# Graph-Based Resource Allocation for Integrated Space and Terrestrial Communications
## 4 Graph-Based RA in Cellular Homogeneous and Het-Nets
### Lessons Learned and Trends in Development  
The summary presented in this section shows several features of the research in graph-based RA algorithms for homogeneous and Het-Nets, which provide hints for their further development. They are as follows:  
* In this deployment scenario, the most prominent problem is RA because it is usually assumed that: (1) UA is performed independently through traditional methods, and (2) there is no inter-network spectrum sharing (SSh), i.e., the spectrum access is static, which is the usual case for this type of network. Due to the cell-edge throughput degradation problem, however, UA and RA should be considered together, and graph-based algorithms are a potent solution [28; 29; 37]. _A significant challenge of designing them lies in how to properly group the different sets of vertices (UEs, macro cells, small cells, and APs), so that their constraints on the transmission power may be satisfied. In addition, the users in each tier are influenced by interference in an non-reciprocal manner (a small-cell user may receive more interference from a nearby macro cell than a user served by that cell)._ This also needs to be considered in the way that the graph's edges are defined. Furthermore, the optimization problems will involve multiple independent variables such as the channel gains, power allocations, available channels, and distance between the nodes. * When it comes to graph-based RA algorithms for Het-Nets, little attention has been given to the influence of the channel's impairments (such as UE movement speed, fading, and noise distributions) on the received SINR, and thus to the throughput. _Consequently, there is worth in considering whether a particular graph topology can take advantage of the dynamic network conditions (for example, the graph's edges may need to be readusted often with the movement of the UE). Modeling the Het-Net via GNN and considering these conditions as attributes of the vertices and edges can provide effective solutions in this case._ These challenges become even more intricate when it comes to whether a particular set of channels is available or not, i.e., when multiple radio access technologies (RATs) coexist in the same or adjacent portions of the spectrum [20]. Then, dynamic and intelligent multiple access needs to be implemented through CR methods (discussed in Section 6). * Several noteworthy works [44; 45; 46; 47; 48; 49] have expanded the scope of RA methods for communications between UAV and satellite nodes, as well as their connection to terrestrial Het-Nets. The presented graph models are hereby briefly reviewed, and the relevant challenges described. Reliable connectivity between satellite nodes, considering the temporal changes due to their varying movement velocities, is considered in [44; 45; 46; 47]. The creation of edges is determined by the duration of direct visibility between two communication nodes and their link latency. Network topology is adapted in time by the vertices' attributes, according to the shortest path length and relevance (number of shortest paths going through a vertex) [44]. Alternatively, an RA solution is proposed in [45] which constructs the time-varying graph using two types of edges. They represent caching links and communication links, with each type of edge connecting two vertices (satellites/high-altitude platforms) that are related by either the cached data at one of the nodes, or by their information exchange. This approach is built upon in [46] by a commodity-flow algorithm for frequency scheduling of ground-based, UAV, and satellite nodes, and the links between those that communicate directly for the provision of a particular service. In [47], both the change in the nodes' available bandwidth and their traffic load are modeled by a Markov model, jointly with the time-varying graph that models the space and terrestrial nodes' movements.