# Graph-Based Resource Allocation for Integrated Space and Terrestrial Communications
### Article Structure  
The structure of this article is illustrated in Figure 2. Section 1 contains the introduction to the topic. A motivation for why its contribution is significant is provided in Section 2. Then, Section 3 reviews the essential characteristics and types of graphs for the application of such methods in wireless communications. The reviewed graph-based RA methods are classified in the three categories of network types (homogeneous and Het-Nets, D2D and CR), each in a separate Section. For each category, there are sub-categories (each representing a Subsection within the Sections), defined as common features of research that include papers with similar problem formulations or system model scenarios. These sub-categories are presented in Figure 2 as pie-charts to illustrate their relative influence within each category. The volume of each sub-category in the chart is related to the number of papers that are reviewed in it. In each Section, particular emphasis is placed on the way in which the optimization problems are defined via graphs, and in what manner they are solved. This is an important consideration for such problems, because they need to be formulated through appropriate graph structures that usually are not trivial to define. On the basis of the review of the graph-based RA methods for the three categories of  
Figure 1: Summary of PRISMA flowchart of the article selection process for this review. networks in Sections 4-6, the GRIST concept is introduced in Section 7. Its implementation challenges and a graph-based model for algorithm design are also introduced. The contributions of this article can be summarized as follows:  
* The methods for RA are categorized according to their network scenario (HetNet/D2D/CR). The graph-based solutions and their performance are thoroughly surveyed. For each category, a tabular outline of the main design characteristics (such as the graph type and its formulation, and the optimization method used for the solution) is given. Analysis of these characteristics provides features of the research in every type of network scenario, as well as directions for future development. * On the basis of the surveyed methods and existing visions for 6G ISTNs, the concept for Graph-based Resource allocation for Integrated Space and Terrestrial communications (GRIST) is introduced. It conceptualizes the design paradigm for the interconnectivity and coexistence of different networks within ISTN by describing their properties via a hypergraph and its attributes. The challenges in implementing GRIST are also given. Then, the design process of RA algorithms in GRIST is itself modeled by  
Figure 2: Graph-based RA in advanced wireless networks. a simplified variant of a complete graph, thus presenting a scheme for determining the appropriate balance between the different design aspects (such as delay and computational complexity). Lists of acronyms and notations, with their definitions, are given in Tables 1-4, respectively. The rest of this article is organized as follows. The findings of relevant survey papers are summarized in Section 2, to illustrate the motivation for this work. The main types and properties of graphs, as well as their capabilities for signal processing in telecommunications, are provided in Section 3. Graph-based methods for RA in homogeneous and Het-Nets, D2D, and CR networks are reviewed in Sections 4-6, respectively. The proposed GRIST concept, challenges for its implementation in 6G ISTNs, as well as the model for the design process of RA algorithms are conceptualized in Section 7. Several directions for future research in the area are identified in Section 8. Lastly, the conclusions of this article are summarized in Section 9.