# Graph-Based Resource Allocation for Integrated Space and Terrestrial Communications
## 2 Motivation  
In the recent years, substantial research efforts have been made to develop different kinds of algorithms for RA, including those that utilize graph-based design, for various network scenarios. To the best of the authors' knowledge, there are no other surveys in the literature that aim attention directly at graph-based RA, but there are others that are relevant to the present analysis. This section reviews contemporary surveys in the field and briefly explores the topics that they are focused on, as well as their limitations, which are summarized in Table 5. Accordingly, the contributions presented in this article are additionally emphasized. A comprehensive survey of deep-reinforcement-learning (DRL)-based solutions for RA in Het-Nets is presented in [5]. The advantages of employing DRL (the chief of them being that they better facilitate the solution of the complex optimization problems involved) in contrast with non-learning algorithms, as well as challenges for future research, are described. Categorization is performed of both the types of Het-Nets, and of the relevant techniques for DRL employed in the literature. Nonetheless, graph-based solutions are not considered. A broad survey of graph neural network (GNN)-based algorithm designs for various procedures in wireless communications and image processing is presented in [11]. The basics of the most commonly used GNN structures and their applications in telecommunications are described. Key issues in employing these algorithms are outlined; however, the discussion on RA is limited only to those that are based on GNNs. The article [12] presents a more expansive view on the applications of GNNs by including problems in not just wireless, but also wired and software-defined networks. For each kind of network, the relevant problems and the types of GNNs are categorized. Nevertheless, the discussion on RA based on GNNs is limited. The authors in [4] provide a comprehensive survey of the challenges related to RA in the different types of Het-Nets. Consequently, the analytical definitions of wireless system capacity for each type are presented, and methods for solving the RA problem are reviewed. An in-depth \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Definition** \\ \hline \(N_{PU}\) & Number of PUs \\ \(N_{CAP}\) & Number of CAPs \\ \(N_{c}\) & Number of channels \\ \(N_{AP}\) & Number of APs \\ \(N_{RRH}\) & Number of RRHs \\ \(N_{AUE}\) & Number of AUEs \\ \(N_{RB}\) & Number of RBs \\ \(P^{out}\) & Outage probability \\ \(P_{FA}\) & Probability of false alarm \\ \(P_{MD}\) & Probability of misdetections \\ \(P_{T}\) & Total network transmit power \\ \(P_{CR,T}\) & Total CR network transmit power \\ \(r\) & Distance \\ \(R_{CR,T}\) & Total CR network throughput \\ \(R_{u}\) & Average throughput per user \\ \(R_{c}\) & Average throughput per cell \\ \(R_{SU}\) & Average throughput per SU \\ \(R_{AUE}\) & Average throughput per AUE \\ \(R_{AUE}\) & Average throughput per MUE \\ \(R_{AP}\) & Average throughput per AP \\ \(R_{T}\) & Total network throughput \\ \(R_{T}\) & Average network throughput \\ \(T_{RB}\) & Maximum number of RBs per D2D link \\ \(\mathcal{V}\) & Set of vertices \\ \(W\) & Weight matrix \\ \(\Omega_{1l}\) & Utility function \\ \(\Omega_{s}\) & Social utility function \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Table of notations. classification of the RA algorithms is made, according to the tasks they complete (such as UA and beamforming), and the performance parameters by which they are evaluated. Finally, the authors propose two paradigms for RA in future cellular Het-Nets. Advancements in D2D communications are thoroughly analyzed in [6]. The modes of implementation and integration of D2D nodes with existing and upcoming networks within the context of 5G are explored in detail, together with several important factors for their realizations (such as device discovery, RA, security, economics-related benefits, and others).