# SINDVis: User-Centered Dynamic Interactive Visualization System for Space Information Networks
## 3 Theory and Methods
### TopoView Methods
#### 3.4.2 FAT Visualization
The multi-centroid constraint diagram is shown in Figure 9.  
Figure 8: The local-world and edge-weight evolution mechanism of the space information network (SIN): (**a**) node increase mechanism; (**b**) node deletion mechanism; (**c**) edge increase mechanism; (**d**) edge deletion mechanism; (**e**) edge-weight evolution mechanism (adopted from [PERSON] et al. [16]). Figure 9: Multi-centroid constraint diagram. The centroid constraint is implemented based on FDA and the effect of clustering and clustering is formed under the action of nodal centroid attraction [22]. Therefore, the centroid constraint concretely implements the content same strain with FDA, which also provides the possibility for practice in this paper. Considering the local-world has \(C_{r}=\{v_{i},\ldots,v_{k}\}\), and the location of its arbitrary node \(v_{t}\) is \((x_{t},y_{t})\), the position of the centroid \(v_{r}\) inside the local-world \(C_{r}\) is \(v_{r}=(x_{r},y_{r})\), and there is  
\[\left\{ \begin{array}{l}x_{r}=\frac{1}{k}\sum_{i=1}^{k}x_{t}\\ y_{r}=\frac{1}{k}\sum_{i=1}^{k}y_{t}\end{array} \right.. \tag{7}\]  
Assuming that the constrained attraction of the node \(v_{t}\) is \(f(v_{r})\), and the direction of the current node points to the centroid, there is  
\[f(v_{r})=d\times strength(C_{r}) \tag{8}\]  
where, \(d\) is the distance from the node \(v_{t}\) to the centroid \(v_{t}\) of the local-world, and \(strength(C_{r})\) is the attraction strength. Therefore, there is  
\[d=||v_{t}-v_{r}||=\sqrt{dx^{2}-dy^{2}}=\sqrt{(x_{t}-x_{r})^{2}-(y_{t}-y_{r})^ {2}}. \tag{9}\]  
Combining Equation (8), the constrained attraction of \(v_{t}\) in the x-direction and y-direction, respectively, gives  
\[\left\{ \begin{array}{l}f(v_{r})_{X}=\frac{dx}{d}\times f(v_{r})=dx\times strength (C_{r})\\ f(v_{r})_{Y}=\frac{dy}{d}\times f(v_{r})=dy\times strength(C_{r})\end{array}  \right.. \tag{10}\]  
Along the same vein, we can obtain the constrained attraction of other nodes \(v_{j}\) of \(C_{r}\) and other local-world regions. Therefore, the resultant force of the \(v_{t}\) after considering the constrained attraction is  
\[f=f(v_{r})+f_{a}+f_{r}. \tag{11}\]  
For the phenomenon of the edge-weight evolution of SIN, we intended to construct the visual view by double coding including the thickness and color of the connection curve. Because this part is easier, we do not discuss it in detail. In the visual view of GeoView, the time of the current scene is displayed and controlled by the \"widget Cesium timeline\". In order to correspond to the view of GeoView, we also added a timeline in the visual view of TopolView for the user to manipulate and track the details of the network evolution of interest to the user in different moments or periods. Therefore, we designed and added the timeline module in TopolView, as shown in Figure 10.  
As shown in Figure 10, the timeline module of TopolView consists mainly of two parts: A is the control module including play/pause and fast forward/rewind, and B is the time bar which is used to display moment and time fragments. To sum up, this section discussed the theory of FAT visualization including the specific theoretical components and implementation methods. Since the implementation of FAT is based on improved FDA, the FAT visualization algorithm is no longer specific. Figure 10: The timeline schematic of TopolView.