# Multi-View Network Representation Learning Algorithm Research
## 3 Our Method
### Feature Weighting for Different \(k\)-Step Networks  
For the different \(k\)-step networks, the same vertex pairs should be given the different weight values. However, DeepWalk, GraRep and node2 vec neglect the weight information for higher order neighbors. Therefore, the proposed MVNR in this paper solves this problem by following procedures. We first introduce Matrix Forest Index (MFI) to evaluate the weights between vertices for 1-step network. For the weights between vertices on \(k\)-step network. We compare the performance of the MFI algorithm with some classical link prediction algorithms, such as the algorithms based on common neighbor, the algorithms based on random walk and the algorithms based on path. We find that MFI achieves the best link prediction performance on several real citation network datasets. In addition, MFI can be calculated only through the Laplacian matrix of the network. The input of Laplacian matrix is the adjacency matrix of the network. The input matrix is exactly the same as that of the proposed MVNR algorithm. Unlike other link prediction algorithms, the input, calculation procedure and result of MFI can be embedded in the learning framework of the MVNR algorithm. Therefore, we choose the MFI algorithm to measure the weights between vertices. MFI is formulated as follows:  
\[S=\left(I+L\right)^{-1} \tag{7}\]  
Here, \(S=\left[s_{ij}\right]\) is a matrix which is constructed by the matrix forest index. \(L\) is the Laplacian matrix of \(G\), \(I\) is an identity matrix. \(L\) can be calculated based on adjacency matrix, its detailed calculation method can be found in the Algorithm 1. We do not use MFI algorithm and adjacency matrix of \(k\)-step to calculate the weights between vertices in \(k\)-step network. We calculate the weights between vertices in \(k\)-step network by using the weights between vertices in 1-step network and the adjacency matrix of \(k\)-step network. Because the weights calculated by MFI in 1-step network include the weights of existing edges and the future connection probabilities of non-existing edges between vertices. Only through such calculation can the weights of different \(k\)-step networks be different and hierarchical and also play a role in adjusting the weights of different \(k\)-step network representations. For the \(k\)-step network, we define its weight matrix as follows:  
\[W^{(k)}=[w^{(k)}_{ij}],\ \ \ w^{(k)}_{ij}=s_{ij}\times a^{(k)}_{ij}, \tag{8}\]  
where \(a^{(k)}_{ij}= \begin{cases}0,&a^{k}_{ij}=0\\ 1,&a^{k}_{ij}\geq 1\end{cases} ,s_{ij}\in S\), \(a^{(k)}_{ij}\in A^{(k)}\). \(a^{(k)}_{ij}\) is the element of the adjacency matrix \(A^{(k)}\). As mentioned above, the link weights consist of the certainty degrees of existing edges and the link probabilities of non-existing edges. The certainty degrees of existing edges are applied to 1-step network and the link probabilities of non-existing edges are used to \(k\)-step networks where \(k>2\). Through the Equations (7) and (8), we find that we only compute the value of MFI for one time to different \(k\)-step networks and then through the MFI matrix, we construct the weight matrices of different \(k\)-step networks. For the 1-step network, we only retain the similarity values between two vertices with one edge in the MFI matrix and delete the similarity values between two vertices without any edge in the MFI matrix. Therefore, we define the similarity value between two vertices with one edge as the certainty degrees of existing edges. For the 2-step network, we only retain the similarity values between two vertices, where these two vertices are reachable within two steps and the similarity value is the future link probabilities of non-existing edges of the 1-step network. Specifically, the weights of the 2-step network are the similarity values between the current central vertex and the neighbor's neighboring vertices. Therefore, we regard this kind of similarity value as the future link probabilities between two vertices without any edge.