# A Graph Convolutional Network-Based Method for Congested Link Identification
## 3 The GCN Approach  
### Bayesian Network Model  
According to the previous description, the congestion probability of a measured route of an end-to-end path is determined by the congestion probability of the links it traverses through, and the congestion probability of a link is determined by both the channel factors and the node factors. In this paper, we construct a BNM as shown in Figure 4 to describe the interaction between different factors in the congested link identification problem. When constructing the BNM for congested link identification, consider the graph \(\mathbf{DG}=(\mathbf{V},\mathbf{E})\), where the set \(\mathbf{V}=\{v_{i}\}\) represents three distinct sets of variables. Each set of variables in \(\mathbf{V}\) can take on two possible states: 1, representing congested, and 0, representing non-congested. These sets are as follows:  
* \(\mathbf{Y}=\{y_{1},\ldots,y_{m}\}\) is the the set of end-to-end measured route states (representing the status of the measured routes in the network), where \(y_{i}\) represents the status of the \(i\)th route in the routing table (i.e., \(y_{1}\) is the status of \(r_{1}\), \(y_{2}\) is the status of \(r_{2}\), and so on). * \(\mathbf{X}=\{x_{1},\ldots,x_{n}\}\) is the set of link states in the network, where \(x_{i}\) represents the status of the \(i\)th link in the network (i.e., \(x_{1}\) is the status of link \(l_{1}\), \(x_{2}\) is the status of link \(l_{2}\), and so on). * \(\mathbf{N}=\{n_{1},\ldots,n_{l}\}\) is the set of node states in the network, where \(n_{i}\) represents the status of the \(i\)th node in the set of network nodes (i.e., \(n_{1}\) is the status of network node \(s\), \(n_{2}\) is the status of network node \(a_{1}\), \(n_{3}\) is the status of network node \(a_{2}\), and so on). \(\mathbf{E}=\{e_{ij}\}\) is the set of edges connecting nodes \(v_{i}\) and \(v_{j}\), representing the causal influence among the variables in \(\mathbf{V}\). The graph \(\mathbf{DG}\) captures the first-order causal dependency between nodes (i.e., either \(v_{i}\) influences \(v_{j}\) or \(v_{j}\) influences \(v_{i}\)). In the Bayesian network constructed in this paper, the state of a route is related to the states of the links it traverses, and the state of a link is related to the state of the node it flows into. If congestion is detected on route \(r_{i}(i=1,2,\ldots,m)\) with \((y_{i}=1)\), then the problem of inferring congestion on each link traversed by the congested route \(r_{i}\) can be transformed into a problem of selecting the most likely set of values for the hidden variables \(\mathbf{X}\), given the observed variables \(\mathbf{Y}\) in the BNM. The joint probability of the nodes in the BNM is described as follows:  
\[P(\mathbf{X})=\prod_{j=1}^{m}P(x_{j}|p_{a}(x_{j})) \tag{8}\]  
Figure 4: BNM of the network. where \(p_{a}(x_{j})\) is the parent node of node \(x_{j}\) in the BNM. Based on the detection results \(y_{i}\) for each route \(r_{i}\), we can utilize the BNM to infer the most likely set of links experiencing congestion. This can be solved using the \(argmax\) function:  
\[\arg\max_{\mathbf{X}\subseteq\mathbf{L}}P(\mathbf{X}|\mathbf{Y}) \tag{9}\]  
where \(\mathbf{Y}\) is the set of measured routes. We define an \(m\)-dimensional indicator vector \(\mathbf{X}\) such that \(x_{i}=1\) when the link \(l_{i}\in\mathbf{X}\) is congested and, conversely, \(x_{i}=0\) when the link \(l_{i}\) is not congested. According to [PERSON] theorem:  
\[\arg\max_{\mathbf{X}\in\mathbf{L}}P(\mathbf{X}|\mathbf{Y})=\arg\max_{\mathbf{ X}\in\mathbf{L}}\frac{P(\mathbf{Y}|\mathbf{X})P(\mathbf{X})}{P(\mathbf{Y})} \tag{10}\]  
Since \(P(\mathbf{Y})\) is only related to the network status and detection results and is independent of the selection of links--that is, \(P(\mathbf{Y})\) is independent of \(\mathbf{X}\)--it can be regarded as a constant.