# Vehicle Cooperative Network Model Based on Hypergraph in Vehicular Fog Computing
## 4 Description of Vehicle Assistance Network Evolution Model Based on Hypergraph  
Among the many complex networks, cooperative networks are popular among researchers, due to their universality in the real world, such as corporate cooperation networks, industry-university-research cooperation networks, actor cooperation networks, scientific research cooperation networks, etc. [45]. A lot of research has been done on network model construction and feature analysis. But most of them are based on the topology of ordinary graphs. The nodes in the graph represent different individuals in the system, and the relationships between individuals are represented by edges. One edge of a normal graph can only connect two nodes and cannot describe many features of the real network. Some cooperative networks are described by bipartite graphs. However, the two types of nodes in the bipartite graphs are heterogeneous and cannot effectively analyze the network's connectivity, degree distribution and other topological features. With the continuous expansion of the network scale, the types of nodes are diversified, and the relationship between nodes is more complicated. Network system problems that surpass ordinary networks have arisen. Therefore, scholars have tried to apply the principles of super-networks to complex networks. There are two main types of super-networks: hypergraph-based super-networks and network-based super-networks [46]. The superstructure of a hypergraph-based network is a hypergraph, which pays more attention to the dynamic evolution process, so it can be used as a tool for studying the dynamic analysis of complex networks. Hypergraph theory can not only guarantee the homogeneity of points and edges, but also clearly express the relationship between nodes and nodes, and nodes and edges. The super-network studied in this paper belongs to the hypergraph based on hypergraph [47]. The super-edge in the hypergraph can connect two or more nodes [48]. Any network that can be represented by a hypergraph is a super-network, the definition of a hypergraph:  
Let \(V=\{v_{1},v_{2},v_{3},v_{4},\ldots,v_{n}\}\) (\(n\) is a positive integer) be a finite set. (1) \(e_{i}\
eq\phi(i=1,2,3,\ldots,m)\)  
(2) \(U_{i=1}^{m}e_{i}=E\)  
The binary relationship H = (\(E\),\(V\)) is called a hypergraph [49]. The elements \(v1,v2,v3,v4,\ldots,vm\) of \(V\) are called vertices of the hypergraph, \(E=\{e_{1},e_{2},e_{3},e_{4},\ldots,e_{m}\}\) are the edge sets of the hypergraph, and the set \(e_{i}=\{e_{1},e_{2},e_{3},e_{4},\ldots,e_{m}\}\) (\(i=1,2,3...m\)) are super-edges of the hypergraph. A vertex belongs to two or more super-edges, which is called super-edge adjacency. The vertex, \(v_{k}\), belongs to the super-edge number, which is called the super-degree of the vertex, \(v_{k}\), and is represented by \(d_{H}(v_{k})\). If the two vertices belong to the same super-edge, then the two vertices are said to be adjacent. If the intersection of the two super-edges is not empty, the two super-edges are said to be adjacent [50]. Common representations are graphical and matrix forms, including the matrix, adjacency matrix and Laplacian matrix [51]. In the super-network, the nodes in the super-edge adopt the full connection mode, and the node degree is the same as the node degree definition of the conventional network, that is, the number