# Graph-Based Resource Allocation for Integrated Space and Terrestrial Communications
## 4 Graph-Based RA in Cellular Homogeneous and Het-Nets
### RA Methods Based on GSP  
The authors in [30] explore the application of GNNs for modeling cellular networks and solving popular problems for them. A general variant for graph construction is adopted--BSs and UEs are the vertices, connected via edges if there is an association between a BS and a UE, or two BSs interfere with each other's transmissions. Edges are not drawn between nodes which are over a certain distance apart from each other because their signals' strengths are neglected. The edges' weights are represented as the channel gains for each link. At a particular vertex, information from its neighbors is aggregated, and then combined via convolution to extract the graph's topological features. The Adam optimization algorithm [41] is applied for the GNN training procedure. It converges in a few iterations, and the throughput results remain stable. The GNN for Het-Nets is further adapted for learning optimal power allocation when the adjacency matrix does not exhibit universal permutation equivalence and invariance [31]. The authors propose a more advanced user clustering by dividing the UEs into \(N_{BS}\) subsets (all users served by the same BS compose one subset) and processing them separately through the GNN's layers. To achieve this, the GNN's weight matrix is formed by \(N_{BS}\times N_{BS}\) sub-matrices, which process the information of each UE subset. The bipartite graph is therefore composed of both the BSs and the UEs as vertices (all BSs have \(P_{max}\) as a vertex attribute), with their associated and interference links being the edges (their attribute is the channel gain). The RMSProp [42] optimization algorithm is applied. Substantial computational efficiency for the scenario with fixed numbers of UEs, macro BSs and small-cells, in comparison with the alternatives, has been achieved. A novel GNN architecture via algorithm unfolding for power allocation in an ad hoc network is proposed in [32]. The objective function of the throughput is decomposed into several parameters which process the vertices' information iteratively, and are dependent on the channel gain (which is the vertex attribute). Every connection between a transmitter and a receiver comprises a vertex, with the edges being the interfering connections. Optimization is performed through stochastic gradient descent (SGD). A significant reduction in the computational complexity compared to the alternatives is reported. An alternative solution to the power allocation in ad hoc networks is proposed in [33; 43]. The RA policy (i.e., the processing layer's parameters) is trained for the whole graph, whereas it is implemented locally (for each transmitter-receiver pair). The graph is composed of the transmitters as vertices, and the interfering links between them as edges (the channel gains being their weights/attributes). For each vertex, the information from its immediate neighbors is aggregated to capture the asynchronism of the interference, or the synchronism of the associated communication links. The result is then filtered through the GNN's layers to obtain the power allocations. The layers' parameters are obtained via alternating the gradients between the variables of an Lagrangian function (through a convolutional neural network) dependent on the channel gain, and the adjacency matrices. Asynchronous SGD is utilized to train the model. Close to linear gain in \(\bar{R}_{u}\) occurs with the increase in the number of transmitter-receiver pairs in the testing stage. Further developments are made by the DRL framework for channel allocation, based on GNNs in wireless local area networks (WLANs) with reduction in the number of states, as proposed in [34]. The graph is formed from the APs, and all of them that are able to detect each other's signals are connected via edges. At each instance, the Q-learning processes each AP's current state via the GNN and chooses the channel allocations (from the replay buffer of stored policies), such that the reward (a function of the throughput) is maximized.