# Identifying Influential Nodes of Complex Networks Based on Trust-Value
## 2 Methods
### The Similarity Ratio  
In social networks, if two individuals have common hobbies, they have quite a few similarities, to some extent, which means they interact with each other more frequently and exchange more  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \(H(i)\) & The hobby set of node \(i\). \\ \(s(i,j)\) & The similarity between nodes \(i\) and \(j\). \\ \(S_{\triangleright}\) & The sum of similarity between node \(v\) and its neighbor nodes. \\ \(N_{v}\) & The adjacent nodes’ set of \(v\). \\ \(|N_{v}|\) & The number of node \(v\)’s neighbors. \\ \(R_{s(i,j)}\) & The ratio of the similarity between nodes \(i\) and \(j\) to the ratio of the sum of the similarity between node  
information with each other. Conversely, the similarity between individuals is relatively low if there is no intersection between them; accordingly, they may not interact frequently. Thus, we define the similarity ratio as:  
**Definition 1**.: _The similarity ratio of a node to an adjacent node is the value of the similarity between the two nodes divided by the sum of the similarity between the adjacent node and its neighbor nodes. The similarity ratio of node \(i\) to adjacent node \(j\) is defined in Equation (2)._  
\[R_{s(i,j)}=\frac{s(i,j)}{\sum_{k\in N_{j}}s(j,k)} \tag{2}\]  
Assume there is a network as shown in Figure 1a. The hobby sets of nodes \(v\), 1, 2, 3, and 4 are \(H(v)\), \(H(1)\), \(H(2)\), \(H(3)\), and \(H(4)\), respectively. Their values are shown in Table 2.  
If the similarity \(s(i,j)\) of nodes \(i\) and \(j\) is defined as  
\[s(i,j)=|H(i)\cap H(j)|, \tag{3}\]  
where \(s(1,v)=2\), \(s(2,v)=1\), \(s(3,v)=0\), and \(s(4,v)=1\), and the ratios of the similarity of nodes 1, 2, 3, and 4 to the summation of node \(v\) are \(R_{s(1,v)}=0.5\), \(R_{s(2,v)}=0.25\), \(R_{s(3,v)}=0\), and \(R_{s(4,v)}=0.25\), respectively.  
If each hobby set of one node is treated as a set of neighbor nodes, as shown in Figure 1b, the similarity between the nodes is determined by their common neighbor nodes. The more similar the neighbor nodes of two nodes, the more similar these two nodes. In addition, the purpose of Equation (3) is only to illustrate the idea that the similarity of nodes depends on the similarity of their neighbor nodes. Here, we use the SimRank [28] algorithm for calculating similarity.  
In SimRank, two objects are similar if they are related to similar objects. Just like a social network, if two individuals have more common or similar neighbors, they will be more similar to each other. In SimRank, the similarity between \(a\) and \(b\) is defined as  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Hobby Set** & **Value** \\ \hline \(H(v)\) & \{basketball, pingpong, run\} \\ \(H(1)\) & \{basketball, run\} \\ \(H(2)\) & \{pingpong\} \\ \(H(3)\) & \{tennis\} \\ \(H(4)\) & \{run\} \\ \hline \hline \end{tabular}
\end{table}
Table 2: The hobby sets of nodes \(v\), 1, 2, 3, and 4.  
Figure 1: Hobby sets and the network of nodes’ similarity. (**a**) A network of nodes with hobby sets. (**b**) Each hobby set of one node is treated as a set of neighbor nodes.  
\[s(a,b)=\left\{\begin{array}{cc}1&a=b\\ \frac{C}{|N_{a}||N_{b}|}\sum_{i=1}^{|N_{a}|}\sum_{j=1}^{|N_{b}|}s(N_{a}(i),N_{b}( j))&a\
eq b\end{array}\right. \tag{4}\]  
where \(C\) is a constant that represents an attenuation factor. The value of \(C\) does not affect the results, since only the ratio of similarity is calculated in the experiments. In this paper, \(C\) is taken as 1. \(N_{a}(i)\) represents the \(i\)th neighbor node of node \(a\). If \(N_{a}\) or \(N_{b}\) is an empty set, then \(|N_{a}||N_{b}|=0\). To prevent the case where the divisor is 0 in Equation (4), \(s(a,b)=0\) is defined. As we can see from Equation (4), how to calculate \(s(a,b)\) in SimRank is an iterative process. Thus, before starting the iteration, we initialize \(s(a,b)\) to 1 if \(a=b\) while \(s(a,b)\) to 0.1 if \(a\
eq b\).