## 3 Proposed Methodology
### Justification of Betweenness Centrality Based Training
Thus, we can say that node \(a\) necessarily has more neighbor nodes of path length \(p\) compared to node \(b\). This is because \(a\) participates in more shortest paths compared to \(b\), and from this it can be deduced that:  
\[\textit{node coverage}(A^{\prime\prime})>\textit{node coverage}(A^{\prime})\]  
Let us consider an example to verify this. Figure 4 illustrates the computational graphs corresponding to a node for the graph defined in the first subfigure of Figure 4. Suppose that for this graph \(G\), the size of the training set is 2. Consider the two subsets \(A^{\prime}\) (random selection) and \(A^{\prime\prime}\) (proposed selection) as:  
\[A^{\prime}=\{E,G\} \tag{12}\]Thus, extending the computational graphs of the selected training nodes in \(A^{\prime}\) to the neighbors up to path length 2 (fixed size \(z\) path length), the node coverage of \(A^{\prime}\):  
\[A^{\prime}=\left\{A,B,C,E,G\right\} \tag{14}\]  
On the other hand, node coverage for \(A^{\prime\prime}\) using the training nodes \(B\) and \(C\) is:  
\[A^{\prime\prime}=\left\{A,B,C,D,E,F,G\right\} \tag{15}\]  
This small illustration shows that \(A^{\prime\prime}\) has a larger node coverage compared to \(A^{\prime}\). Moreover, each node \(a_{i}\in A^{\prime\prime}\) is a feature matrix row associated with it as an \(A^{\prime\prime}\subset V\), and for each node \(v_{i}\in V\), there is a feature vector \(x_{j}\in X\) such that \(1<i<n\), and \(1<j<m\), \(\forall\)\(i,m>1\). Each node of A\({}^{\prime\prime}\) delivers its feature vector to as many nodes as it can reach within the defined constrained path length \(z\). This implies a higher neighborhood contribution compared to the nodes of the subset \(A^{\prime}\). In this way, we finally have a greater availability of feature vectors and labeling information for training the GAT model. From this, we can deduce that the training efficiency of the model (GAT /GCN) is higher when the selection of the training nodes is based on the betweenness centrality model, and therefore, the assignment function (class prediction) obtained for each node in \(V\) will eventually be close to the optimal assignment function \(L\). Next, we will try to show that probability of selection of this subset of nodes (\(A^{\prime\prime}\)) randomly is near equal to zero. Definition 1: The probability that the GAT model (random selection of training nodes by default) selects the same subset of nodes approaches zero. In GAT, the selection of training nodes is conducted randomly from the set of available nodes. In our proposed method, on the other hand, these training nodes are selected based on their betweenness centrality scores. Thus, the subset \(A^{\prime\prime}\) contains the best \(w\) central nodes with the highest betweenness centrality such that \(w<n\), where '\(n\)' is the total number of nodes in the network. This is due to the fact that in GCN we have very few nodes that can be used for training, since no information (features and labels) is available. The number of ways to select a subset of length \(w\) (subset of \(w\) nodes) is given as \({}^{n}C_{w}\). Our goal is to find the probability of choosing the subset \(A^{\prime\prime}\) from these \({}^{n}C_{w}\) subsets. Thus, let us consider an event \(Q\) as: _Selection of the subset \(A^{\prime\prime}\) of the set \(V\)_, where \(V\) is the set of all vertices of the graph and \(|V|=n\). The probability of this event will be:  
\[P(Q)=1/^{n}C_{w} \tag{16}\]  
Figure 4: Node selection: a1, a2 (traditional random selection) vs b1, b2 (proposed selection). Assuming that 60% of the data is used for the training, we then obtain \(w=3n/5\). Thus:  
\[\textit{betweenness centrality score}(a)>\textit{betweenness centrality score}(b)\]0  
If we solve the problem, we obtain \(P(Q)\approx 0\).