## 3 FL-Ahneap for Mobile App Start-Up Prediction under Privacy Protection
### Network Representation Learning under Federated Learning  
The representation learning model for the attributed heterogeneous network in AHNEAP uses the following formula to calculate the node's embedding:  
\[v_{i,r}=h_{z}(x_{i})+\alpha_{r}M_{r}^{T}U_{i}a_{i,r}+\beta_{r}D_{z}^{T}x_{i} \tag{1}\]  
where \(r\) represents the edge type in the attribute heterogeneous network built in AHNEAP, and there are three types of node: time, location, and application, and three types of edge: time application, location application, and preamble application. Therefore, \(r\) is a positive integer and satisfies \(1\leq r\leq 3\). \(x_{i}\) denotes a feature of the node \(i\). \(h_{z}(x_{i})\) is a transformation function, whose role is to calculate the influence of \(x_{i}\) in the embedding representation of the node, and is the base embedding for node \(i\). According to the self-attention mechanism [37], \(a_{i,r}\) is the weight of the edge embedding vector of node \(i\) in the sub-network of edge type \(r\), and is computed according to \(a_{i,r}=softmax\big{(}w_{r}^{T}tanh(W_{r}U_{i})\big{)}\), and is actually a simple feedforward neural network. \(w_{r}^{T}\), \(W_{r}\), represent the transformation probability matrix, which requires the training of the model, and optimization \(U_{i}\) represents the hidden layer state in the self-attention mechanism. \(tanh()\) is the activation function of \(W_{r}U_{i}\), and \(softmax()\) is a normalized function, which converts the result from negative infinity to positive infinity to the probability 0-1. \(U_{i}\) is a vector composed of the edge embedding vectors of node \(i\) in three sub-networks, which represents the relationship between node \(i\) and its neighbor nodes. Each node aggregates the mean value of the features of a certain number of neighbor nodes in each sub-network, as the edge embedding vector of the node in the sub-network \(U_{i}\) is formed by concatenating the edge embedding vectors of node \(i\) in three sub-networks. \(M_{r}\) is the transformation matrix of the edge embedding vector; \(M_{r}^{T}\) represents the transpose of \(M_{r}\), and the \(i\)th one-dimensional vector in \(M_{r}\) represents the influence of the edge embedding vector of the node \(i\) in the sub-network of the edge type \(r\) on the embedding representation of node \(i\). \(\alpha_{r}\) represents the influence coefficient of the edge embedding vector of each node on the embedding representation of the node in the sub-network with edge type \(r\). \(D_{z}\) is a feature transformation matrix on node \(i\)'s corresponding node type \(z\), designed for the purpose of calculating the embedding representation vector of the new node, and the new nodes are isolated from all nodes in the network. \(D_{z}\) represents the similarity relation between node \(i\) and all nodes in the network from the perspective of features similarity; \(D_{z}^{T}\) is the transpose of \(D_{z}\). \(\beta_{r}\) represents the  
Figure 1: Framework of the FL-AHNEAP method. influence coefficient of the similarity relation between features in the sub-network with edge type \(r\) on the node embedding. In this section, we use the FederatedAveraging algorithm to realize the parameter aggregation of the network representation learning model. In Equation (1), \(h_{z}(x_{i})\) is actually obtained by multiplying a feature matrix and a transformation matrix. There is a transformation matrix \(H\), and there are transformation matrices \(M\) and \(D\), and two transformation matrices \(w\) and \(W\) in \(a_{i,r}\), and finally there is a transformation matrix \(T\) for node edge embedding. The updates of model parameters are actually the updates of the above six transformation matrices. Algorithm 1 is pseudo code of network representation learning based on federated learning. Inputs: _Nodes, Features, Neighbors_. Model parameters: transformation matrix in network representation learning model \(H_{0}\), \(M_{0}\), \(D_{0}\), \(w_{0}\), \(W_{0}\), \(T_{0}\).