# Vehicle Cooperative Network Model Based on Hypergraph in Vehicular Fog Computing
## 5 Model Analysis  
The stochastic process theory and the mean field method are used to theoretically analyze the evolution law of the vehicular fog computing cooperative network to obtain the super-degree distribution of the vehicles. For ease of analysis, \(t_{i}\) is the time when the \(i\)-th batch of vehicles enters the network, \(d_{H}(i)\) is the super-degree of the \(i\)-node at time \(t\). Assuming that \(d_{H}(j)\) is a continuous real-valued variable, the average field method is used to know that the dynamic equation of vehicle \(j\) should be satisfied. \[\frac{\partial(d_{H}(j))}{\partial t}=\lambda_{1}mk_{2}f_{i}^{2} \tag{5}\]  
where \(m\) is due to the generation of m super-edges, \(k2\) is the old node in \(k2\) networks in the process of forming the super-edge, the existing vehicle \(i\) may be formed into the first super-edge, the second super-edge,... \(m\) super-side selection. When \(t\) is large enough, \(m_{0}\ll\lambda_{1}tm(k+k1)\), \(m_{0}\ll\lambda t\), we can obtain the following:  
\[\frac{\partial d_{H}}{\partial t}\approx\frac{\varphi k2(d_{H}(i)+\alpha)}{t( k+k1)}+\frac{\lambda_{1}mk2(1-\varphi)}{t} \tag{6}\]  
After simplifying all terms, we can obtain the following:  
\[d_{H}(t)=(m+\frac{(1-\varphi)m(k1+k2)}{\varphi})(\frac{t}{t_{1}})^{(\frac{ \varphi k2}{k1+k2})}-\frac{(1-\varphi)m(k1+k2)}{\varphi} \tag{7}\]  
Since the new joining process of the network node satisfies the Poisson distribution with the constant \(\lambda 1\), and the addition of the super-edge in the super-network is in accordance with the random process, the instantaneous average super-distribution distribution of the network satisfies the following equation. \[P_{H}(d_{H})\approx\frac{k1+k2}{mk2\varphi+(1-\varphi)mk2(k1+k2)}(\frac{m+\frac{(1 -\varphi)m(k1+k2)}{\varphi}}{d_{H}(t)+\frac{(1-\varphi)m(k1+k2)}{\varphi}}) \tag{8}\]  
It can be seen from the expression of super-degree distribution that the vehicle cooperative model has both the characteristics of random networks and scale-free networks. The strength of the characteristics depends on the preferred connection probability, \(\varphi\), and the random connection probability, 1-\(\varphi\), of the vehicles. When \(\varphi=0\), it means that the vehicles are all randomly connected during the network growth process, forming a random network, and the average super-degree distribution of vehicle conforms to the exponential distribution. When \(\varphi=1\), it means that the nodes are all optimally connected during the network growth process, forming a scale-free network, and the average super-degree distribution is in accordance with the power law distribution. In order to better understand the nature of the network, we performed numerical simulation analysis on the formed over-degree distribution. At \(t=1000\), \(m=3,k1=4,k2=2,\varphi=0.2,\varphi=0.4,\varphi=0.6\) and \(\varphi=0.8\) respectively, a comparison with \(\varphi=1\) super-degrees distributed in double logarithmic coordinates is shown in Figure 2.  
It can be seen from Figure 2 that the characteristics of the network presented by \(\varphi=1\) are those of the standard scale-free network. The degree distribution is a power-law distribution, and the characteristics of the network represented by \(\varphi=0\) are standard random networks, whose degree distribution is bell-shaped distribution. As the value of \(\varphi\) decreases, scale-free network characteristics begin to weaken, and random network characteristics begin to appear. As can be seen from Figure 2a, \(\varphi=0.2\), the network exhibits the characteristics of a random network, and the scale-free characteristic is weak. It can be seen from Figure 2b,c that, when \(\varphi\) is relatively close to 0.5, the characteristics of the network are not so obvious. In fact, it is a combination of exponential distribution and power law distribution. In fact, the distribution law of many real networks is difficult to fit or predict with a single  
Figure 2: The characteristics with different \(\varphi\).