# SINDVis: User-Centered Dynamic Interactive Visualization System for Space Information Networks
## 3 Theory and Methods
### TopoView Methods
#### 3.4.1 FDA Layout
For the phenomenon of the increment and deletion of edges in dynamic problems, we designed fade-in and fade-out functions and made the network edges increase or disappear in an orderly manner to allow the layout to be smoothly and dynamically changed. This follows the same principle as the FR algorithm, and assumes that the edges only affect attraction and do not affect the repulsion. \(M\) represents the number of iterations, and \(l\) represents an iterative index. The expressions of the fade-in and fade-out functions are  
\[f_{in}(d)=\frac{d^{2}(l-1)}{k(M-1)},1\leq l\leq M \tag{1}\]  
\[f_{out}(d)=\frac{d^{2}(M-l)}{k(M-1)},1\leq l\leq M \tag{2}\]  
where, for the fade-in function \(f_{in}(d)\), the value of attraction is 0 when there is no connection, and the value of attraction is \(f_{a}(d)\) in the last iteration, and for the fade-out function \(f_{out}(d)\), the value of attraction is \(f_{a}(d)\) at the initial moment, and the value of attraction is 0 in the last iteration. For the phenomenon of the increment and deletion of nodes in dynamic problems, because attraction and repulsion exist in each pair of nodes, the importance of nodes was considered in the improved algorithm. The attraction and repulsion between nodes depend not only on their distance, but also on the importance of nodes. When the new node \(v_{i}\) joins, there is a connection with its neighboring node \(v_{j}\), and the attraction and repellant between the two nodes is affected by the node's importance. Therefore, we introduced the clustering coefficients of complex networks [20] to calculate the importance of nodes, and the clustering coefficient was calculated as follows:  
\[C_{i}=\frac{2E_{i}}{k_{i}(k_{i}-1)} \tag{3}\]  
where, \(k_{i}\) indicates that there are \(k_{i}\) edges connected to node \(i\), and \(E_{i}\) represents the number of edges between \(i\) and \(k_{i}\). Meanwhile, when \(k_{i}=0\) or \(k_{i}=1\), then \(C_{i}=0\), and in order to avoid this situation, we improved \(C_{i}\) by adding the number 1. The expression is  
\[NC_{i}=\frac{2E_{i}}{k_{i}(k_{i}-1)}+1. \tag{4}\]  
\(IM_{ij}\) denotes the importance of nodes \(v_{i}\) and \(v_{j}\), and the calculation formula is  
\[IM_{ij}=\sqrt{NC_{i}+NC_{j}}. \tag{5}\]  
After the introduction of \(IM_{ij}\), the attraction and repulsion between the nodes will change with the clustering coefficients, which is well in line with the real network situation. The greater the \(IM_{ij}\) is between nodes \(v_{i}\) and \(v_{j}\), the greater the attraction between them is and the smaller the repulsion is, or vice versa. Therefore, the attraction between the nodes should be proportional to the \(IM_{ij}\), and the repellant should be inversely related to the \(IM_{ij}\). Therefore, to improve the Formula (1), there is  
\[\left\{ \begin{array}{c}f_{a_{in}}(d)=IM_{ij}\cdot d^{2}/k\\ f_{r_{in}}(d)=-\frac{1}{IM_{ij}}\cdot k^{2}/d\end{array} \right. \tag{6}\]  
where, \(f_{a_{in}}\) and \(f_{r_{in}}\) respectively represent attraction and repulsion, considering the nodes' levels of importance. When node \(v_{i}\) is deleted, the attraction and repulsion between the nodes connected to it disappear, which is relatively simple and has not been explored much. According to the dynamic evolution rule of SIN, we proposed an improved FDA layout method based on the FR algorithm. Similar to other improved methods of the FDA layout, we think that this improved FDA layout method also belongs to the FDA family. Therefore, we categorized this section as having an FDA layout. In the implementation algorithm of the improved FDA layout method, we separated the node and the edge, and this part did not have a specific display. This section, combined with the characteristics of the dynamic change of SIN, we have described the design of the improved FDA layout based on the FR algorithm and on the foundation for the SIN dynamic visualization. Next, we describe the visualization method.