# A Link Prediction Method Based on Neural Networks
## 3 Problem Description and Formula
### Reliability Indexes
#### 3.2.2 Global Network Structure Reliability (P)  
[PERSON] and [PERSON] [48] first proposed the generalized exponential distribution of reliability index and applied it to the machine, components and other equipment [49]. Its form is:  
\[P=e^{-\lambda_{t}} \tag{10}\]  
here, \(\lambda\) is the failure rate. The failure rate of the systems is difficult to obtain, so the formula evolves into:  
\[P=1-e^{-\lambda C} \tag{11}\]  
where \(\lambda\) is the reliability factor and \(C\) is the equivalent system cost of destructions after achieving the object.  
In researches, many complex systems are regarded as networks. Therefore, we consider changing some expression forms to realize the calculation of global network structure reliability to comprehensively characterize the optimization degree in the process of network evolution. Generally, the parameter \(\lambda\) is related to the knowledge level, the computing resources, human resources, time and other measurable factors [50]. Similarly, the critical network structural characteristic indexes include average shortest path length, betweenness centrality structural entropy and average clustering coefficient [51; 52]. Therefore, we quantify global network structure reliability within a weighted summation method to describe the parameter \(\lambda\). One of the most straightforward indexes to measure a network is the degree. We consider that the average degree (\(K=\frac{\sum_{x=1}^{N}k_{x}}{k_{x}}\)) has the same effect as \(C\) in calculating the global network structure reliability, so the global network structure reliability can be expressed in the following form:  
\[P=1-e^{-(\varepsilon_{1}L+\varepsilon_{2}E_{B}+\varepsilon_{3}C)\times K} \tag{12}\]  
where \(L\) is the average shortest path length. \(E_{B}\) is the betweenness centrality structural entropy. \(C\) is the average clustering coefficient. \(\varepsilon_{1}\), \(\varepsilon_{2}\), \(\varepsilon_{3}\) are the weights of parameters, and \(\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}=1\). Here, we assume that \(\varepsilon_{1}=\varepsilon_{2}=\varepsilon_{3}=1/3\) in a general scale-free network. The following is the specific description and formula explanation for these indexes.  
A. Average Shortest Path Length (\(L\)).  
The average shortest path length is an important feature measure in network analysis, which is the average distance of all node pairs. For an undirected network, it is defined as follows:  
\[L= \frac{2\sum_{x<y}d_{xy}}{N(N-1)} \tag{13}\]  
The smaller the \(L\), the greater the connection intensity between nodes. In order to solve the difference between \(L\) and other indexes, let \(L=1/L\).  
B. Betweenness Centrality Structural Entropy (\(E_{B}\)).  
Node betweenness centrality characterizes the importance of the nodes. Network structure entropy is introduced to characterize the dispersion of the nodes' importance [51]. Let \(d_{yk}\) denote the total number of the shortest path starting from the nodes \(y\) to \(k\); \(d_{yk}(x)\) is the number of the shortest path through the node \(x\) from node \(y\) to \(k\). If it goes through the node \(x\), then \(d_{yk}(x)=1\), otherwise \(d_{yk}(x)=0\). The betweenness centrality of node \(x\) is \(B_{x}\):  
\[B_{x}=\frac{\sum_{y\
eq x\
eq k}\frac{d_{yk}(x)}{d_{yk}}}{(N-1)(N-2)} \tag{14}\]  
The betweenness centrality structural entropy is expressed as:  
\[E_{B}= -\sum_{x=1}^{N}B_{x}lnB_{x} \tag{15}\]  
C. Average Clustering Coefficient (\(C\)).  
The clustering coefficient is a characteristic index to describe the tightness of the network [52]. A node with a larger clustering coefficient is closer to its neighbor nodes in the network. The clustering coefficient of node \(x\) (\(C_{x}\)) is:  
\[C_{x}=\frac{2E_{x}}{k_{x}(k_{x}-1)} \tag{16}\]  
Therefore, the network's average clustering coefficient is:  
\[C= \frac{1}{N}\sum_{x=1}^{N}C_{x} \tag{17}\]  
here, \(E_{x}\) is the number of edges connected to neighbor nodes of node \(x\).