\mathbf{h},\mathbf{h}^{*})\) are not included because they are static without random processes.

Figure 4. Visualizations of the distribution shifting issue with t-SNE on six datasets. The points are generated with the output before the last layer of the model to be explained \(f\), which is then plotted with t-SNE. The red points mean original graphs \(G\), the blue points mean substructure explanations \(G^{*}\), and the green points mean mixup explanations \(G^{(mix)}\). Green dots align well with red dots, while blue dots shift away from red dots.
\(\Delta_{\text{Euclidean}}\). Figure 5 shows the correlation between \(\Delta_{\text{AUC}}^{\text{GNNExplainer}}\), \(\Delta_{\text{AUC}}^{\text{PExplainer}}\), \(\Delta_{\text{Cosine}}\), and \(\Delta_{\text{Euclidean}}\). We can see that all these four improvements strongly correlated to each other with statistical significance, indicating the improvements achieved by MixupExplainer in explanations accuracy own to the successful alleviation of the distribution shifting issue.

### Parameter Study (RQ3)

In this section, we investigate the hyperparameters of our approach, which include \(\lambda\) and \(\eta\), on the BA-2motifs dataset. The hyperparameter \(\lambda\) controls the weight on the original graph during the mixup process. We find the optimal value of \(\lambda\) by tuning it within the \([0,1]\) range. Note that, with \(\lambda=0\), Eq. (7) is trivial and doesn't help explain \(G_{a}\) with only \(\mathbf{A}_{b}\). The experimental results can be found in Figure 6. We can see that the best performance is achieved with \(\lambda=0.1\) and that the approach consistently outperforms the best performance from baselines with \(\lambda\in[0.05,1]\). The hyperparameter \(\eta\) is the number of cross-graph edges during mixup, indicating the connectivity between label-dependent explanations and label-independent subgraphs. We tune it within the \([1,20]\) range on the BA-2motifs dataset. The results in Figure 7 show that the best performance is achieved with \(\eta=1\). With different \(\eta\), our approach shows stable and consistently better performance than the best baseline.

## 6. Conclusion

In this work, we study the distribution shifting problem to obtain robust explanations for GNNs, which is largely neglected by the existing GIB-based post-hoc instance-level explanation framework. With a close analysis of the explanation methods of GNNs, we emphasize the possible distribution shifting issue induced by the existing framework. We propose a simple yet effective approach to address the distribution shifting issue by mixing up the explanation with a randomly sampled base graph structure. The designed algorithms can be incorporated into existing methods with no effort. Experiments validate its effectiveness, and further theoretical analysis shows that it is more effective in alleviating the distribution shifting issue in graph explanation. In the future, we will seek more robust explanations. Increased robustness indicates stronger generality and could provide better class-level interpretation at the same time.

###### Acknowledgements.

 The work was partially supported by NSF award #2153311. The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing any funding agencies.

Figure 5. Correlation between improvements of AUC-ROC scores in explanation performance and the improvements of distribution distances on different datasets. The value of \(r\) indicates the Pearson correlation coefficient, and the values with \(*\) indicate statistical significance for correlation, where \(***\) indicates the p-value for testing non-correlation \(p\leq 0.001\).

Figure 6. Hyper parameter analysis of \(\lambda\) on BA-2motifs with Mixup-PGExplainer. (a) The performance of explanation w.r.t AUC. The blue line represents the mean AUC score with standard deviations over ten runs with different random seeds on each \(\lambda\) value. The red line represents the performance of the baseline PGExplainer. (b) The distances between \(h\) and \(h^{(\text{mix})}\) for different \(\lambda\). The blue and yellow lines represent the mean of the Cosine score and Euclidean distance with standard deviations, respectively.

Figure 7. Hyperparameter analysis of \(\eta\) on BA-2motifs with Mixup-PGExplainer. (a) The performance of explanation w.r.t AUC. (b) The distances between \(h\) and \(h^{(\text{mix})}\) for different \(\eta\).