# A Comprehensive Failure Risk Assessment Method of
## 2 Comprehensive Risk Assessment of the Machining Center Component Based on Topological Analysis
### Analysis of the Topological Characteristics of the Machining Center Component  
#### 2.2.1 Analysis of Multi-Dimensional Topological Structure of Machining  
Center Components  
At present, there are certain limitations in the modeling of the failure propagation model of the machining center. For example, the failure propagation model is only considered from the perspective of the failure mechanism or only based on the function flow. It is only an abstraction of the overall function structure of the machining center system and cannot accurately represent the topological structure of the system. Therefore, on the basis of obtaining the ANP model of the machining center, this paper transforms it into a topological structure model from the component level and uses the centrality measure of network theory to analyze its topological structure [25]. 1. Degree centrality Degree is a simple and important concept used to describe the attributes of individual nodes. Degree Centrality (DC) is the most direct metric used to characterize node centrality [26]. The number of adjacent nodes of node \(v_{i}\) in the network is called the degree of node \(v_{i}\), and its calculation equation is as follows: \[D_{i}=\sum\limits_{j}^{n}a_{ij},i\
eq j,\] (4) where \(i\) is the currently concerned node \(v_{i}\). \(j\) is any node \(v_{j}\) in the network. \(n\) is the number of nodes in the network. \(a_{ij}\) is the element in the adjacency matrix \(\mathbf{D}\). When node \(v_{i}\) and node \(v_{j}\) are directly connected, \(a_{ij}=1\), otherwise \(a_{ij}=0\). Then, the degree centrality of a node is:  
\[DC_{i}=\frac{D_{i}}{n-1}. \tag{5}\]  
The node degree is to characterize the importance of a node in the local network from the network topology. When the degree of a node is larger, it means that the degree centrality of the node is higher. That is, the greater the number of neighboring nodes directly connected to the node, the more important the node, and the more important its position in the network. 2. Betweenness centrality Node betweenness centrality (\(BC\)) is used to measure the importance of a node's position in the entire network [27]. Much of the literature believes that the greater the betweenness centrality of a node, the more critical it is in the network. The calculation equation is as follows: \[BC_{i}=\frac{\sum\limits_{j=1}^{n}\sum\limits_{k=1}^{n}\frac{N(j,k)}{N(j,k)}}{ n(n-1)/2},\] (6) where \(N(j,k)\) is the number of shortest paths between component nodes \(v_{j}\) and \(v_{k}\). \(N(j,i,k)\) is the number of shortest paths passing through node \(v_{i}\) between nodes \(v_{j}\) and \(v_{k}\). 3. Closeness centrality Closeness Centrality (CC) is used to measure the ability of a node to influence other nodes through the network [28], and its measurement standard is the topological distance between nodes. The smaller the distance between a node and other nodes, the greater its closeness centrality value, which means that the node is at the center of the network model, and its importance is greater. The closeness centrality of node \(v_{i}\) is defined as the reciprocal of the sum of the shortest paths between all nodes in the topological network, namely:  
\[CC_{i}=\left[\sum_{j=1}^{N}d_{ij}\right]^{-1},\ i\
eq j, \tag{7}\]  
where \(d_{ij}\) is the shortest distance between component node \(v_{i}\) and \(v_{j}v_{i}\
eq v_{j}\). 2.2 Calculation of the Structure Comprehensive Importance of the Component Based on the Topology Analysis  
In order to coordinate the multi-dimensional structural importance index of the machining center component and identify the comprehensive importance of the topological structure of the machining center components, it is necessary to calculate the structure comprehensive importance of the component. The specific calculation steps are as follows:  
(1) Problem setting  
Suppose there are \(n\) components to be evaluated and the number of indicators to measure their structural importance is \(g\).