## 3 Dynamic Network Layout Algorithm
### Structure-Based Influence Computation and Propagation  
We introduce a novel node movement constraint strategy based on the factor that nodes are influenced by structural changes. When the network transitions from state \(N_{i-1}\) to \(N_{i}\), the network changes directly affect one or more nodes within the network, requiring adjustments to their positions due to alterations in their connectivity. We quantify the network structural influence factor on different nodes and propagate this influence factor through the network using a depth-first-search (DFS) graph traversal approach. This allows more nodes to have the opportunity to readjust their positions, ultimately leading to the attainment of an ideal layout. (1) Node Influence Calculation  
Figure 3: SSBM-based initial positioning for new nodes. The new node may: (**a**) have edges with multiple existing nodes, (**b**) have one edge with a existing node, and (**c**) have one edge with a newly added node. Figure 2: Centroid-based initial positioning for new nodes. A new node may: (**a**) be a isolated node, (**b**) have one edge with a existing node, (**c**) have edges with multiple existing nodes, and (**d**) have one edge with a newly added node. In previous methods, affected nodes were typically labeled as movable nodes without considering the varying degrees of impact different changes have on different nodes. As illustrated in Figure 4, when removing an edge from the network (indicated by dashed lines in the figure), it affects two vertices connected by this edge, denoted as nodes \(u\) and \(v\), rendering their existing positions no longer suitable, thus necessitating a recalculation of their positions. Following the removal of this edge, the degree of node \(v\) becomes 1, meaning it has only one neighboring node left, whereas the degree of node u remains 8. The re-positioned nodes should appear as depicted in Figure 4. Comparatively, the position change of node \(v\) is much greater than the change of node \(u\). This discrepancy arises because a node's position is determined collectively by its neighboring nodes. The node with the higher degree should experience a smaller impact than the node with the lower degree. Hence, in this study, when calculating the influence of network structural changes on different nodes, the structural information of nodes is taken into account. The influence value \(I_{i}\) on node \(i\) (\(n_{i}\)) are computed based on the degree information before and after the changes as Formula (2). \[I_{i}(t)= \begin{cases}min(\frac{d_{i}^{add}(t)+d_{i}^{add}(t)}{d_{i}(t-1)},1)& n_{i}\in N(t-1)\\ 1&n_{i}\
otin N(t-1)\end{cases}  \tag{2}\]  
where \(d_{i}^{add}(t)\) and \(d_{i}^{del}(t)\) represent the deletion and addition degrees of \(n_{i}\):  
\[d_{i}^{add}(t)=|j\in N_{i}(t-1)\backslash N_{i}(t)| \tag{3}\]  
\[d_{i}^{add}(t)=|j\in N_{i}(t)\backslash N_{i}(t-1)| \tag{4}\]  
(2) Influence propagation  
Nodes within the network are not isolated. The movement of one node will result in changes to all nodes connected to it. Similarly, localized changes within the network not only lead to alterations in their neighborhood structure but also impact the overall network structure. The influence degree that a node's movement has on its neighboring node is determined by the neighbor's degree. As illustrated in Figure 5, for node \(c\), its movement affects node \(d\) with a degree of 1 because node \(d\) is only connected to node \(c\). Conversely, the movement of \(d\) has an impact of only \(\frac{1}{3}\) on node \(c\) because \(c\) has three connections. Therefore, when node \(u\) moves, the calculation of the influence degree \(I_{uv}\) on node \(v\) is as  
Figure 4: The influence of structural changes on different nodes: (a) before the change, (b) after the change. Formula (5), where \(d_{v}\) represents the degree of node v. In the case of an weighted network, the degree can be replaced with edge weights. \[I_{uv}= \begin{cases}\frac{1}{d_{v}}&uv\in E\\ 0&uv\
otin E\end{cases}