# Vehicle Cooperative Network Model Based on Hypergraph in Vehicular Fog Computing
## 4 Description of Vehicle Assistance Network Evolution Model Based on Hypergraph
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Definition** \\ \hline m0 & The number of cooperation vehicles in the initial network. \\ k1 & Number of newly added nodes in the process of forming super-edges. \\ K2 & Number of nodes in the network during the formation of super-edges. \\ pi & Probability of number k1 of vehicles newly joining the vehicle cooperation network \\ \(\lambda 1\) & the Poisson process of the constant. \\ pi & Probability of selecting the number of nodes in the network during the formation of a super-edge. \\ \(f_{1}^{i}\) & Meet and assist probability. \\ \(d_{H}(i)\) & Super-degree of the vehicle, i. \\ \(\alpha\) & Harmonic parameter. \\ \(f_{j}^{2}\) & Probability that vehicle i is selected for connection. \\ \(n(t)\) & Total number of nodes in the network at time, t. \\ \(\varphi\) & Percentage of connections by super-degree. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Table of notations. of edges connected in the entire network. The super-degree of the node, that is, the number of super-edges where the node is located, the point over-degree is more meaningful in the real network. Therefore, the analysis of the evolution model focuses on the node overdistribution. In general, a larger node is more important means that the status in the entire network is more important. The super-degree distribution \(P_{H}(d_{H})\) of the node is the proportion of nodes with the degree of \(d_{H}\) occupying the nodes in the whole network:  
\[P_{H}(d_{H})=\frac{N_{d_{H}}}{N} \tag{1}\]  
where \(N_{d_{H}}\) represents the number of nodes with a degree of \(d_{H}\), and \(N\) represents the total number of network nodes. Therefore, the analysis and evolution model of this paper focuses on analyzing the distribution of node super-degrees. At the same time, the hypergraph-based super-network has the characteristics of a super-edge connecting multiple nodes, which makes it well described for complex systems with multi-node connections [52]. The evolution of the super-network can also be used to analyze the topological and evolutionary rules of complex systems with multi-node structures. Therefore, this section presents a dynamic evolution model of a vehicle-based fog computing node based on hypergraph. Table 1 outlines a list of the main notations to be used throughout this paper. Assume that the initial join of the collaborative network is some cooperation nodes with a small number of m0, and the following rules are executed at regular intervals, to expand the network. The symbols and meanings used below are as follows. (1) Network initialization  
Assume that there are m0 cooperation vehicles in the initial network, and jointly complete a certain cooperation task to form a super-edge. (2) Network growth  
With the passage of time, \(k1\) new vehicles are added to the previously formed vehicle cooperative network, at regular intervals, to form m super-edges, together with \(k2\) existing vehicles in the previous period, and there are m cooperative tasks. The rules added are as follows. The \(k1\) vehicles are added together with the probability \(p_{i}\) to form the super-edge. Some existing network evolution models assume that the nodes are discretely entering the network at the same time interval to form the super-edge, but in the real network, the nodes joining the network are not equal time intervals. There are relatively many nodes added in some time periods, so the nodes are described in terms of the growth model of the network according to the Poisson process. The arrival process of the node satisfies the Poisson process of the constant, \(\lambda_{1}\). \[p_{i}=P[X=k1]=\frac{(\lambda_{1}t)^{k1}e^{-\lambda_{1}t}}{k1!}(k1=2,3,4,\cdots) \tag{2}\]  
The \(k1\) new vehicle nodes select \(k2\) (\(k2<\) m0) vehicles in the preexisting network with probability, \(p_{j}\), to form m super-edges at time, \(t\), and there is no identical super-edge, \(p_{j}\) satisfy \(\sum_{j=0}^{m}p_{j}=1\).