I. Introduction
In recent years, deep learning (DL) has achieved many impressive results in computer vision, natural language processing and other application fields. Inspired by this, many researchers have tried to apply DLbased methods, especially “learning optimization” methods, to solve tricky optimization problems in wireless network. This method aims to achieve suboptimal performance in realtime without relying on the domain knowledge of experts, that is, to automate the algorithm design process. This method usually has two kinds. The first is “endtoend learning”, which directly uses neural networks to approximate the optimal solution of the optimization problem. For example, in order to solve the problem of power control, multilayer perceptrons (MLPs) are used to approximate the input and output mapping of the weighted minimum mean square error (WMMSE) algorithm
[learnOpt2018]. The second paradigm is “learning and optimization”, which uses neural networks to replace traditional algorithms to learn more difficult strategies or unfolds traditional iterative algorithm into several learnable layers. For example, A. Bora et al. used generative model from neural networks instead of standard sparsity model to represent data distributions [CSGM2017]. N. Shlezinger et al. designed a deep neural network (DNN) to implement the specific parts of Viterbi algorithm that are channelmodelbased, leaving the rest of the algorithm structure intact [ViterbiNet2020]. K. Gregor et al. designed a nonlinear, parameterized, feedforward architecture with a fixed depth that can be trained to approximate the optimal sparse code [LISTA2010].A key design element of the aforementioned two “learning optimization” methods is the neural network architecture. At present, most of the existing works uses MLPs [OptPowerContrl2020] or convolutional neural networks (CNNs) to generate the learning model [PowerContrlCNN2018]. Although they have achieved good performance in smallscale wireless networks, they fail to use the topology information of wireless networks. When the network scale becomes larger, the performance of these methods drops sharply, because MLPs and CNNs don’t have the ability to learn and exploit the network topology information of wireless networks. In order to improve the performance and generalization ability of learning models, an effective idea is to incorporate the network topology information into the neural network architecture, so that the neural network does not need to learn the topology structure of communication networks.
In wireless networks, although the device is located in Euclidean space, channel state cannot be embedded in Euclidean space. Therefore, the data in wireless networks is also nonEuclidean. When using the “learning optimization” method in wireless networks, it is necessary to use a neural network that works in nonEuclidean space. There are some overviews on the application of machine learning (ML) or DL in wireless networks, such as the latest application of DL in mobile or wireless networks [DLwireless2018], DL applied in different wireless network layers [DLmobilewireless2018], the application of ML in improving the qualityofservice (QoS) and qualityofexperience (QoE) of wireless networks [MLwirelessLayerSurvey2021], the application of ML in improving the QoS and QoE of 6G networks [ML6GSurvey2021]. However, the application of DL or ML in wireless network are mainly utilizing the classical DL or ML methods that cannot effectively capture the nonEuclidean information of wireless networks.
In general, the optimization problems of wireless networks can be formulated as graph optimization problem. There are two types of neural networks that can take advantage of the permutation invariance, namely, graph neural networks (GNNs) [GNNsurvey12020] and deep sets [DeepSets2017]. Compared with deep sets, GNNs not only have the permutation invariance properties, but also can simulate the interaction between devices in wireless networks. Attracted by the powerful performance of GNN, many researchers have began to use GNNs to solve problems of wireless networks, and these works are comprehensively reviewed in this paper.
There are some overviews about the paradigms and applications of GNNs in the past few years. Four basic paradigms of GNNs were introduced specifically in [GNNsurvey2019, GNNsurvey12020]. Besides, graph reinforcement learning and graph adversarial methods were reviewed in [graphDLSurvey2020]. The GNN model is divided into two categories, i.e., spatialbased GNNs and spectralbased GNNs, and the connections between categories are revealed in [SurveySpatialSpectral2020]. The expressive power of GNNs and some GNN variants was introduced comprehensively in [SurveyGNNexpressPower2020]. The current methods of using software or hardware to accelerate GNNs were reviewed from a computational perspective [computingGNNsurvey2020]. The interpretability of GNNs was introduced deeply in [ExplainGNNsurvey2020]. In addition, there are also several overviews on the applications of GNNs, such as the method of solving the problems in traffic domain [graphTrafficDomain2020], the application of GNNs in power system [graphPowerSystem2021], GNNs for recommender system [graphKGRS2021]. The relationship between GNNs and the current development of neuralsymbolic computing was introduced in [GNNNeuSymbol2020].
However, as far as we know, there is no overview of the application of GNNs in wireless networks. In this paper, we give out a comprehensive overview of GNNs and their applications in wireless networks, providing a definite guide and future directions for researchers who are interested in this topic. The main contributions of this overview are summarized as follow

We introduce several classical paradigms of GNNs that applied in wireless networks. Each paradigm is summarized with the help of the existing results, it’s easy for interested researchers to start.

A comprehensive review of GNNs in wireless networks is summarized with the existing directions (e.g., power control, link scheduling, channel control, wireless traffic prediction, vehicular communication, point cloud, etc.).

We summarize the existing work and puts forward several challenges and some meaningful research directions about the application of GNNs in wireless network.
The rest of this paper is organized as follows: In Section II, we review the basic definitions of graphstructured data and several classical paradigms of GNNs that applied in wireless networks. In Section III, we introduce the application of GNNs in wireless networks. In Section IV, we discuss a few valuable directions for the application of GNNs in wireless networks. Finally, in Section V, we conclude this work.
II. Definition And Paradigms
In this section, due to the basic paradigms of GNNs have been fully elaborated in [GNNsurvey12020, graphDLSurvey2020], we only simply introduce several classical models for each popular paradigm of GNNs, which have been applied in the surveyed wireless network studies, mainly including graph convolutional neural networks, graph attention networks, graph auto encoders, graph recurrent networks, graph reinforcement learning, and spatialtemporal graph neural networks. The overview of these paradigms are shown in Fig. 1. We assume that graphs considered in this section are undirected and define the basic concepts of graphs. For ease of reading, the symbols commonly used in graphs are summarized in Table I.
Graph  
The number of nodes  
The set of nodes  
A node  
The set of neighbor nodes of  
The set of nodes  
An edge connecting nodes and  
Adjacent matrix  
The transpose of matrix  


The degree matrix of matrix  
power of the elements in the matrix  
Feature matrix  
The th column of matrix  
Denotes the th element of vector 

The dimension of the feature vector  
The sample size  
The maximum depth of GNNs  
Multivariate Gaussian 
A. Basic Definitions of GraphStructured Data
Before introducing the paradigms of GNNs, we would like to describe several basic definition of graphstructured data firstly. Graph structural information is a kind of data in nonEuclidean space, which is commonly expressed as , where is the set of edges and is the set of nodes. Let to denote a node and to denote an edge pointing from to . The adjacency matrix of graph is represented as . If , , otherwise, . If adjacency matrix is asymmetric, the graph is directed graph, otherwise, the graph is undirected graph. The degree of a node is defined as the number of all 1hop neighbor nodes. For undirected graphs, the degree matrix of a graph is a diagonal matrix, where . The Laplacian matrix of an undirected graph is defined as . The symmetric normalized Laplacian matrix is defined as . Note that the symmetric normalized Laplacian matrix is a real symmetric semipositive definite matrix, it can be decomposed into , where
is the eigenvector matrix and
is the diagonal matrix of withbeing the eigenvalue. In a graph, each node may have its own attribute information, which is represented by feature matrix
, where is the dimension of feature vectors. If feature matrix changes over time, the graph is defined as a spatialtemporal graph.B. Graph Convolutional Neural Networks
Graph convolutional neural networks (GCNs) implement convolutional operation on graphstructured data. In other words, the convolution operation of GCNs is transformed from Euclidean space to nonEuclidean space[GeoDL2017]. The core idea of GCNs is to learn a mapping function, which can combine the neighbor nodes’ information with its own feature information to generate a new node representation. According to different convolution methods, GCNs can be divided into spectralbased GCNs [GCN, ChebNet, 1stChebNet, DCN, AGC, Cayleyn] and spatialbased GCNs [SpatialGCN, MPNN2017, GraphSage2017, DCNN]. In the sequel, we simply introduce several classical models in terms of spectralbased GCNs and spatialbased GCNs.
1) Spectralbased GCNs
. Since the number of neighbors of each node is not fixed, a fixed convolutional kernel cannot be implemented on graph, but the convolutional operation can be carried out when the graphstructured data is converted to the frequency domain. Given the input graph feature matrix
and a graph filter , the graph convolution to can be defined as [GNNsurvey12020](1)  
where denotes the graph convolution operation,
denotes the graph Fourier transform,
denotes the inverse graph Fourier transform, denotes the Hadamard product, and .Different spectralbased GCNs can be defined by changing . For example, Bruna et al. proposed Spectral convolutional neural network (SpectralCNN) in which is learnable [GCN]. However, due to the existing of the eigendecomposition of , SpectralCNN faces several challenges, such low efficiency of computation, etc. To overcome these limitations, Defferrard et al. proposed Chebnet via redefining the graph filter with Chebyshev polynomials [ChebNet]. Kipf et al. further constrains the number of parameters and proposed a model, i.e., GCN, which might be effective in solving the overfitting, to minimize the number of operations at each layer, i.e.,
(2) 
where is learnable weight matrix. In order to solve the situation that may lead to gradient explosion, the authors further transform into , where .
By comparing Chebnet with GCN, Chebnet has higher computational complexity and a large number of parameters, but it has stronger expression ability. Chebnet’s order convolution operator can cover maximum steps neighbor nodes of the central node, while GCN only covers the firstorder neighbor nodes. However, the perception domain of graph convolution can be expanded by stacking multiple GCN layers, so the flexibility is relative high. The most important thing is that GCN with lower complexity is easier to train, faster, more effective and more practical, so it has become the most typical method.
Remark 1.
In the practical application of using spectralbased GCNs to solve wireless network problems, which will be introduced in Section III, the definition of graph filters for spectralbased GCNs usually combines the adjacency matrix of wireless network topology and the channel state information , which can make full use of complex radio information. On the other hand, the introduction of channel state information changes the adjacency relationship of the adjacency matrix, that is, exploits the information hidden in the wireless networks.
2) Spatialbased GCNs. The spatialbased graph convolution is similar to the image convolution. The two convolution operations all extract the neighbor information of a node in a graph or a pixel in an image to obtain a richer feature representation of the node or the pixel. The difference between image convolution and spatialbased graph convolution is that the nodes in a graph are unordered while the pixels in an image are irregular, and the number of neighbors of each pixel in an image is limited while the number of neighbors of each node in a graph is not sure. So the definition of spatialbased convolution operation cannot use a fixedsize convolution kernel like the image convolution operation. Thus, the key of spatialbased GCNs is to define convolution operation with different neighborhood numbers and keep local invariance.
According to the overview result, the most widely used spatialbased GCNs in wireless network problems are mainly including Message passing neural network (MPNN) and diffusionconvolutional neural networks (DCNNs). MPNN was proposed in [MPNN2017], which is a unified framework of spatialbased GCNs. MPNN decomposes the spatialbased graph convolution into a message aggregation phase and a combination phase, i.e.,
(3)  
where is the feature representation of the edge between node and node , and are the aggregation function and the combination function in the th iteration, respectively. is the message aggregated from node ’s neighbors and is the new feature representation of node in the th iteration. W. L. Hamilton further proposed GraphSage model via fixing the number of neighbors for message passing [GraphSage2017], which can overcome the shortcomings of MPNN, such as the computation efficiency is low when the number of neighbors of a node is too large. The graph convolution operation of GraphSage is implemented by
(4) 
where is an aggregation function in the th layer, is a random sample of the node ’s neighbors.
The diffusionconvolution operation in the the DCNNs model builds a potential representation by scanning the diffusion process of each node through probability transition matrix in the graph structure, i.e,
, where is the hidden state in the th layer, is the probability transition matrix and denotes the power of .Remark 2.
The GCN models introduced above meet graph structures with the same node type and edge type. However, in wireless networks, the types of communication nodes and the types of communication between nodes may be rich and diverse, that is, the corresponding wireless network topology graph is a heterogeneous graph. Therefore, how to design an effective graph convolution method with more heterogeneous network topology information is very important. In section III, several works on designing graph convolution for heterogeneous graphs is introduced. However, the existing methods are designed based on spatialbased GCNs, and there is no work on designing heterogeneous graph convolutions based on spectralbased GCNs.
C. Graph Attention Networks
In deep neural networks, attention mechanism [Attention] has been regarded as a more expressive means of information fusion, and it has been widely used in computer vision and natural language processing. The core of attention mechanism is to assign weight to the given information, and the information with high weight means that the system needs to focus on processing. Different neighbor nodes may have different influence on the central node. As illustrated in Fig. 2, Velickovic et al. introduced the attention mechanism into graphstructured data and performed the aggregation operation on neighbor nodes to realize the adaptive allocation of weight to different neighbors [GAT], i.e., Graph Attention Network (GAT), which is defined as follows
(5) 
where is the attention mechanism, is the hidden state of node at the th layer. Multihead attention mechanism can be implemented to further improve the expression ability of the attention layer, that is, independent attention mechanisms can be utilized and then the output results are concatenated together, i.e.,
(6) 
where denotes the concatenation operation and is the th attention mechanism.
Remark 3.
There are few applications of graph attention mechanism in wireless networks. However, this mechanism is very helpful to improve the performance of the model. The interaction between communication nodes in wireless networks is different, especially in heterogeneous wireless networks.
D. Graph AutoEncoder
Inspired by the conventional autoencoders, graph autoencoders (GAEs) utilize GNNs as encoders to learn lowdimensional latent representations (or embeddings) of nodes. The goal of encoders in GAEs is to encode the structural information of nodes. While decoders in GAEs aims at decoding the structural information about the graph from learned latent representations [RepresentLearnGraph2017]. The general overview of GAEs is shown in Fig. 3. Specifically, the encoder maps the node to a lowdimensional vector embedding based on the node’s structural information, and the decoder extracts the information that the user interested from the lowdimensional vector embedding.
GAEs have been used in many fields by virtue of their concise encoderdecoder structure and efficient encoding ability. Kipf et al. proposed the variational GAE (VGAE) using a GCN encoder and a simple inner product decoder, which aims at link prediction in citation networks [VGAE]. The encoder maps each node to a lowdimensional latent representation using GCN, then a network embedding can be obtained. The decoder computes the pairwise distance given network embedding and applies a nonlinear activation. Finally, the decoder outputs the reconstructed adjacency matrix.
Remark 4.
The main advantage of the graph autoencoder is to mine the topological information in the graph, and then learn an effective low dimensional feature vector representation for each node or the whole graph. This feature vector representation can reflect the characteristic that can separate from other nodes or graphs to a certain extent.
E. Graph Recurrent Networks
In order to cope with longterm information propagation across the graphs, there is a trend to utilize the gate mechanism from RNNs, such as longshort term memory (LSTM) and gated recurrent unit (GRU). LSTM is a popular type of RNNs that has gate units to adjust the information flow inside the unit. GRU also has gate units similar to LSTM, which is proposed to solve the problems of longterm memory and gradients in back propagation, and also to enable each recurrent unit to adaptively capture the dependencies of different time scales. The difference is that GRU does not have a sperate memory unit.
Y. Li et al. utilized GRU that implemented in the propagation step to design a gated GNN, i.e., GGNN [GraphGRU2016]. GGNN unrolls the recurrence to a fixed number of steps
and use backpropagation through time in order to compute gradients. The basic recurrence paradigm is designed as follows,
(7a)  
(7b)  
(7c)  
(7d)  
(7e)  
(7f) 
where determines how nodes in the graph communicate with each other, are the two columns of blocks in and corresponding to node , is a bias item. contains activations from edges in both directions. and are weight matrices. N. Peng et al. also explored a general framework for crosssentence ary relation extraction based on graph LSTMs [GraphLSTM2017].
Remark 5.
There is less work to solve wireless network problems only using graph recurrent networks. Generally speaking, in some application scenarios that need consider the temporal dependency in wireless networks, temporal dependency and spatial dependency should be considered at the same time. Therefore, the existing work is to capture the spatial and temporal relationship of wireless networks at the same time, namely, spatialtemporal GNNs, to be introduced in subsection G.
F. Graph Reinforcement Learning
Reinforcement learning (RL) builds a model of environment and learning via exploring the unknown environment to get an optimal strategy [RL]. The learner (or agent) explores the operating environment itself and learns the best action based on trial and error. Generally speaking, traditional deep learning is difficult to deal with the problem with nondifferentiable objective function or constraints, while RL can solve this problem by learning from feedback. In recent years, RL has been gradually applied to graphstructured tasks, such as graph generation, graph classification, and graph reasoning tasks, etc.
J. You et al. considered the problem of generating directed molecular graphs with nondifferentiable objective functions and constraints [GCPN2018]
. In order to solve this problem, the authors proposed a model named graph convolutional policy network (GCPN) based on RL and GCNs. Specifically, the graph generation is modeled as a Markov decision process of adding nodes and edges, and then the generative model in the graph generation environment is regarded as an RL agent. The node representations are learned by GCNs, the action of the agent is designed as a link prediction, and the reward is defined as a sum over domainspecific rewards and adversarial rewards. GCPN is trained in an endtoend manner via using proximal policy gradient.
Graph attention model (GAM) utilized RL to solve graph classification task based on random walks
[GAM2018]. The generation of random walks was modeled as a partially observable Markov decision process. The RL agent should performs two actions at each time step, i.e., predicts the label of input graph and generates the rank vector using designed rank network. The reward is designed as , whereif the GAM classified the graph correctly, otherwise,
. is the environment. W. Xiong et al.proposed a model, i.e., DeepPath, for knowledge graph reasoning
[DeepPath2017]. Specifically, DeepPath targets at finding the most informative path between two target nodes. The action of RL agents is to predict the next node in the path at each step and output a reasoning path in the knowledge graph. The reward functions include the scoring criteria: global accuracy, path efficiency and path diversity.Remark 6.
Although RL has been widely used in wireless networks, the application of graph RL is still in its infancy. Generally speaking, wireless network problems may have some resource constraints, while RL has more advantages than traditional machine learning methods in dealing with constraints. Therefore, the application of graph reinforcement learning in wireless networks will have a good prospect.
G. SpatialTemporal Graph Neural Networks
For some wireless network problems, the graph structure and graph input may be dynamical. Spatialtemporal graph neural networks (STGNNs) play an important role in dealing with graphs that have dynamic node inputs while connected nodes are interdependent. There are two categories for STGNNs from the perspective of capturing temporal dependency, RNNbased methods and CNNbased methods.
C. Chen et al. utilized residual recurrent graph neural networks (ResRGNN) to predict the traffic flow in traffic network [GraphGRUTP2019]. ResRGNN can utilize the spatial attributes to capture spatial features using diffusion convolution. In addition to using graph convolution, ResRGNN also uses GRU to discover temporal dependency for each node. Then ResRGNN can jointly capture the spatialtemporal dynamics. Specifically, the implementation of RGNN unit at time is
(8a)  
(8b)  
(8c)  
(8d)  
(8e) 
where and denote graph signal, external feature and the outputted hidden state at time , respectively. and represent the reset gate and update gate at time , respectively. and are learnable graph filters, is the learned weights of the output layer. denotes the output at time .
S. Guo et. al proposed the attention based spatialtemporal graph convolutional networks (ASTGCN) based on GCNs and CNNs to predict traffic flow [STGCNtf2019]. Specifically, ASTGCN utilizes spectralbased GCNs, i.e., ChebNet [ChebNet] to capture the spatial dependency among different nodes in traffic network graph. In addition, 1D CNN is utilized to capture temporal dependency for each node in time series. The implementation details of capturing spatial and temporal dependencies are illustrated in Fig. 4.
Remark 7.
Traffic prediction is the most popular application of spatialtemporal GNNs in wireless network, which will help the management of wireless networks resources. With the development of communication technology and the rise of wireless services, the traffic of wireless services is complex and huge, and the management of wireless networks resources will become more and more difficult. Therefore, wireless network traffic prediction will be a promising and challenging direction.
III. Applications in Wireless Networks
In this section, we focus on introducing the works on the application of GNNs in wireless networks. As shown in Table II, the applications of GNNs in wireless networks mainly cover power control, link scheduling, channel control, wireless traffic prediction, vehicular communication, and point cloud, etc. To serve as a catalyst for further study of the applications of GNNs in wireless networks, this section reviews a large amount of available literatures and discusses oriented researches.
Area  Year  Application  Algorithm  Scheme  
Power Control  2019 

Spatialbased GCNs  Y. Shen et al. [gnnWPC2019]  
2021 

Spatialbased GCNs  Y. Shen et al. [gnnResourceMng2021]  
2020  Power allocation in adhoc wireless networks  Spectralbased GCNs  M. Eisen et al. [REGNN2020]  
2019 

Spectralbased GCNs  M. Eisen et al. [LSWgnn2019]  
2020 

Spectralbased GCNs  M. Eisen et al. [TPgnn2020]  
2021 

Spectralbased GCNs  I. Nikoloska et al. [FastPowerControl2021]  
2020  Downlink power control in wireless networks  Spectralbased GCNs  N. Naderializadeh et al. [wpcCOgnn2020]  
2020 

Spectralbased GCNs  A. Chowdhury et al. [FpaGNNdau, UnfoldGnnPA]  
2021  Power control in multicell cellular networks  Spatialbased GCNs  J. Guo et al. [LpcCSgnn]  
2021 

Spatialbased GCNs  X. Zhang et al. [JUAPAinHUDN]  
2021 

Spatialbased GCNs  X. Zhang et al. [PowerContrlBF2021]  
2020 

Spectralbased GCNs  Z. Gao et al. [ResAllocGNN]  
2020 

Spectralbased GCNs  Z. Wang et al. [UspLrnARAadhoc]  
Link Scheduling  2019 

Graph embedding  W. M. Lee et al. [graphEmbedWLS]  
2020 

Spectralbased GCNs  Z. Zhao et al. [DistributeSchedgnn]  
2019 

Spectralbased GCNs  K. Lei et al. [GcngcnLP2019]  
2021 

Spatialbased GCNs  S. He et al. [he2021gblinks]  
Channel Control  2020 


K. Nakashima et al. [DeepRLCAgnn2020]  
2020 

Spatialbased GCNs  Y. Yan et al. [GnnCTMIMO2020]  
2020 

GAT  K. Tekbıyık et al. [ChannelEstGAN]  
2020 

Spatialbased GCNs  A. Scotti et al. [GnnMIMOdetect]  
2021 

Spatialbased GCNs  T. Jiang et al. [BeaformIRS2021]  
Traffic Prediction  2020 


S. Zhao et al. [CellNetTPgnn2020]  
2020 


L. Yang et al. [StlNetTPgcngru2020]  

2020 


J. Dong et al. [GCQNcavNet]  
2020 


PYJ. Ha et al. [cavMARL]  
2020 


Z. He et al. [RAgnnVC2020]  
Point Cloud  2020 

Dynamic GCNs  J. Shao et al. [BranchyGNN]  
2020 

GAE  T. Fujihashi et al. [Wireless3Dpcgnn]  
2021 

3D GNN  C. S. Jeong et al. [ARanchor2021]  
Others  2019 

Spectralbased GCNs  S. Lohani et al. [FSOAMC2019]  
2021 

Spectralbased GCNs  W. Yan et al. [NetworkLoc2021]  
2021 

Spectralbased GCNs  M. Lee et al. [DecentralizedInfer2021] 
A. Power Control
Power control is a basic problem of resource management in wireless networks. A large mount of works has studied the power control problem using traditional optimization methods and DNNs. However, the traditional optimization methods face high computation complexity and DNNs have poor scalability and generalization as the scale of wireless network increases. To cope with these barriers, GNNs are utilized by many researchers to solve power control problem in wireless networks, which have the natural characteristics of solving the problem with graph structure.
To develop the scalable methods for solving the power control problem in user interference channels, Y. Shen et al. proposed an interference graph convolutional neural network (IGCNet) [gnnWPC2019]. In this scheme, the user interference channel is modeled as a complete interference graph with node and edge labels, which is shown in Fig. 5. The th transmitterreceiver pair is viewed as the th node. The node label contains the state of direct channel and the weight of the th pair. One edge between two nodes indicates an interference link, with label as the interference channels and . The aggregation and combination rules of IGCNet are designed as follows
(9)  
where is to take the largest value in a set, and denote two different , denotes the operation that concatenates two vectors together. denotes the feature vector of the edge connecting node and node in the th iteration, is the aggregated information from the neighbor nodes to the central node in the the iteration, and
is the updated hidden representation of node
in the th iteration. The IGCNet is trained in an unsupervised manner to learn the optimal power control.Y. Shen et al. also modeled wireless networks as wireless channel graphs and formulated the radio resource management problems as graph optimization problems [gnnResourceMng2021]. Then they proposed a family of neural networks, i.e., message passing graph neural networks (MPGNNs) to solve the largescale radio resource management problem. It demonstrates that MPGNNs satisfy the permutation equivariance property and have the ability to address the largescale problems while enjoying a high computational efficiency. For an effective implementation, this work proposed a wireless channel graph convolution network (WCGCN) belonging to the MPGNNs class. Then, the authors tested the effectiveness of WCGCN for power control and beamforming problems. It demonstrates that WCGCN matches or outperforms classic optimizationbased algorithms without domain knowledge, and with significant speedups.
M. Eisen et al. also considered the problem of optimal power allocation across a set of transmitters and receivers in wireless networks and proposed a GNNbased scheme, i.e., the random edge graph neural networks (REGNN) [REGNN2020]. Different from the work in [gnnWPC2019], REGNN performs convolutions over random graphs formed by the fading interference patterns in wireless networks. Specifically, suppose there are filter coefficients defining graph filters in the th layer. The intermediate feature in the th layer is produced as follows
(10) 
The REGNN operator is defined by , where . The final problem to be solved by this work is
(11)  
where is a function with respect to the joint space of resource allocations and states to the space of rewards. and denote the utility function and constraint function, respectively. The constrained objective function cannot be directly used to train REGNN, so the authors further presented an unsupervised modelfree primaldual learning algorithm to train the weights of the REGNN [LSWgnn2019]. REGNN is also utilized to solve power control in decentralized wireless networks [FastPowerControl2021]. In order to enable the fast adaption of power control policy to timevarying topologies, the authors applied firstorder metalearning on data from multiple topologies for a fewshot adaptation to new network configurations.
N. Naderializadeh et al. considered the problem of downlink power control in wireless networks, consisting of multiple transmitterreceiver pairs communicating with each other over a single shared wireless medium [wpcCOgnn2020]. The counterfactual optimization technique is applied to guarantee a minimum rate constraint, which adapts to the network size, hence achieving the balance between average and th percentile user rates throughout a range of network configurations. The GNNbased model is also trained based on primaldual learning framework.
The previous methods are all based on datadriven neural networks, which have poor interpretability and scalability. Different from the previous works, A. Chowdhury et al. proposed a hybrid datamodel driven neural architecture inspired by the algorithmic unfolding of the iterative WMMSE, i.e., unfolded WMMSE (UWMMSE), to solve the optimal power allocation problem in a singlehop adhoc wireless network [FpaGNNdau]. The optimization problem should be solved is
(12)  
where and are variables should be optimized. The allocated power is computed by a function of the channel state matrix through a layered architecture with trainable weights . Precisely, setting , the th layer of UWMMSE is implemented as follows
(13a)  
(13b)  
(13c)  
(13d) 
and the output power is determined as . simply ensures that . The function parameterized by is chosen to be GCNs. The whole workflow of UWMMSE is shown in Fig. 6. Numerical experiments demonstrates that UWMMSE significantly reduced the computational complexity compared to the conventional WMMSE [UnfoldGnnPA]. Compared with neural network models, unfolding methods have better interpretability and scalability via stacking several trainable layers that correspond to modeldriven iterations.
J. Guo et al. considered the power control problem in multicell cellular networks [LpcCSgnn]. Specifically, this work models the cellular networks as a heterogeneous graph, i.e., wireless interference graph (WIG), and then proposed a heterogeneous GNN (HetGNN), called PGNN that satisfies the permutation invariance (PI) and the permutation equivalence (PE), to learn to optimize power control in multicell cellular networks. It’s worth noting that this work first finds the PI and PE properties of the relationship between the optimal transmit powers and channels, inspired by the finding that the parameter sharing scheme determines the invariance or equivalence relationship [EqParaShare]. For the WIG, the aggregator and the combiner of PGNN are designed as follows,
BSs aggregating information from UEs
(14)  
UEs aggregating information from BSs
(15)  
where and are model parameters in the combination function, and are used to aggregate the information from UEs and BSs, and and are used to aggregate the information from edges, respectively. To distinguish the hidden outputs between BSs and UEs, and are the hidden output of and , respectively. Numerical results illustrated that both the sample complexity for training and the model size of PGNN for learning the optimal power control policy in multiuser multicell networks are much lower than the existing DNNs, when achieving the same sum rate loss from the numerically obtained solutions.
X. Zhang et.al considered the joint user association and power control problem in heterogeneous ultradense networks (HUDNs) [JUAPAinHUDN]. The HUDNs are modeled as a heterogeneous graph, which is shown in Fig. 7. A heterogeneous GraphSAGE (HGSAGE) that extended from GraphSAGE [IdcRLLG2018], used to extract the latent node representations. To embrace both the generalization of the learning algorithm and higher performance of HUDNs, the learning process of HUDNs is divided into two parts, i.e., the generalizationrepresentation learning (GRL) part and the specializationrepresentation learning (SRL) part. In the GRL part, the GNN learns a representation with a tremendous generalized ability to suit any scenario with different user distributions, which processes in an offline manner. Based on the learned GRL representation, the SRL finely tune the parameters of GNN online to further improve the performance for quasistatic user distribution.
X. Zhang et al. focused on power control/beamforming in heterogeneous devicetodevice (D2D) networks [PowerContrlBF2021]. Different from the previous works, this work considers a heterogeneous D2D network with two types of links. Each kind of links holds different features, so does the interference it cause. The heterogeneous D2D network and corresponding heterograph is depicted in Fig. 8. Suppose the interference from link type to link type is characterized as . Take node as an example, the update rules in relation is defined as follows
(16a)  
(16b) 
where is an edge update function of relation , is a vertex update function of relation . denotes the th vertex with link type . is the attribute vector of vertex , is the attribute vector of the edge between vertex and . Then, the aggregation rules is defined as follow
(17) 
where is the number of relations causing interference to link . is the aggregation function.
Z. Gao et al. investigated the optimal power assignment and node selection in free space optical fronthaul networks based on the instantaneous channel state information of the links [ResAllocGNN]. GNNs are utilized to exploit the FSO network structure with smallscale training parameters. Then a primaldual learning algorithm is developed to train the GNN in a modelfree manner.
Z. Wang et al.
addressed the asynchronous decentralized wireless resource allocation problem with a novel unsupervised learning approach
[UspLrnARAadhoc]. Specifically, the interference patterns between transmitting devices are modeled as a graph capturing the asynchrony patterns via the activation of the graph edges on a highly granular time scale. A decentralized learning architecture, i.e., the aggregation graph neural networks (AggGNNs) is designed based on the graph representation of interference and asynchrony. The AggGNN policy leverages successive local state exchanges to build local embeddings of the global network state at each node, which is then processed through CNNs to determine resource allocation decisions. Such a policy is trained offline in an unsupervised manner that captures network performance under asynchronous conditions without explicit modeling.B. Link Scheduling
Wireless communication links are affected by signal fading, interference and noise, etc. Although power control can improve the overall performance of wireless networks, when the wireless network is a ultradense network, power control alone is not enough to eliminate the strong interference between shortrange links. In this case, link scheduling is one of the effective means to further improve the performance of the wireless networks.
To overcome the high computational complexity of the traditional methods and eliminate the costly channel estimation, M. Lee et al. proposed a novel DL based graph embedding for link scheduling in D2D networks [graphEmbedWLS]. In detail, this work first models the D2D network as a fullyconnected directed graph, then computes a lowdimensional feature vector based on the distances of both communication and interference links without requiring the accurate channel state information for each node. Finally, a multilayer classifier is utilized to learn the scheduling policy in a supervised manner and unsupervised manner, respectively.
A distributed schedule transmission scheme for wireless networks is proposed to overcome the difficulty encountered in solving the maximum weighted independent set (MWIS) problem [DistributeSchedgnn]. The authors first proposed a GCNbased distributed MWIS solver for link scheduling by combining the learning capabilities of GCNs and the efficiency of greedy MWIS solvers. The proposed solver can be trained on simulated networks and generalizes well across different types of graphs and utility distributions.
In order to tackle the challenging temporal link prediction task of dynamic networks, K. Lei et al. introduced a novel nonlinear GCNGAN model by leveraging the benefits of GCNs, LSTM as well as the GANs [GcngcnLP2019]. Consequently, the dynamics, the topology structure and evolutionary patterns of dynamic networks can be fully exploited to improve the temporal link prediction performance. In particular, GCNGAN firstly utilizes the GCN to explore the local topology characteristics of each single graph snapshot to obtain comprehensive representations. Then the comprehensive representations are fed into an LSTM network to capture the evolving patterns of dynamic graph. Finally, GAN is applied to generate highquality predicted graph snapshot with an adversarial process, where generative network consists of the GCN and the LSTM, discriminative network consists of a fullyconnected network.
In ultradense D2D mmWave communication networks, S. He et al. considered the problem of joint beam selection and link activation across a set of communication pairs to effectively control the interference between communication pairs via inactivating part communication pairs [he2021gblinks]. The resulting optimization problem is formulated as an constrained integer programming problem that is nonconvex and NPhard. To solve this problem efficiently, an unsupervised learning model, i.e., GNNbased Beam selection and Link scheduling (GBLinks), which based on spatialbased GCNs. To satisfy the constraints of the optimization problem, GBLinks is trained using Lagrangian dual learning framework. Numerical results show that the proposed GBLinks model can converges to a stable point with the number of iterations increases, in terms of the average sum rate. It also shows that GBLinks can reach nearoptimal solution through comparing with the exhaustively search in smallscale D2D mmWave communication networks and outperforms greedy search scheme.
C. Channel Control
To improve the spectral efficiency in densely deployed wireless local area networks (WLANs), K. Nakashima et al. proposed a GCNbased deep RL model for channel allocation [DeepRLCAgnn2020]. The idea behind their work is that the objective function is modeled as a parametric function of topologies, channels and communication quality. Specifically, graph convolutional layers are adopted to extract the features of channel vectors with topology information, which is the adjacency matrix of the graph depending on the carrier sensing relationships. Then, a selective data buffering scheme was proposed to prevent overfitting by reducing the duplication of the sampling data specific to WLAN channel allocation problems. The main learning algorithm of this method was double DQN employing dueling network and prioritized experience replay.
Generally speaking, channel tracking for massive MIMO highmovement networks is a challenging task. Y. Yan et al. proposed a new channel tracking method based on GNNs [GnnCTMIMO2020]
. The authors first used a small number of pilots to achieve initial channel estimation, and represented the obtained channel data in the form of a graph. Then, they described the spatial correlation of channels through the weight of the edges in the graph. Finally, this work also designed a GNNbased channel tracking framework, which includes encoder, core network and decoder. The numerical results confirmed that the GNNbased scheme proposed can achieve better performance than the scheme using feedforward neural network.
K. Tekbıyık et al. first used GAT for channel estimation [ChannelEstGAN]. The performance of the proposed method is studied on the twoway backhaul link of highaltitude platform stations with reconfigurable intelligent surfaces. The numerical results show that for the fullduplex channel estimation, the performance of the GAT estimator is better than the least squares. Contrary to the previous studied method, GAT has the ability to estimate the concatenated channel coefficients at each node separately. Therefore, there is no need to use the time division duplex mode during the pilot signaling in the fullduplex communication. Moreover, the numerical results also show that even if the training data does not include all changes, the GAT estimator is robust to hardware impairments and smallscale fading characteristics changes. A. Scotti et al. considered the inference task of massive MIMO detection under timevarying channels and higherorder qadrature amplitude modulation and proposed a messagepassing solution based on GNNs, i.e., MIMOGNN [GnnMIMOdetect].
D. Wireless Traffic Prediction
As we all know, wireless network resources are limited. Effective resource management can improve the utilization of network resources. Traffic prediction is a key scientific issue, which is helpful to manage the network resources. However, due to the network traffic has high interdependencies of spatial and temporal, traffic prediction is a challenging issue. There are less works on wireless network traffic prediction using GNNs, mainly including cellular network traffic prediction and satellite network traffic prediction.
Due to the existing works that do not explicitly consider the impact of handover on the spatial characteristics of cellular traffic, which may result in lower prediction accuracy. S. Zhao et al. proposed a new prediction model STGCNHO, which uses the transition probability matrix of the handover graph to improve the accuracy of traffic prediction [CellNetTPgnn2020]. STGCNHO builds a stacked residual neural network structure that combines graph convolution and CNN with gated linear units [GLU2017] to capture the spatial and temporal relationships of traffic. Unlike RNN, STGCNHO trains quickly and predicts the traffic demand of all base stations at the same time based on the information collected from the entire graph. Unlike the CNN grid, STGCNHO can predict not only the base station, but also the cells within the base station.
In reference to the satellite network traffic prediction, L. Yang et al. pointed out that the traditional network traffic prediction model could not effectively extract the spatiotemporal characteristics of network traffic, and then proposed a combination of GCNs and GRU network traffic prediction model [StlNetTPgcngru2020]. On the basis of using the GCN model to learn the satellite network topology and extracting the spatial characteristics of the satellite network traffic, the data with the spatial characteristics is used as the input of the GRU model to discover the temporal change of the satellite node attributes, and then extract the temporal characteristics of the satellite network traffic, and finally predict satellite network traffic through the fully connected layer.
E. Vehicular Communication
GNNs have relatively few applications in vehicle communication networks. Recently, GNNs have been used to control the connected autonomous vehicles (CAVs) lane changing decisions for a road segment, to mitigate highway bottleneck congestion, and to allocate spectrum in vehicletoeverything (V2X) networks.
J. Dong et al. proposed a DL model that combines GCN and a deep Q network, i.e., GQN, to control multiple CAVs to make cooperative lane change decisions [GCQNcavNet]. From the perspective of CAV operations, the proposed model not only enables CAV to successfully carry out lane changes to meet its personal intention of merging from the prescribed ramp, but also guarantees safety and efficiency. The authors also demonstrated the effectiveness of the model in the following aspects: (a) Solving the dynamic digital agency problem, specifically for driving tasks where the model has high flexibility; (b) Integrating relevant local and global information obtained through cooperative perception Information; (c) Make safe and collaborative decisions based on the fused information; (d) Have sufficient robustness under the condition of different traffic densities, and make consistent decisions without retraining the model decisionmaking.
P. Y. J. Ha et al. applied RL algorithms to train CAV driving behaviors, which can be used to relieve highway bottleneck congestion [cavMARL]. In the highway environment, the challenge for RL is the dynamic number of vehicles and continuous vehicle control parameters. In order to solve the continuous actions required for CAV control, the authors used a deep deterministic policy gradient algorithm to train the RL agent. Due to the dynamic input length caused by the large change in the number of vehicles observed, GCN is used. This method has been applied to alleviate highway congestion, especially those caused by physical highway bottlenecks caused by continuous lane descent. Multiple CAVs could be trained to perform collaborative operations to increase throughput and reduce speed deviations in traffic before the bottleneck.
Z. He et al. studied the spectrum allocation in V2X networks [RAgnnVC2020]. The authors first modeled the V2X network as a graph, where each vehicletovehicle link is a node in the graph. Then, GNNs are applied to learn the lowdimensional representations of each node from graph structure information. According to the learned characteristics, multiagent RL is used for spectrum allocation. DQN is used to learn to optimize the total capacity of the V2X network.
F. Point Cloud
There are several works solving some tasks in point cloud using GNNs. J. Shao et al. proposed BranchyGNN for efficient point cloud processing [BranchyGNN]. BranchyGNN uses branch network and source channel coding to reduce the computational cost and intermediate feature transmission overhead on the device, respectively. Numerical results show that the proposed framework can ensure lower inference delay than other benchmarks. T. Fujihashi et al. proposed a novel soft point cloud transmission scheme for future wireless streaming of holographic and threedimensional data [Wireless3Dpcgnn]. Specifically, the proposed scheme combines GNNbased point cloud coding and nearanalog modulation to simultaneously achieve: 1) prevent cliff effect, 2) prevent leveling effect, 3) high energy compression, and 4) low communication overhead. In addition, the endtoend DL framework based on GAE solves the random distortion caused by the fading channel by using pre/post equalization and precoding technology. C. S. Jeong et al. proposed a system that can provide AR services via 3D GNN using cameras and sensors on mobile devices [ARanchor2021].
G. Others
In addition to the aforementioned related work, GNNs are also used to solve some other problems in wireless networks. S. Lohani et al. introduced a method for maximizing the throughput of unmanned aerial vehicle (UAV)assisted ground network [FSOAMC2019]. Throughput maximization involves optimizing UAV trajectory to minimize delay and packet loss, thereby enhancing congested nodes and transmission channels. Nodes, links, and overall topology are characterized by delay, loss, throughput, and distance enables active enhancement strategies. Location awareness GNN is used for characterization, prediction and dynamic UAV trajectory enhancement.
W. Yan et al. solved the network localization of a wireless network in twodimensional (2D) space using GCNs [NetworkLoc2021]. The adjacent matrix of an undirected graph corresponding a wireless network is constructed as follows
(18) 
where is a Euclidean distance threshold to determine whether there is an edge between two nodes or not. is the distance between node and . The corresponding augmented adjacency matrix is defined as . The graph convolution operation in the th layer is defined as , where ,
is a nonlinear activation function, and
. The initialization is , where . M. Lee et al. analyzed and enhanced the robustness of the decentralized GNN in different wireless communication systems, making the prediction results not only accurate but also robust to transmission errors [DecentralizedInfer2021].IV. Key Issues and Future Development
According to the current review results, we summarize some ongoing or future research directions that are worth exploring:
It can be found that most of the works based on GNNs focus on solving wireless network resource management problems. However, other topics in wireless networks are still in their infancy. In addition, most of the current works solve wireless network problems on a smallscale, how to effectively apply GNNs to solve largescale wireless network problems is a topic worth exploring.
As we all know, wireless network related topics have extremely high requirements for the stability and effectiveness of algorithms, which is also a big obstacle facing deep learning algorithms at present. Therefore, models with high efficiency and strong robustness are important and urgent. Effectively solving this problem will be a milestone achievement.
The corresponding optimization problem of wireless network usually are constrained. Although there are related solutions to solve the constrained optimization problems, such as Lagrangian multiplier method and reinforcement learning, the training of models based on these methods is difficult, and the processing of constraints may not be completely effective. How to effectively deal with complex or large scale constraints is still a longterm topic.
Deep unfolding is an effective combination of modeldriven methods and datadriven methods, which not only effectively utilizes the interpretability and scalability of modeldriven algorithms, but also uses the expressive power of datadriven methods. Although deep unfolding has been studied a lot, the problem to be solved has fewer simple constraints, and there is a little work combining GNNs to design unfolding models. There is still a lot of work to be explored.
As we all know, high dynamics is a major feature of wireless networks, and currently there are few GNNbased work to solve some problems in high dynamic wireless network scenarios. The reasonable and effective use of some inherent characteristics of GNNs will likely contribute to such problems.
V. Conclusions
In this paper, we first introduce some basic paradigms of GNNs that have applied in wireless networks, and make a classified introduction to the application of GNNs in wireless network scenarios, mainly including power control, link scheduling, channel control, wireless traffic prediction, vehicular communication, point cloud, and other directions. From the review results, the application of GNNs in wireless networks is still in its infancy. There are many problems to be solved and further improved, and it will also face many challenges. Finally, based on the existing results, we suggest some future directions for participators interested in this domain.
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