## 3 Proposed Work
### Node Selection and Slice Mapping Model  
The node selection method is based on network theory [27, 28] and uses node degree and centrality values to select candidate nodes for \(NSR_{i,t}\). These parameters are chosen to select the nodes with the optimal position in the network. Accordingly, the degree of a node refers to the number of connections adjacent to other nodes in the network. Each node \(n\ \in N\) has two degrees: an out-degree, which refers to the number of outgoing edges,  
\[k_{i}^{in}=\sum\limits_{j=0}^{I}\varepsilon_{j,i} \tag{1}\]  
and an in-degree of the number of incoming edges onto \(n\),  
\[k_{i}^{out}=\sum\limits_{j=0}^{I}\varepsilon_{i,j} \tag{2}\]  
where \(\varepsilon\) is equal to 1 if node \(i\) is directly connected to node \(j\), and 0 if otherwise. The total degree of a node is then calculated as,  
\[\sum\limits_{i=0}^{I}k_{i}=k_{i}^{in}+k_{i}^{out} \tag{3}\]  
A centrality analysis is then performed for each node to determine its degree of importance to the information flow in the network. In this study, the betweenness centrality of the node is the chosen measure. We are interested in selecting candidate nodes based on how frequently they are used for network information flow. As such, the betweenness centrality of node \(i\) is estimated as,  
\[\beta(n_{i})=\sum\limits_{j=1}^{J}\frac{g_{j}(n_{i})}{g_{j}} \tag{4}\]  
where \(g_{j}(n_{i})\) is the number of all paths passing through node \(n_{i}\), and \(g_{j}\) is the sum of all the shortest paths from nodes \(j\) to \(k\). The node's viability \(\xi\) is then calculated using the values for \(\sum\limits_{i=0}^{I}k_{i}\) and \(\beta(n_{i})\) as in Equation (5),  
\[\xi=\frac{\sum\limits_{i=1}^{I}C(n_{i})\ *\ \left(\sum\limits_{i=1}^{L_{NSR}}B(n_{ i})\right)}{\sum\limits_{i=0}^{I}k_{i}+\beta(n_{i})} \tag{5}\]  
where the node resource values \(\sum\limits_{i=1}^{I}C(n_{i})\) for computing capacity, and \(\sum\limits_{i=1}^{L_{NSR}}B(n_{i})\) for the total link capacity for all links of \(n_{i}\) are considered. All \(\xi\) values are stored and arranged (in descending order) in the node viability array (NVA) to be used later for candidate node selection. Moreover, sorting from the highest value to the lowest means that nodes with higher \(\xi\) values are given priority for candidate node selection (Algorithm 1). For the mapping of paths that interconnect network nodes, conventional approaches use the shortest path methods [20, 23, 28] such as breadth-first search (BFS) and \(k\)-shortest path algorithms. This study implements the maximum flow approach utilizing the Edmonds-Karp method [29]. The network's maximum flow indicates the amount of data that can flow through the network's connections at a given time. This approach calculates the network's flow by taking the sum of available bandwidth for all edges in the network. Additionally, this process aims to find all feasible paths or augmenting paths \(\psi(n_{\alpha},n_{\Omega})\) from source node \(n_{\alpha}\) to target node \(n_{\Omega}\) for every \(NSR_{i,t}\) while considering the link capacity of every edge in the network. Such an approach reduces the chances of congestion when there is an influx of requests in the network. The nodes ranked in the NVA are used as target nodes for the link mapping process using the Edmonds-Karp method.