# Graph Dilated Network with Rejection Mechanism
## 6 Experiments
### Evaluation of Rejection Mechanism  
Actually, the proposed rejection mechanism (RM) is a general method to address the over-smoothing problem. Therefore, we apply the RM to other GNN models to evaluate whether RM can help these GNNs to prevent the over-smoothing problem (i.e., answering the question **Q2**). Additionally, we also provide the results of GraphDRej to show the effectiveness of RM. Furthermore, to avoid the influence of multiple dilated convolution kernels, for GraphDRej, we apply a single dilated convolution kernel with \(\gamma=1\) (i.e., a GCN-like kernel). The results on Cora in terms of classification accuracy are presented in Figure 5 (similar performance trends are also observed on other data sets). Some observations can be summarized as follows. **Evaluation with the standard GNNs. ** First, we evaluate the RM with the standard GNNs, which suffer the over-smoothing problem. As shown in Figure 5a-d, when we gradually increase the number of layers, most of the original models fail in the over-smoothing problem. Specifically, when it comes to a deeper model (e.g., the layer of GNNs is 9 or 10 in Figure 5), the performance of most of these models sharply decreases, which also supports the discussion mentioned in Section 4. When we apply RM to these failed models, all of these models in a deep version obviously overcome the over-smoothing problem and achieve high performances. Thus, it indicates the rejection mechanism is indeed beneficial to solving the over-smoothing problem. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method\(\backslash\)Data Sets** & **Cora** & **Citeseer** & **Pumbed** \\ \hline GraphSAGE & 76.20 & 71.00 & 84.54 \\ GCN & 79.98 & 71.90 & 80.70 \\ GAT & 76.66 & 70.85 & 84.62 \\ JK & 74.82 & 71.53 & 84.75 \\ PN & 75.46 & 71.83 & 83.64 \\ MADReg & 76.38 & 71.07 & 84.56 \\ GG-NN & 73.99 & 70.85 & 84.62 \\ GraphDRej & **81.00 (\(\gamma=[1,2]\))** & **72.58 (\(\gamma=[1,2]\))** & **85.39 (\(\gamma=[1,2]\))** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Node classification accuracy (%) for two-layer graph neural networks (GNNs). The list in parentheses next to the accuracy of GraphDRej refers to the value set of \(\gamma\) adopted by the corresponding GraphDRej on different data sets. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method\(\backslash\)Data Sets** & **Cora** & **Citeseer** & **Pumbed** \\ \hline GraphSAGE & 80.54 (6) & 71.45 (4) & 84.54 (2) \\ GCN & 80.26 (4) & 71.90 (2) & 80.70 (2) \\ GAT & 80.54 (6) & 70.85 (2) & 84.62 (2) \\ JK & 81.83 (9) & 71.53 (2) & 85.32 (4) \\ PN & 79.34 (6) & 71.83 (2) & 83.67 (2) \\ MADReg & 81.37 (6) & 71.30 (4) & 84.56 (2) \\ GG-NN & 77.40 (4) & 70.85 (2) & 84.85 (4) \\ GraphDRej & **83.39 (3, [1,2])** & **72.58 (2, [1,2])** & **85.47 (2, [1,2,3])** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The best performance for GNNs on the node classification task. For baselines, the number in parentheses next to the accuracy indicates the best performing number of layers. For GraphDRej, the first number in parentheses has the same meaning as that in baselines, and the second list in parentheses refers to the value set of \(\gamma\) adopted by the corresponding GraphDRej. **Evaluation with the GNNs, which also address the over-smoothing problem. ** Then, we evaluate the RM with the GNNs (i.e., JK [21], PN [29], and MADReg [30]), which also address the over-smoothing problem. (1) As seen in Figure 5e-f, both JK and PN can well solve the over-smoothing problem, as the performances do not drop sharply with the increasing of the layer number. (2) As shown in Figure 5g, it seems that the over-smoothing problem still exists in MADReg. Similar conclusions can be found in the original paper of MADReg [30]. Specifically, from Table 5 in [30], we can find that although MADReg can improve the performance of the standard GNNs, the performance of a GNN with the MADReg in a deep version is far away from the performance of that in a shallow version. (3) Then, we apply RM on these GNNs (including JK, PN, and MADReg).