# A Graph Convolutional Network-Based Method for Congested Link Identification
## 3 The GCN Approach
### The Network Structure of a GCN  
The graph we constructed is inherently heterogeneous, meaning it comprises nodes and edges of diverse types and attributes. To address the complexities of such graphs, we drew inspiration from relational graph convolutional networks (RGCNs), introduced in [34]. The RGCN extends traditional graph convolutional networks (GCNs) to efficiently handle large-scale multi-relational graphs by adapting GCN principles specifically for realistic knowledge bases. It inherits the GCN's convolutional architecture for graphs while incorporating mechanisms to model the complex interactions between different types of nodes and relations. Motivated by these strengths of the RGCN, we reformulated the congested link identification problem as a (semi-)supervised classification task on the nodes within our heterogeneous graph.  
Relational graph convolutional networks (RGCNs) have the following advantages when dealing with network node classification problems. The RGCN excels at capturing the relationship between nodes in the network, whereas traditional graph neural networks cannot. By considering the relationships between nodes, the RGCN understands the role and importance of each node, thereby improving the accuracy of node classification. It is a scalable model which can handle networks of any size, enabling it to achieve good performance on large-scale network data and networks of different sizes. The RGCN is also highly robust to noise and outliers in the network, maintaining high accuracy even in the presence of such disturbances. Furthermore, it has high computational efficiency, processing large-scale network data in a short time. In short, the RGCN effectively tackles the network node classification problem by leveraging the relationship between nodes and its scalability, robustness, and computing efficiency.  
The following propagation model is defined to compute the forward pass update of the entity or node indicated by \(\
u\) in the previously constructed graph:  
\[h_{i}^{(l+1)}=\sigma\left(\sum_{r\in R}\sum_{j\in N_{i}^{r}}\frac{1}{c_{i,r}}W _{r}^{(l)}h_{j}^{(l)}+W_{0}^{(l)}h_{i}^{(l)}\right) \tag{13}\]  
where \(N_{i}^{r}\) denotes the set of neighbor indices of node \(i\) under the relation \(r\in R\) and \(c_{i,r}\) is a normalization constant. In this paper, we set \(c_{i,r}=\frac{1}{|N_{i}^{r}|}\) for convenience. Figure 8 shows the calculation chart of single-node updating in a GCN model.  
Figure 7: Transforming the task of congested link identification into graph node classification.  
In this paper, we stack the GCN layers as shown in Equation (13) and activate the final layer via softmax activation (\(\bullet\)). The cross-entropy loss is as shown in Equation (14). Then, the task of the training phase is to minimize the cross-entropy loss:  
\[L=-\sum_{i\in Y}\sum_{k=1}^{K}t_{ik}\ln h_{ik}^{(L)} \tag{14}\]  
The cross-entropy loss function is particularly suited for multi-class classification problems, as it combines the log-softmax operation and the negative log-likelihood loss in a single step. Specifically, the cross-entropy loss computes the softmax activation over the logits (raw output of the network) and then calculates the cross-entropy loss between the predicted probabilities and the target labels. By minimizing this loss function during training, we aimed to optimize the model's parameters such that it could accurately classify the input data into the correct classes. The optimization of the model is carried out using an optimization function (e.g., stochastic gradient descent or its variants) which iteratively updates the model's weights and biases in the direction which minimizes the loss.  
In Equation (14), \(Y\) is the set of indices of the labeled nodes, \(h_{ik}^{(L)}\) is the \(k\)th entry of the network output for the \(i\)th labeled node, and \(t_{ik}\) is the ground truth label. In this paper, we train the model using (full-batch) gradient descent techniques. Figure 9 shows the schematic depiction of the congested link identification model.