## 4 Non-Cooperative Game of Customers
### Iterative Distributed Scheme Based on Gossip Algorithm
These weights are determined by the Laplacian matrix \(\mathbf{L}\), which is defined as:  
\[\mathbf{L}=\mathbf{D}-\mathbf{A} \tag{25}\]  
\(\lambda_{i}(\cdot)\) denotes \(i\)th largest eigenvalue of a matrix, and defines a constant \(\alpha\):  
\[\alpha=\frac{2}{\lambda_{1}(\mathbf{L})+\lambda_{I-1}(\mathbf{L})} \tag{26}\]  
The weights for customer \(i\in\mathcal{I}\) can be expressed as:  
\[\omega_{ii}=1-\alpha d_{ii}\omega_{ij}\left\{ \begin{array}{ll}\alpha&j\in \mathcal{N}_{i}\\ 0&j\
otin\mathcal{N}_{i}\end{array} \right. \tag{27}\]  
Thus, the iterative step is as follow:  
1. Initial \(k_{\text{ave}.i}^{(r-0)}=k_{i}^{opt(m)}\), and calculate weights;Exchange information with others and update the estimating average value of \(\hat{R}_{i}(t)\) according to (24). The converge condition is:  
\[\left|k_{\mathrm{ave}.i}^{(r+1)}-k_{\mathrm{ave}.i}^{(r)}\right|<\zeta \tag{28}\]  
where \(\zeta\) is a small positive constant. However, there still exists a problem. Note that the Laplacian matrix \(\mathbf{L}\) is hard to obtain not only for customers but also for LSE. In addition, different from [49; 50], the network in our scheme is not physically connected but is socially connected, and customers only know the connecting information regarding themselves. That is, there is no one who can gather and calculate an adjacent matrix \(\mathbf{A}\) and degree matrix \(\mathbf{D}\). To solve this problem, this paper improves the approach to calculating the heights to be adjusted to our DR scheme. In practice, the number of familiar customers that will exchange information with a certain customer is roughly fixed, or we can regulate a certain constant for each customer in the network to exchange information with, that is, the degree for everyone can be a constant \(d^{*}\) in our social network. We randomly create 2000 graphs according to the following rules:  
* Each graph has a certain degree \(d^{*}\) for every node, and \(d^{*}=\{1,2,3,\cdots,20\}\);
* The total nodes number \(I\) range from 100 to 1000, i.e., \(I=\{100,200,\cdots,1000\}\);
* In a graph with a certain degree and total nodes number, randomly change \(\mathcal{E}\) 10 times, namely, choose the link ways randomly for 10 times. Then, we calculate the value of \(\alpha\) of each graph. The results are illustrated as Figure 2. It is easy to find that \(\alpha\) mainly depends on the degree value \(d^{*}\) but not \(I\) and \(\mathcal{E}\). Thus, we can use the least-square method to fit a curve based on these samples. The fitted curve is calculated as (29) and is shown in Figure 2. It is fitted well with \(R^{2}=0.9892\). \[\left\{ \begin{array}{ll}\alpha(d^{*})=h_{3}+h_{1}e^{(-h_{2}d^{*})}\\ h_{1}=1.62\quad h_{2}=0.6018\quad h_{3}=0.0817\end{array} \right. \tag{29}\]  
As previously described, the degree number of every customer can be regulated as a constant. As a result, \(\alpha\) can be estimated easily using (29) based only on the degree number but not detailed global information of adjacent matrix \(\mathbf{A}\), which ensures the distributedly integrative process. The process for each customer to estimate \(\hat{R}_{i}\) can be characterized by the Algorithm 2. Figure 2: Least square method fitted curve and actual value sample.