## 3 Network Models and Problem Description  
### Network Models  
Given a network topological graph \(G=(V,E,W)\), where V represents the set of network nodes, E represents the set of edges between nodes, and W represents the weights on the edges. Network topology models can be applied to simulate various complex real-world systems, such as social networks, logistics networks, power systems, and information networks. Assuming that node \(d\) is the destination node and node \(s\) is the source node in the topology, the Dijkstra algorithm, also known as the shortest path algorithm, \(sp(s,d)\) is used in this paper to calculate the path, denoted by the symbol shortest path, abbreviated as \(sp\). According to the shortest path algorithm, it represents the optimal path with the minimum cost from node \(s\) to node \(d\), where the path \(sp(s,d)\) is a sequence \(s=s_{1},s_{2}\ldots\ldots s_{m}=d\), and there exists an edge \((s_{i},s_{i+1})\in E\) between any two nodes \(s_{i}\) and \(s_{i+1}\). \(sp^{\prime}(s,d)\) represents the backup path from node \(s\) to node \(d\), and where the path \(sp^{\prime}(s,d)\) is a sequence \(s=d_{1},d_{2}\ldots\ldots\ldots d_{n}=d\) and there exists an edge \((d_{i},d_{i+1})\in E\) between any two nodes \(d_{i}\) and \(d_{i+1}\), then the following definition applies:  
**Definition 1** (Independent Paths). : _In paths \(sp(s,d)\) and \(sp^{\prime}(s,d)\), except source node \(s\) and destination node \(d\), \(sp(s,d)\) and \(sp^{\prime}(s,d)\) do not share any node with each other, \(sp(s,d)\) and \(sp^{\prime}(s,d)\) occur at the same time, and \(sp(s,d)\) and \(sp^{\prime}(s,d)\) do not have any shared edges, in this case, \(sp(s,d)\) and \(sp^{\prime}(s,d)\) are said to be two independent paths._  
_In a topology \(G\), if any two paths from node \(s\) to node \(d\) do not satisfy the definition of independent paths, we define these two paths as intersecting, meaning there is an overlap between them. The cross-degree is defined as follows._  
**Definition 2** (Cross-Degree). : _Given a topological graph \(G=(V,E,W)\), with source node \(s\) and destination node \(d\), there are two paths \(sp(s,d)\) and \(sp^{\prime}(s,d)\). The set of edges shared by these two paths is defined as the crossing number, which can be expressed as follows:_  
\[K(sp(s,d),sp^{\prime}(s,d))=\{sp(s,d)\cap sp^{\prime}(s,d)\} \tag{1}\]  
_In this Equation (1), \(sp(s,d)\cap sp^{\prime}(s,d)\) denotes the intersection of \(sp(s,d)\) and \(sp^{\prime}(s,d)\), while \(|K(sp(s,d),sp^{\prime}(s,d))|\) represents the value of the cross-degree, which is defined as the number of common edges in the two paths. This is expressed in Equation (2) as follows:_  
\[|K(sp(s,d),sp^{\prime}(s,d))|=\left\{ \begin{array}{cc}\sum\limits_{i=1}^{i= w}(v_{i},v_{i+1})\in sp(s,d)\cap sp^{\prime}(s,d)&w=m-1\\ 0&sp(s,d)\cap sp^{\prime}(s,d)=\emptyset\end{array} \right. \tag{2}\]  
_In Equation (2), \(m-1\) denotes the penultimate node number in the optimal path \(sp(s,d)\). That is, iterating over all the edges on the best path to find the edges that are common to the backup path \(sp^{\prime}(s,d)\) yields the number of common edges, which represents the value of the crossover degree. The following example illustrates this definition._  
Figure 1 shows five nodes and five edges in the topology graph \(G=(V,E,W)\), and the number on each edge is the cost of the edge. Assuming that node \(s\) is the source node and node \(d\) is the destination node, calculating the cost on the path from node s to node d yields two paths. A sequence of paths \(sp(s,d)=\{s,b,e,d\}\) with a path cost of 16, and a sequence of paths \(sp^{\prime}(s,d)=\{s,a,e,d\}\) with a path cost of 19. The two paths are compared, and since 16 is less than 19, the best path is \(sp(s,d)\), and the backup path is \(sp^{\prime}(s,d)\). The cross-degree  
of these two paths is \(K(sp(s,d),sp^{\prime}(s,d))=\{(e,d)\}\), \(|K(sp(s,d),sp^{\prime}(s,d))|=1\), i.e., the cross-degree is \(1\).