# Multi-NETVIS: Visual Analytics for Multivariate Network
## 4 Research Methods
### Louvain-3D FDA Layout  
The 3D space can accommodate a large network structure, however, the network topology containing all nodes and edges is not conducive to quickly observing the backbone structure. The combination of community mining and the layout algorithm can clearly  
Figure 2: Layout transformation process. (**a**) 3D network topology; (**b**) Network community division; (**c**) 2D projection; (**d**) 2.5D network topology  
present the community-based backbone structure in the network. Therefore, a 3D force-directed layout algorithm based on community structure (Louvain-3D FDA) is proposed by combining the Louvain community detection algorithm and force-directed layout for the 3D layout. The implementation process of the Louvain-3D FDA is shown in Figure 3, including the backbone structure extraction, 3D force-directed layout, community-based layout, and adding association edge between communities. Backbone structure extraction: in this study, we extract the network backbone topology based on the community structure. The Louvain algorithm [5] is a community discovery algorithm based on the optimization module; the concept is to divide the community through iteration in order to maximize the overall module of the network. The modularity is defined as  
\[Q=\frac{1}{2m}\sum_{i,j}\left[A_{i,j}-\frac{k_{i}k_{j}}{2m}\right]\delta\left( c_{i},c_{j}\right) \tag{5}\]  
where \(A_{ij}\) indicates whether nodes \(i\) and \(j\) are connected (if there is a connected edge, the value is 1; otherwise it is 0), \(m\) indicates the number of all edges in the network, \(k_{i}\) stands for the degree of node \(i\), \(c_{i}\) represents the community to which node \(i\) belongs, and \(\delta\left(c_{i},c_{j}\right)\) indicates whether nodes \(i\) and the node \(j\) are divided into the same community. We assume that network \(G\) has \(n\) communities after community division; then, \(G\) can be expressed as \(G=G_{0}{}^{*}=\left(c_{1},c_{2},\cdots,c_{n}\right)\). The community \(c_{i}\) is defined as a virtual node \(N_{i}\), and a set of virtual nodes can be represented as \(V^{\xi}=\left(N_{1},\cdots,N_{n}\right)\). We use an edge set to replace the associated edges among communities, which can be expressed as \(E^{\xi}=\left\{l_{N_{i}\leftrightarrow N_{j}}\middle|N_{i}\in c_{i},N_{j}\in c _{j},i\
eq j\right\}\), where \(l_{N_{i}\leftrightarrow N_{j}}\) represents the associated edge connecting the virtual nodes \(N_{i}\) and \(N_{j}\). Finally, the backbone network of \(G\) can be expressed as \(G^{\xi}=\left(V^{\xi},E^{\xi}\right)\). 3D force-directed layout: we select the force-directed layout [21] to calculate the layout of the backbone network and the community internal network. In the network \(G^{\xi}\), the position of a virtual node \(i\) can be represented as \(p_{i}\in P(i=1,\cdots,n)\). The core concept of a force-directed layout is that adjacent nodes are close to each other due to mutual gravitation and non-adjacent nodes are moved away from each other due to mutual repulsion. The calculation formula for the gravitational force \(F_{ni}\), repulsive force \(F_{ri}\), and moving step length \(V_{i}\) of node \(i\) in each iteration are as follows:  
\[\left\{ \begin{array}{l}F_{ni}=\sum\limits_{j\in N_{i}}\frac{\left\|p_{j}-p_{ i}\right\|\left(p_{j}-p_{i}\right)}{k}\\ \\ F_{ri}=\sum\limits_{j=1,\cdots,n,j\
eq i}k^{2}\frac{\left(p_{j}-p_{i}\right)}{ \left\|p_{j}-p_{i}\right\|^{2}}\\ \\ V_{i}=F_{ni}+F_{ri}\end{array} \right. \tag{6}\]  
where \(N_{i}\) is the set of neighbor nodes of node \(i\), while \(p_{i}\) and \(p_{j}\) indicate the positions of nodes \(i\) and \(j\), respectively. We define the adjustable parameters for the control distance as  
Figure 3: Louvain-3D FDA implementation process. \(k=\sqrt{\frac{2V}{|N|}}\), where \(V\) indicates the volume of a sphere with a custom radius for the layout and \(|N|\) indicates the number of virtual nodes.