# Research on the Prediction of Operator Users' Number Portability Based on Community Detection
## 2 Modeling and Experiment Comparison
### Build a Community Detection Network
For example, the auxiliary node (hexagonal node) in Grid 3_1 can be interpreted as an exclusive customer manager for area-A, which is widely connected with all the users in Grid 3_1. Therefore, the hexagonal nodes in Grid 3_1 are connected to all circular nodes in the same grid, and the same is true for Grid 3_2, and so on. An auxiliary node is constructed separately in each grid, which is fully connected to the users belonging to that grid. Figure 5: Layout of the multilayer community network. \begin{table}
\begin{tabular}{l} \hline
1 Input network G; \\
2 Conduct eigendecomposition of network G: \(D^{-1/2}AD^{-1/2}\approx U_{h}\Lambda_{h}U_{h}^{\top}\); \\
3 Approximately estimate the value of M by \(\hat{M}=\frac{\text{vol}(G)}{b}D^{-1/2}U_{h}(\frac{1}{\hat{T}}\sum_{r=1}^{T} \Lambda_{h}^{r})U_{h}^{\top}D^{-1/2}\); \\
4 Calculate \(\hat{M}^{\prime}=\max(\hat{M},1)\); \\
5 Use singular value decomposition (SVD) to perform log \(\hat{M}^{\prime}=U_{d}\Sigma_{d}V_{d}^{\top}\) low-rank approximation on logM “; \\
6 Obtain \(U_{d}\sqrt{\Sigma_{d}}\) converted by the network G as the network embedding. \\ \hline \end{tabular}  
* A: \(\mathbf{A}\)\(\mathbf{\in}\)\(k_{+}^{|V|\times|V|}\) is the adjacency matrix of network G (the adjacency matrix of the graph is a two-dimensional array vertex relation), and \(A_{ij}\) is the edge weight between vertex i and j; D: D = diag(\(d_{1},\ldots,d_{|V|}\)), where d denotes the generalization degree of vertex i (the sum of the number of times all edges are associated with a vertex v becomes the degree of v); Vol(G): Vol(G) = \(\sum_{i}\sum_{j}A_{ij}=\sum_{i}d_{i}=\) volume of weighted graph G; P: P = \(D^{-1}A\); T\(\&\)B: Context window size and the number of negative samples in skip-gram; U and \(\Lambda\) denote the orthogonal eigenvector matrix and the diagonal matrix of eigenvalues, respectively.  
\end{table} 
Table 2: The processing steps of community discovery. Step 4: Traverse the grid of the upper layer, that is, a larger layer of the grid (such as grid 2-1). In the network of this layer, construct a new auxiliary node, which is fully connected with the auxiliary node in the lower grid. Repeat the step until the topmost grid is reached (e.g., grid 1-1) to complete the network construction. This method makes the network structure smaller and more energy-efficient than the fully connected network structure. It can more closely reflect the association between network nodes without being randomly selected. Step 5: Output the graph with users as nodes. Step 6: Use the graph as the input and substitute it into Table 2 for calculation. Output a matrix of x*4, where x is the number of users in the graph, and 4 refers to four new attributes that contain the topological information regarding the user community network graph and hidden related information. The new attributes are all numeric attributes (assuming that both user 1 and user 2 exist in grid 3-1, set the weight of the connection between the auxiliary node and each user in grid 3-1 to be one, then, the distance between user 1 and user 2 is two. In the output new numerical attribute, the new attribute A of user 1 and that of user 2 is calculated by substituting the distance formula to obtain the result of two, which means that the distance between user 1 and user 2 is two. Thus, the new attribute contains the topological information and hidden association information regarding the user community network graph). Table 2 shows a network embedding method based on the matrix factorization framework, which is considered one of the most effective methods at this stage for application scenarios that require fast, high-quality, and large-scale network embedding [16; 17; 18; 19]. Meanwhile, this matrix-factory-based framework is more computationally intensive and efficient and offers significant improvements over DeepWalk and LINE for conventional network mining tasks.