# Graph Dilated Network with Rejection Mechanism
## 5 Graph Dilated Network with Rejection Mechanism
### Graph Dilated Convolution Kernel  
To have a sufficient size of neighborhoods in message propagation, we explore the idea of dilated convolution kernel [32] used in Computer Version to enlarge the reception field. Then, we propose a graph dilated convolution kernel, which changes the standard and single message propagation scheme, i.e., one layer for searching 1-hop neighbors, and brings the diversity of propagation schemes.  
Specifically, we introduce a graph dilation rate \(\gamma\), which refers to the distance between the node and its neighbor nodes. Figure 4 illustrates the examples of graph dilated convolution kernels with \(\gamma\in\{1,2,3\}\). Actually, the standard graph convolution kernels used in GNNs can be taken as a special case of our graph dilated convolution kernel where \(\gamma=1\). Obviously, a larger value of \(\gamma\) allows the model to explore nodes from a larger neighborhood. For example, a two-layer stacked graph dilated convolution (\(\gamma=3\)) network can view neighbors within six hops, whereas for a standard two-layer stacked graph convolution network, only neighbors in two hops are considered. Therefore, we can enlarge the neighborhood size via graph dilated convolution kernels.  
As presented above, different graph dilated convolution kernels take nodes in different hops as the neighborhoods. To have a diverse and sufficient size of neighborhoods, multiple types of graph  
Figure 4: An example for graph dilated convolution kernel with different values of \(\gamma\). The red node refers to the center node, and the corresponding neighborhood of the center is constructed by the blue nodes. Note that when \(\gamma=2\), only the neighbors in exact 2-hops are considered.  
dilated convolution kernels are simultaneously applied to the center node. Then, we further propose Multi Graph Dilated Convolution Kernels (MGDCK). At each \(k\)-th layer, multiple types of graph dilated convolution kernels are adopted. Each node \(v\) may aggregate the messages not only from \(1\)-hop neighbors (i.e., \(N(v)\)), but also from \(2\)-hop neighbors (i.e.,\(N_{2}(v)\)). Formally, when each layer contains \(T\) types of graph dilated convolution kernels, the _AGGREGATE_ and _COMBINE_ functions can be expressed as  
\[m_{v}^{(k,t)}=AGGREGATE\Big{(}\big{\{}h_{u}^{(k-1,t)}:u\in N_{t}( v)\big{\}}\Big{)} \tag{10}\] \[h_{v}^{(k,t)}=COMBINE\big{(}m_{v}^{(k,t)},h_{v}^{(k-1,t)}\big{)} \tag{11}\]  
where \(t\in\{1,2,3,\ldots,T\}\), _COMBINE_ refers to the combination function with RM mentioned in Section 5.1, and the implementation of _AGGREGATE_ can be various. In this paper, we adopt a mean aggregation strategy. Other aggregation functions can also be applied. Then, the representation of node \(v\) at the \(k\)-th layer is the mean of all the representation vectors produced by \(T\) graph dilated convolution kernels  
\[h_{v}^{(k)}=mean\Big{(}\big{\{}h_{v}^{(k,t)}:t\in\{1,2,3,\ldots, T\}\big{\}}\Big{)}. \tag{12}\]  
Note that other pooling strategies such as max-pool, min-pool, and attention-pool can also be considered as the summarization operators on all the representations from these \(T\) graph dilated convolution kernels. Here, we only take the mean-pool as an example.  
Although directly stacking layers can also enlarge the size of neighborhoods, too many layers may introduce the difficulty of the model training. The experimental results also show the effectiveness of graph dilated kernels compared with the directly stacking strategy (see Section 6.7).