## 3 System Model Design
### Fuzzy Pattern Recognition Process for Node Classification in Opportunistic Mobile Social Networks
#### 3.2.2 Fuzzy Pattern Recognition for Node Classification in Opportunistic Mobile Social Networks
For the sake of a more precise decision-making on data transmission, it is essential to execute the process of weight adjustment and fuzzy relationship inference. By synthetically evaluating the control result of pattern recognition and fuzzy relation inference, message application users are able to determine the next hop, even the third hop, from their neighbor nodes efficiently. The procedure of weight adjustment plays a significant role in fuzzy relation evaluation, which aims to reasonably assess the status of each fuzzy input in the whole data transmission. The social attributes of nodes are layered and interlaced, which is specifically reflected in the fact that the social attributes of message application users are related to each other. Therefore, we adopt the analytic hierarchy process (AHP) [30] to evaluate the weight of their social attributes in routing selection and data forwarding. Firstly, we need to determine the comparison weight between \(n\) different fuzzy inputs and construct a \(n\)-order comparison matrix, which is shown as Table 1. \begin{table}
\begin{tabular}{c|c c c c} \hline \hline \(\mathbf{W}\) & \(|ei(\mathbf{X_{1}})|\) & \(|ei(\mathbf{X_{2}})|\) & \(\cdots\) & \(|ei(\mathbf{X_{n}})|\) \\ \hline \(|ei(X_{1})|\) & \(1\) & \(Ra_{(1,2)}\) & \(\cdots\) & \(Ra_{(1,n)}\) \\ \(|ei(X_{2})|\) & \(1/Ra_{(1,2)}\) & \(1\) & \(\cdots\) & \(Ra_{(2,n)}\) \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\ddots\) & \(\vdots\) \\ \(|ei(X_{n})|\) & \(1/Ra_{(1,n)}\) & \(1/Ra_{(2,n)}\) & \(\cdots\) & \(1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Weight scale for \(n\) different fuzzy inputs. AHP directly and effectively combines the results of human experience and objective judgment of machines, and quantitatively describes the importance of pairwise comparison of elements in a hierarchy. Then, the weight value reflecting the relative importance order of elements at each level is calculated by mathematical method, and the relative weight of all elements is calculated and sorted by the total order between all levels [30]. The adjustment of weight parameters in our experiment is based on the characteristics of the real data set. Consequently, from Table 1, the weight ratio \(Ra(i,j)\) is the ratio of all the elements in the first row to all the elements in the first column, which could be set as  
\[Ra_{(i,j)}=\frac{|ei(X_{i})|}{|ei(X_{j})|}. \tag{13}\]  
Then, we normalize each of the columns in the above matrix \(CM\), which can be shown as  
\[Rb_{(i,j)}=\frac{Ra_{(i,j)}}{\sum_{i=1}^{n}Ra_{(i,j)}} \tag{14}\]  
where \(\sum Ra(i,j)\) denotes the sum of the \(Ra(i,j)\) for each column and \(Rb(i,j)\) is the new comparison weight. After that, the eigenvector \(w_{i}\) of the \(i\)-th fuzzy input is summed from each row of the new matrix, which can be formalized as  
\[B_{(i)}=\sum_{j=1}^{n}Rb_{(i,j)} \tag{15}\]  
\[w_{i}=\frac{B_{(i)}}{\sum_{i=1}^{n}B_{(i)}}. \tag{16}\]  
To reasonably record the weight value of each social attribute of mobile nodes in the networks, the proposed algorithm loads the weight value corresponding to each fuzzy input into a row vector. Then, the weight vector \(W_{(}w_{1},w_{2}\cdots w_{n})\) for \(n\) different fuzzy inputs could be defined as  
\[W_{(}w_{1},w_{2}\cdots w_{n})=[w_{1},w_{2},w_{3}\cdots w_{n}]. \tag{17}\]  
Based on the above formulas, \(n\) weight values for \(n\) different fuzzy inputs (social attributes of nodes) can be obtained from \(w_{i}\). After determining the importance of each social attribute of message application users on the process of message routing and forwarding, the system comes to assign each fuzzy input an appropriate weight automatically, which is in favor of the final comprehensive evaluation of the fuzzy relationship between fuzzy input factor sets and fuzzy subsets. Additionally, the fuzzy relationship matrix \(R_{f}\) could be induced by \(f\), which can be represented as  
\[R_{f}=\left[