# PDASTSGAT: An STSGAT-Based Multipath Data Scheduling Algorithm
## 3 PDASTSGAT
### Spatiotemporal Synchronous Graph Attention Neural Networks  
Assuming the MPTCP subflows are feature-rich nodes, we establish a model capable of directly capturing the influence exerted by each node on its neighbors, including those  
Figure 2: PDASTSGAT architecture. Our PDASTSGAT consists of an input layer, an output layer, and multiple spatiotemporal synchronous graph attention neural networks (STSGATs) with aggregating and cropping layers.  
in the current time step as well as adjacent time steps. The most straightforward approach to accomplish this is to establish connections between each node and its counterparts in the neighboring time steps. By establishing connections between each node and its corresponding self in the preceding and subsequent time steps, a localized spatiotemporal graph can be formed. Leveraging the topology of this localized spatiotemporal graph, we can directly extract the correlations between each node and its spatially and temporally adjacent neighbors [14].  
The adjacency matrix of the spatial graph is denoted as \(A\in\mathbb{R}^{N\times N}\). Meanwhile, the adjacency matrix of the localized spatiotemporal graph, which is constructed by considering three consecutive spatial graphs, is represented as \(A^{{}^{\prime}}\in\mathbb{R}^{3N\times 3N}\). For a given node, \(i\), within the spatial domain graph, its updated representation in the localized spatiotemporal graph can be computed using \((q-1)N+i\), where \(t(0<q\leq 3)\) denotes the time step within the localized spatiotemporal graph. If there exists a substantial correlation between two nodes within the localized spatiotemporal graph, the corresponding entry in the adjacency matrix tends to approach a value of 1, \(A^{{}^{\prime}}_{i,j}\in[0,1)\).  
The adjacency matrix, \(A^{{}^{\prime}}\), comprises \(3N\) nodes. The diagonal elements of the adjacency matrix represent the adjacency matrices of the spatial networks encompassing three consecutive time steps. Additionally, the off-diagonal regions of the matrix depict the connectivity between each node and itself across the neighboring time steps.  
Compared to previous spatial-temporal synchronous graph convolutional networks, directly analyzing features using a mask matrix [11] consumes significant computational resources, especially when the number of nodes reaches a certain level and the spatial structure becomes complex. This results in increased training time and a higher number of iterations. To optimize spatial feature analysis, we introduce a graph attention network to model spatial features. The attention mechanism is formulated as follows [15]:  
\[e_{ij}=LeakyReLU(\overrightarrow{\mathbf{a}}^{T}(W\overrightarrow{\mathbf{h} }_{i}\parallel W\overrightarrow{\mathbf{h}}_{j})) \tag{2}\]  
\[A=[\alpha_{ij}]=softmax(e_{ij})=\frac{\exp(e_{ij})}{\sum_{j\in N_{i}}\exp(e_{ ij})} \tag{3}\]  
where \(e_{ij}\) represents the correlation between node \(j\) and node  
To enhance the stability of the self-attention mechanism, we incorporate the multi-head attention mechanism [16] for spatial feature extraction. This integration serves to mitigate any potential loss in the spatial information captured by the graph attention network (GAT), ensuring a more robust and reliable performance. The output features acquired from the \(K\) attention heads are concatenated according to the following expression:  
\[\overrightarrow{\mathbf{h}}_{i}^{K}=||_{k=1}^{K}\sigma(\sum_{j\in N_{i}}\alpha_ {ij}^{k}W^{k}\overrightarrow{\mathbf{h}}_{j}) \tag{5}\]  
where \(\overrightarrow{\mathbf{h}}_{i}^{K}\) denotes the output feature of node \(i\) in the multi-head attention mechanism of the STSGAT network. \(K\) represents the number of self-attention mechanisms, \(\alpha_{ij}^{k}\) signifies the correlation between node \(j\) and node \(i\) under the \(k\)th self-attention mechanism, and \(W^{k}\) denotes the weight matrix corresponding to the \(k\)th self-attention mechanism.