SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Corollary 1. Assume that  $\delta(h) = c\gamma^h$  for some constants  $c > 0$  and  $\gamma < 1$ , if  $d > 0$ , then the loss decreases polynomially fast:
$$
r _ {n} \leq (\frac {C}{1 - \gamma^ {d}}) ^ {1 / d} n ^ {- 1 / d}
$$
If  $d = 0$ , then the loss decreases exponentially fast:
$$
r _ {n} \leq c \gamma^ {n / C - 1}
$$
Proof. From Theorem 1, whenever  $d > 0$  we have
$$
n \leq C \sum_ {l = 0} ^ {h (n)} \delta (l) ^ {- d} = C c ^ {- d} \frac {\gamma^ {- d (h (n) + 1) - 1}}{\gamma^ {- d} - 1} \tag {1}
$$
thus  $\gamma^{-dh(n)}\geq \frac{n}{Cc^{-d}} (1 - \gamma^d)$ , therefore we can also have:
$$
r _ {n} \leq \delta (h (n)) \leq c \gamma^ {h (n)} \leq \left(\frac {C}{1 - \gamma^ {d}}\right) ^ {1 / d} n ^ {- 1 / d} \tag {2}
$$
Now, if  $d = 0$  then  $n \leq C\sum_{l=0}^{h(n)} \delta(l)^{-d} = C(h(n) + 1)$ , and we deduce that the loss is bounded as  $r_n \leq \delta(h(n)) = c\gamma^{n/C-1}$ .
B.4 Analysis of Time Complexity
Given a graph  $G = (V, E)$ , we take a single iteration as an example to analyze the time complexity of the expansion-based MCTS in our proposed SAME. A single iteration includes four steps: selection, expansion, simulation and backpropagation [1].
B.4.1 Important Substructure Initialization Phase
In this phase, the goal of our SAME is to figure out a group of important substructures whose size is smaller than  $\gamma$ . In the selection step, our method chooses an unvisited node from the graph or chooses the node with the highest reward if all nodes have been visited ( $\mathcal{O}(|V|)$ ). Then, in the expansion step, our method selects the node within 1-hop neighbors whose value is largest
according to  $G$ , which requires  $\mathcal{O}(1)$ , and adds it to the tree. In the simulation step, only the adjacency node will be appended each time. For the graph  $G$ , in the worst case SAME will append all the nodes by visiting all the edges in the graph. Thus, the time complexity of simulation is  $\mathcal{O}(|V| + |E|)$ . For the backpropagation step, since the maximum size of a substructure is  $\gamma$ , the depth of the MCTS tree will not exceed  $\gamma$ . The time complexity of reward backpropagation is bounded by  $\mathcal{O}(\gamma)$ . Since we perform  $M_1$  iterations, according to the law of multiplication, the time complexity of the first phase is  $\mathcal{O}(M_1) \times \mathcal{O}(\text{selection}) \times \mathcal{O}(\text{expansion}) \times \mathcal{O}(\text{simulation}) \times \mathcal{O}(\text{backpropagation}) = \mathcal{O}(M_1\gamma |V|^2 + M_1\gamma |V| \times |E|)$ .
B.4.2 Explanation exploration phase
In the selection step, the MCTS is initialized with an unvisited substructure from Important Substructure Set (size  $= K$ , where  $K \ll |V|$ ) or chooses the substructure with the highest reward if all substructures have been visited ( $\mathcal{O}(K)$ ). The expansion step is  $\mathcal{O}(1)$  as we only need to append the substructure with the largest value to the current state. In the following simulation step, our method will append other substructures to the current state until the size exceeds the sparsity limit, which is bounded by  $\mathcal{O}(2^K)$  for finding all possible combination. Notice that  $\mathcal{O}(2^K)$  can be very large, we set a maximum simulation time  $t_s$  as budget which requires  $\mathcal{O}(t_s)$ . Since the final explanation is highly related to the sparsity, the cost of backpropagation step is bounded by  $\mathcal{O}(\frac{|V| \times (1 - \text{spsarsity})}{\gamma})$ , where  $\gamma$  is the maximum size of each substructure. With  $M_2$  iterations, the time complexity in this phase is  $\mathcal{O}(M_2) \times \mathcal{O}(\text{selection}) \times \mathcal{O}(\text{expansion}) \times \mathcal{O}(\text{simulation}) \times \mathcal{O}(\text{backpropagation}) = \mathcal{O}(M_2 K t_s \frac{|V| \times (1 - \text{spsarsity})}{\gamma})$ .
Overall, time complexity of SAME is  $\mathcal{O}(M_1\gamma |V|^2 + M_1\gamma |V| \times |E| + M_2Kt_s\frac{|V| \times (1 - sparsity)}{\gamma})$ . As the  $t_s$ ,  $\gamma$ ,  $K$ ,  $M_1$ ,  $M_2$  and sparsity is predefined, therefore SAME is a polynomial-time method under these constraints.
C Detailed Pseudo Code of SAME
In this section, we are going to provide a detailed overview of the processes of SAME which includes important substructure initialization and explanation exploration.
Algorithm 1 outlines the process of using Monte Carlo Tree Search (MCTS) to find a set of important substructures, as mentioned in Section 3.1 of the main manuscript. Algorithm 2 and 3 illustrate the method to discover an optimal explanation from these important substructures, as described in Section 3.2. Algorithm 4 provides a detailed description of how the importance of an explanation is assessed using the structure-aware Shapley value, which was proposed in Section 2.1 of your document. This algorithm outlines the steps necessary to approximate the Shapley value for a given explanation, effectively measuring the contribution of each individual component via sampling.
D Dataset Description
We provide details of the datasets used in our experiments, including BBBP [11], MUTAG [2], BA-2Motifs [8], BA-Shapes [12], Graph-SST2 [7], and Graph-SST5 [7].
Molecular graphs. We use the BBBP [11] containing approximately 2000 molecular graphs, which are classified into two classes over the property of blood-brain barrier penetration (BBBP). Another dataset MUTAG [2] is a collection of molecules with  $-NO_{2}$  functional groups. The goal is to predict whether these molecules are mutation-induced.
Barabási-Albert graphs. The BA-2Motifs [8] and the BA-Shapes [12] are used for graph classification and node classification respectively. For each instance in BA-2Motifs, it is a Barabási-Albert graph attached by motifs with a structure either house-like or five-node-cycle-like. The instances are labelled according to the type of motifs they get. The graph in BA-Shapes is a Barabási-Albert graph with house-structured network motifs. The nodes in the graph will be classified into four classes, with labels 0 for the nodes belonging to the original graph and labels 1-3 for the nodes on the middle, bottom, or top of the house-like structures respectively.
Sentiment graphs. The sentiment graphs are built from real-world text sequences, and labelled according to the semantic meanings. Specifically, the nodes in the graph represent the words with an
Algorithm 1: Important Substructure Initialization
Input:  $f(\cdot)$  well-trained GNN model;  $G$  graph to be explained;  $M_{1}$  max iteration number of MCTS;  $\gamma_{1}$  threshold of max single substructure size;  $\pi$  child selection strategy Output:  $\mathcal{N}$  all nodes in MCTS 1 Initialize the root of the MCTS as  $N_0$    
2  $G_{sub}^{0}\gets \emptyset$    
3  $\mathcal{N}\gets \{N_0\}$    
4 for  $i = 1,2,\ldots ,M_1$  do   
5  $N_{cur}\gets N_0$    
6 curPath  $\leftarrow [N_0]$    
7  $G_{sub}^{cur}\gets G_{sub}^{0}$    
8 while  $|G_{sub}^{cur}| <   N_{max}$  do   
9 forall  $v_{j}$  in the  $\{\mathrm{adj}(G_{sub}^{cur})\}$  do   
10 Expand  $N_{i}$  to get  $N_{j}$    
11  $G_{sub}^{j}\gets G_{sub}^{i}\cup \{v_{j}\}$  // Compute the Shapley-based reward according to Algorithm 4   
12  $R_{j}\gets I(f(\cdot),G_{sub}^{j},G)$    
13  $\mathcal{N}\gets \mathcal{N}\cup \{N_j\}$    
14 end   
15 Select the child  $N_{next}$  and its substructure  $G_{sub}^{next}$  according to the strategy  $\pi$    
16  $N_{cur}\gets N_{next}$    
17  $G_{sub}^{cur}\gets G_{sub}^{next}$    
18 curPath  $\leftarrow$  curPath  $+N_{next}$    
19  $\mathcal{N}\gets \mathcal{N}\cup \{N_{cur}\}$    
20 end   
21  $N_l\gets N_{cur} / / N_l$  is a leaf node   
22forall node  $N_{path_i}$  in the curPath do   
23  $T_{path_i}\gets T_{path_i} + 1$    
24  $W_{path_i}\gets W_{path_i} + I(f(\cdot),G_{sub}^l,G)$    
25 end   
26 end
Algorithm 2: Explanation Exploration
Input:  $f(\cdot)$  well-trained GNN model;  $G$  graph to be explained;  $\mathcal{N}$  set of all the nodes in MCTS;  $K$  number of the top most important structures;  $K_{t}$  threshold to use MCTS;  $M_2$  max iteration number of MCTS;  $\gamma_{2}$  max explanation size;  $\pi$  , child selection strategy. Output:  $G_{ex}$  , the best explanation. 1 Sort the  $\mathcal{N}$  in descending order of the corresponding reward  $R_{i}$    
2  $\mathcal{N}_K\gets$  top  $K$  substructures  $\{N_1,N_2,\dots ,N_K\}$  in  $\mathcal{N}$    
3  $\mathcal{G}_{ex}\gets \{(G_{sub_i},R_i)\}_{i = 1}^{M_2}$  // Find explanations through MCTS in Algorithm 3.   
4  $\mathcal{G}_{ex}^{\prime}\gets \mathrm{MCTS}(f(\cdot),G,M_2,N_{max},\pi ,\mathcal{N}_K)$    
5  $\mathcal{G}_{ex}\gets \mathcal{G}_{ex}\cup \mathcal{G}_{ex}^{\prime}$    
6 Sort the  $\mathcal{G}_{ex}$  in descending order of  $R$  . //  $G_{ex}$  is the explanation at the first element in the sorted  $\mathcal{G}_{ex}$    
7  $G_{ex}\gets \mathcal{G}_{ex}[0]$
Algorithm 3: MCTS Explanation Exploration
Input:  $f(\cdot)$  well-trained GNN model;  $G$  graph to be explained;  $M_2$  max iteration number of MCTS;  $\gamma_{2}$  , threshold of max explanation size;  $\pi$  , child selection strategy,  $\mathcal{N}_K$  , the substructure set. Output:  $\mathcal{G}_{ex}$  , the explanation set found in MCTS 1 Initialize the root of the MCTS as  $N_0^{\prime}$    
2  $G_{ex}^{0}\gets \emptyset ,\mathcal{G}_{ex}\gets \{G_{ex}^{0}\}$    
3 for  $i = 1,2,\ldots ,M_2$  do   
4  $N_{cur}^{\prime}\gets N_{0}^{\prime}$  curPath  $\leftarrow [N_0^{\prime}]$ $G_{ex}^{cur}\gets G_{ex}^{0}$    
5 while  $|G_{ex}^{cur}| <   \gamma_{2}$  do   
6 forall  $N_{j}$  in the  $\{\mathcal{N}_K\backslash N_{cur}^\prime \}$  do // Expand  $N_{cur}^{\prime}$  with  $N_{j}$  to get  $N_j^\prime$  ..  $N_j^\prime \gets N_{cur}^\prime \cup N_j$ $G_{ex}^{j}\gets G_{ex}^{i}\cup G_{subj}$  // Compute reward according to Algorithm 4   
9  $R_j^{\prime}\gets I(f(\cdot),G_{ex}^{j},G)$    
10 end   
11 Select the child  $N_{next}^{\prime}$  and its substructure  $G_{ex}^{next}$  according to the strategy  $\pi$    
12  $N_{cur}^{\prime}\gets N_{next}^{\prime}$ $G_{ex}^{cur}\gets G_{ex}^{next}$    
13 curPath  $\leftarrow$  curPath  $+N_{next}^{\prime}$    
14 end   
15  $N_l^{\prime}\gets N_{cur}^{\prime} / / N_l^{\prime}$  is a leaf node   
16  $\mathcal{G}_{ex}\gets \mathcal{G}_{ex}\cup \{(G_{ex}^{cur},R_{cur}^{\prime})\}$    
17 forall node  $N_{path_i}$  in the curPath do   
18  $T_{path_i}^{\prime}\gets T_{path_i}^{\prime} + 1$    
19  $W_{path_i}^{\prime}\gets W_{path_i}^{\prime} + I(f(\cdot),G_{ex}^{l},G)$    
20 end   
21 end
Algorithm 4: Importance Scoring via Shapley Value
Input:  $f(\cdot)$  , well-trained GNN model;  $G$  graph data;  $G_{ex}^{i}$  , explanation of  $G$  ;  $T$  sampling times;  $k$  , number of neighboring hop Output:  $\mathcal{I}_i$  , importance of  $G_{ex}^{i}$  for GNN  $f(\cdot)$    
1Obtain the  $k$  -hop neighbor nodes of substructure  $G_{ex}^{i}$  , denoted as  $\{v_{n_i + 1},v_{n_i + 2},\ldots ,v_{n_i + k_i}\}$    
2 Then, the players set  $P_{i,khop} = \{G_{ex}^{i},v_{n_i + 1},v_{n_i + 2},\dots,v_{n_i + k_i}\}$    
3 for  $i = 1,2,\ldots ,T$  do   
4 Randomly sample a set  $S_{i}\subseteq \{P\backslash G_{ex}^{i}\}$    
5Obtain  $f(S_{i}\cup \{G_{ex}^{i}\})$  and  $f(S_{i})$  by setting the features of the nodes not in the input structure with zero. //  $m(S_i,G_\epsilon^i)$  denotes the marginalized contribution   
6  $m(S_i,G_{ex}^i)\gets f(S_i\cup \{G_{ex}^i\}) - f(S_i).$    
7 end   
8  $I(f(\cdot),G_{ex}^{i},G)\gets \frac{1}{T}\sum_{t = 1}^{T}m(S_i,G_{ex}^{i})$    
9  $\mathcal{I}_i\gets I(f(\cdot),G_{ex}^i,G)$
initial embedding from the pre-trained BERT [3] and the edges denote the relationships between the words which are extracted by the Biaffine parser [5]. We take experiments on the Graph-SST2 and Graph-SST5 datasets for the sentiment graphs. Note that both Graph-SST2 and Graph-SST5 are the datasets for graph classification of two classes and five classes respectively.
E Metrics
The Fidelity, Inverse Fidelity, and Sparsity are formally defined as:
$$
\operatorname {F i d e l i t y} (G, G _ {e x}) = [ f (G) ] _ {c ^ {*}} - [ f (G / G _ {e x}) ] _ {c ^ {*}} \tag {3}
$$
$$
\operatorname {F i d e l i t y} _ {\operatorname {I n v}} (G, G _ {e x}) = [ f (G) ] _ {c ^ {*}} - [ f (G _ {e x}) ] \tag {4}
$$
$$
\operatorname {S p a r s i t y} \left(G, G _ {e x}\right) = 1 - \frac {\left| G _ {e x} \right|}{\left| G \right|} \tag {5}
$$
Harmonic Fidelity (H-Fidelity) takes a harmonic mean of normalized fidelity (N-Fidelity) and normalized inverse fidelity  $(\mathrm{N - Fidelity}_{\mathrm{Inv}})$ . According to [15], the H-Fidelity can be formally defined as:
$$
\text {N - F i d e l i t y} (G, G _ {e x}): m _ {1} = \text {F i d e l i t y} (G, G _ {e x}) \cdot \text {S p a r s i t y} (G, G _ {e x}) \tag {6}
$$
$$
\mathrm {N} - \text {F i d e l i t y} _ {\text {I n v}} (G, G _ {e x}): m _ {2} = \text {F i d e l i t y} _ {\text {I n v}} (G, G _ {e x}) \cdot (1 - \text {S p a r s i t y} (G, G _ {e x})) \tag {7}
$$
$$
\operatorname {H} - \operatorname {F i d e l i t y} (G, G _ {e x}) = \frac {(1 + m _ {1}) (1 - m _ {2})}{(2 + m _ {1} - m _ {2})} \tag {8}
$$
F Detailed Experimental Settings
In this section, we provide the computational details and detailed hyperparameter settings on different datasets.
F.1 Computation Details
All experiments were performed on a single Nvidia V100 GPU with an Intel Xeon Gold 5218 CPU. Since it takes more than 24 hours for some baselines to explain the whole dataset, our inference time analysis is implemented by randomly sampling from the datasets. For BBBP, BA-2Motifs and BA-Shapes, we use all the test data. And all the data (train, evaluation and test data) is used for MUTAG dataset. For semantic graphs like Graph-SST2 and Graph-SST5, we randomly choose 30 graphs.
F.2 Hyperparameter Settings
We adhere to the hyperparameter settings for GNN training as described in [13, 15], detailed in Table S2. For the qualitative analysis, the GCN in Figures 2, 4, S2, and S3 aligns with the settings depicted in Table S2. Moreover, the GIN featured in Figure 3 is trained for 800 epochs with a batch size of 64 and without adding self-loops, while all other hyperparameters remain constant to Table S2.
Table S2: Hyperparameters for training the GCN models on different datasets. All quantitative results are verified under the following conditions.
Hyperparameter	BBBP	MUTAG	Graph-SST2	Graph-SST5	BA-2Motifs	BA-Shapes
# Layers	3	3	3	3	3	3
Hidden dimensions	[128, 128, 128]	[128, 128, 128]	[128, 128, 128]	[128, 128, 128]	[20, 20, 20]	[20, 20, 20]
Dropout	0.0	0.0	0.0	0.0	0.0	0.0
Readout method	max	mean	max	max	mean	identity
Learning rate	0.001	0.005	0.001	0.001	0.001	0.05
Batch size	32	32	128	128	64	1
# Epochs	200	1000	50	50	800	400
Weight decay	5 × 10-4	0	0	0	0	5 × 10-4
Add self-loop	Yes	Yes	Yes	Yes	Yes	Yes
Table S3: The quality of the explanations from SAME with different hyperparameters.
γ\K	1	3	5	7	9
1	0.2511	0.3319	0.3548	0.3938	0.4026
2	0.2851	0.3121	0.3418	0.3652	0.3786
3	0.2938	0.3180	0.3297	0.3637	0.3741
4	0.3134	0.3355	0.3491	0.3610	0.3540
5	0.3134	0.3387	0.3504	0.3591	0.3598
Table S4: The average inference time (seconds) per sample under different hyperparameters.
γ\K	1	3	5	7	9
1	0.0990	0.3300	1.289	5.178	15.95
2	0.0889	0.3129	1.137	3.817	11.88
3	0.0874	0.3986	0.8344	2.746	7.047
4	0.0932	0.2846	0.8483	2.296	5.330
5	0.0914	0.2822	0.8427	2.191	5.796
The hyperparameters in SAME include  $\gamma$  and  $K$  as we didn't use a time budget in our MCTS. In order to see the effect of these two hyperparameters, we run our SAME by randomly selecting 30 graphs from Graph-SST5 datasets using GCN, the fidelity w.r.t.  $\gamma$  and  $K$  are shown in the Table S3, and we report the average inference time in Table S4. Notice that since we only selected 30 graphs from Graph-SST5 to obtain the results, the results provided in Table S3 might be slightly different from the results given in the main manuscript.
It is noteworthy that  $\gamma = 1$  could lead to optimal fidelity in most benchmarks. However, when  $\gamma$  is small, the final explanation may be composed of disconnected nodes at different positions in the graph, which usually causes the visualization of explanation results not human-understandable. Therefore, we fine-tuned  $\gamma$  and  $K$  according to the visualization of explanation results so that the provided explanations are more human-intuitive. For the inference time, a larger  $K$  leads to a larger search space for the explanation exploration phase. When  $\gamma$  is small, the size of a single candidate substructure is small, indicating that we need to select more substructures from the candidates to obtain an explanation with the desired sparsity. Thus, the inference time increases as the depth of searching grows. The detailed hyperparameter settings are provided in Table S5. For the sentiment network Twitter [13] and molecular network BACE [11], we use them to compare SAME with more SOTA explainers in Section G.3
Table S5: The hyperparameters of SAME for different datasets.
Hyperparameter	BBBP	MUTAG	Graph-SST2	Graph-SST5	BA-2Motifs	BA-Shapes	Twitter	BACE
γ	5	2	3	3	3	5	3	2
K	7	7	7	7	7	7	7	7
G.1 Results under Other Metrics
Here, we present the results over inverse fidelity and harmonic fidelity under the same experimental settings in Table S2. Table S6 demonstrates that SAME is competitive over the 5/6 datasets. Table S7 reports the harmonic fidelity quantities. SAME outperforms the baselines in all the graph tasks.
G.2 Additional Visualization Results
We are presenting additional visualization results for BBBP and Graph-SST2 datasets. Figure S2 presents explanations for the BBBP dataset. SAME identifies critical functional groups (for instance, carbonyl group  $= \mathrm{C} = \mathrm{O}$ ), which result in a high fidelity score. Figure S3 provides a visualization for the negative label of the Graph-SST2 dataset. In this case, SAME uniquely identifies the word "only," which contributes to the negative prediction of the GNN due to the contrasting relationship it establishes.
Table S6: Comparison of our SAME and other baseline explainers using 1 - Inverse Fidelity.
Methods\Dataset	Graph tasks	Node tasks
Molecular graph	Semantic graph	Synthetic graph
BBBP	MUTAG	Graph-SST2	Graph-SST5	BA-2Motifs	BA-Shapes
Grad-CAM [9]	0.897±0.029	0.901±0.048	1.006±0.029	0.937±0.047	0.587±0.092	-
GNNExplainer [12]	0.829±0.012	0.810±0.007	0.855±0.037	0.860±0.034	0.562±0.015	0.953±0.032
PGExplainer [8]	0.838±0.036	0.793±0.030	0.956±0.036	0.850±0.058	0.582±0.021	0.902±0.008
GNN-LRP [10]	0.670±0.044	0.768±0.013	0.784±0.047	0.771±0.041	0.560±0.015	0.988±0.010
SubgraphX [14]	1.010±0.011	1.053±0.017	1.008±0.008	1.022±0.040	0.906±0.162	0.978±0.034
GStarX [15]	0.939±0.011	1.124±0.025	1.072±0.003	1.140±0.031	0.587±0.117	-
SAME	0.993±0.014	1.095±0.008	1.058±0.010	1.039±0.001	0.857±0.046	0.881±0.000
Note: The previous SOTA results on different datasets are marked with an underline. Relative Improve denotes the relative improvement of our SAME method over the SOTA methods.
Table S7: Comparison of our SAME and other baseline explainers using Harmonic Fidelity.
Methods\Dataset	Graph tasks	Node tasks
Molecular graph	Semantic graph	Synthetic graph
BBBP	MUTAG	Graph-SST2	Graph-SST5	BA-2Motifs	BA-Shapes
Grad-CAM [9]	0.520±0.002	0.527±0.002	0.538±0.005	0.529±0.003	0.502±0.014	-
GNNExplainer [12]	0.504±0.002	0.505±0.007	0.507±0.001	0.512±0.001	0.487±0.027	0.514±0.001
PGExplainer [8]	0.511±0.002	0.500±0.006	0.528±0.003	0.516±0.000	0.493±0.023	0.504±0.003
GNN-LRP [10]	0.488±0.003	0.501±0.013	0.495±0.001	0.495±0.002	0.497±0.027	0.518±0.002
SubgraphX [14]	0.560±0.002	0.552±0.002	0.528±0.004	0.541±0.003	0.548±0.007	0.528±0.003
GStarX [15]	0.510±0.002	0.599±0.011	0.533±0.005	0.542±0.005	0.506±0.016	-
SAME	0.571±0.001	0.604±0.022	0.540±0.021	0.566±0.000	0.554±0.007	0.517±0.000
Relative Improve	1.96%↑	0.83%↑	1.31%↑	4.43%↑	1.09%↑	-
Note: The previous SOTA results on different datasets are marked with an underline. Relative Improve denotes the relative improvement of our SAME method over the SOTA methods.
Figure S2: Comparison of the explanations on BBBP with GCN classifier.
Sentence: "shocking only in that it reveals the filmmaker's bottomless pit of self - absorption."
Figure S3: Comparison of the explanations on Graph-SST2 with GCN classifier for a negative prediction.
G.3 Additional Quantitative Analysis
Here we report more quantitative analysis with alternative SOTA methods and datasets. We tested SAME on GAT and GIN, and compared SAME with other SOTA algorithms (GraphSVX [4] and OrphicX [6]). In table S8, We report the averaged fidelity over a sparsity of  $[0.5, 0.6, 0.7, 0.8]$ . For the inference time plus fidelity, they are summarized in Table S9. SAME not only surpasses the benchmarks' performance by alternative GNN architectures, but it also exhibits superior efficiency in elucidating explanations.
Table S8: Comparison of SAME and other baseline explainers with different GNN architecture.
Dataset Methods	GAT	GIN	GCN
Graph-SST2	Graph-SST5	Graph-SST2	Graph-SST5	Twitter	BACE
GraphSVX [4]	-0.0136	-0.0108	0.0839	0.161	0.0691	0.2061
OrphicX [6]	0.071	0.104	0.258	0.176	0.1701	0.3277
SubgraphX [14]	0.060	0.060	0.222	0.254	0.2903	0.3906
GStarX [15]	0.134	0.114	0.274	0.263	0.2501	0.3903
SAME	0.142	0.222	0.400	0.264	0.3127	0.7154
Note: The previous SOTA results on different datasets are marked with an underline.
Table S9: Comparison of SAME and other benchmarks in both fidelity and inference time (per sample). The architecture of our model is GCN.
Methods
Metric	GraphSVX [4]	OrphicX [6]	SubgraphX [14]	GStarX [15]	SAME
Fidelity	Twitter	0.0691	0.1701	0.2903	0.2501	0.3127
BACE	0.2061	0.3277	0.3906	0.3903	0.7154
Time	Twitter	0.0885	0.4942	197.1	36.26	25.85
BACE	0.0915	0.2361	44.41	48.81	26.98
Note: The previous SOTA results on different datasets are marked with an underline.