# An Overview on the Application of Graph Neural Networks in Wireless Networks

Shiwen He, , Shaowen Xiong, Yeyu Ou, Jian Zhang , Jiaheng Wang, , Yongming
Huang, , and Yaoxue Zhang S. He, S. Xiong, Y. Ou, and J. Zhang are with the
School of Computer, Central South University, Changsha 410083, China. S. He is
also with the Purple Mountain Laboratories, Nanjing 210096, China. (email:
{shiwen.he.hn, shaowen.xiong, ouyeyu, jianzhang}@csu.edu.cn).J. Wang and Y.
Huang are with the National Mobile Communications Research Laboratory, School
of Information Science and Engineering, Southeast University, Nanjing 210096,
China. They are also with the Purple Mountain Laboratories, Nanjing 210096,
China. (email: {jhwang, huangym}@seu.edu.cn). Y. Zhang is with the Department
of Computer Science and Technology, Tsinghua University, Beijing 100084,
China. (email<EMAIL_ADDRESS>
###### Abstract

With the rapid enhancement of computer computing power, deep learning methods,
e.g., convolution neural networks, recurrent neural networks, etc., have been
applied in wireless network widely and achieved impressive performance. In
recent years, in order to mine the topology information of graph-structured
data in wireless network as well as contextual information, graph neural
networks have been introduced and have achieved the state-of-the-art
performance of a series of wireless network problems. In this review, we first
simply introduce the progress of several classical paradigms, such as graph
convolutional neural networks, graph attention networks, graph auto-encoder,
graph recurrent networks, graph reinforcement learning and spatial-temporal
graph neural networks, of graph neural networks comprehensively. Then, several
applications of graph neural networks in wireless networks such as power
control, link scheduling, channel control, wireless traffic prediction,
vehicular communication, point cloud, etc., are discussed in detail. Finally,
some research trends about the applications of graph neural networks in
wireless networks are discussed.

###### Index Terms:

Wireless networks, graph neural networks, machine learning, deep neural
networks.

## 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 DL-based methods,
especially “learning optimization” methods, to solve tricky optimization
problems in wireless network. This method aims to achieve sub-optimal
performance in real-time 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 “end-to-end 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, multi-layer perceptrons (MLPs)
are used to approximate the input and output mapping of the weighted minimum
mean square error (WMMSE) algorithm [1]. 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 [2]. N. Shlezinger  _et al._ designed a deep neural network
(DNN) to implement the specific parts of Viterbi algorithm that are channel-
model-based, leaving the rest of the algorithm structure intact [3]. K. Gregor
_et al._ designed a non-linear, parameterized, feed-forward architecture with
a fixed depth that can be trained to approximate the optimal sparse code [4].

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 [5] or convolutional neural networks (CNNs) to generate the learning
model [6]. Although they have achieved good performance in small-scale
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 non-Euclidean. When using the “learning
optimization” method in wireless networks, it is necessary to use a neural
network that works in non-Euclidean 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 [7], DL applied in
different wireless network layers [8], the application of ML in improving the
quality-of-service (QoS) and quality-of-experience (QoE) of wireless networks
[9], the application of ML in improving the QoS and QoE of 6G networks [10].
However, the application of DL or ML in wireless network are mainly utilizing
the classical DL or ML methods that cannot effectively capture the non-
Euclidean 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) [11] and deep sets [12]. 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
[13, 11]. Besides, graph reinforcement learning and graph adversarial methods
were reviewed in [14]. The GNN model is divided into two categories, i.e.,
spatial-based GNNs and spectral-based GNNs, and the connections between
categories are revealed in [15]. The expressive power of GNNs and some GNN
variants was introduced comprehensively in [16]. The current methods of using
software or hardware to accelerate GNNs were reviewed from a computational
perspective [17]. The interpretability of GNNs was introduced deeply in [18].
In addition, there are also several overviews on the applications of GNNs,
such as the method of solving the problems in traffic domain [19], the
application of GNNs in power system [20], GNNs for recommender system [21].
The relationship between GNNs and the current development of neural-symbolic
computing was introduced in [22].

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 graph-structured 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

Figure 1: Overview of the paradigms of GNNs.

In this section, due to the basic paradigms of GNNs have been fully elaborated
in [11, 14], 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 spatial-temporal 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.

TABLE I: Definitions of Graph-Structured Data $G$ | Graph
---|---
$N$ | The number of nodes
$V$ | The set of nodes
$v$ | A node $v\in V$
$\mathcal{N}(v)$ | The set of neighbor nodes of $v\in V$
$E$ | The set of nodes
$e_{ij}$ | An edge connecting nodes $i$ and $j$
$\mathbf{A}$ | Adjacent matrix
$\mathbf{A}^{T}$ | The transpose of matrix $\mathbf{A}$
$\mathbf{A}_{ij}$ |  The element of the $i$-th row and the $j$-th column of matrix $\mathbf{A}$
$\mathbf{D}$ | The degree matrix of matrix $A$
$\mathbf{D}^{-\frac{1}{2}}$ |  $-\frac{1}{2}$ power of the elements in the matrix $\mathbf{D}$
$\mathbf{X}$ | Feature matrix
$\mathbf{X}_{i}$ | The $i$-th column of matrix $\mathbf{X}$
$\mathbf{x}[l]$ | Denotes the $l$-th element of vector $\mathbf{x}$
$d$ | The dimension of the feature vector
$s$ | The sample size
$L$ | The maximum depth of GNNs
$\mathcal{H}\left(\bm{\mu},\bm{\sigma}^{2}\right)$ | Multivariate Gaussian

### A. Basic Definitions of Graph-Structured Data

Before introducing the paradigms of GNNs, we would like to describe several
basic definition of graph-structured data firstly. Graph structural
information is a kind of data in non-Euclidean space, which is commonly
expressed as $G=(V,E)$, where $E$ is the set of edges and $V$ is the set of
nodes. Let $v_{i}\in V$ to denote a node and $e_{ij}=(v_{i},v_{j})\in E$ to
denote an edge pointing from $v_{j}$ to $v_{i}$. The adjacency matrix of graph
is represented as $\mathbf{A}$. If $e_{ij}\in E$, $\mathbf{A}_{ij}=1$,
otherwise, $\mathbf{A}_{ij}=0$. If adjacency matrix $\mathbf{A}$ 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 1-hop neighbor
nodes. For undirected graphs, the degree matrix $\mathbf{D}$ of a graph is a
diagonal matrix, where $\mathbf{D}_{ii}=|\mathcal{N}(v_{i})|$. The Laplacian
matrix of an undirected graph is defined as
$\mathbf{L}=\mathbf{D}-\mathbf{A}$. The symmetric normalized Laplacian matrix
is defined as
$\tilde{\mathbf{L}}=\mathbf{I}_{N}-\mathbf{D}^{-\frac{1}{2}}\mathbf{AD}^{-\frac{1}{2}}$.
Note that the symmetric normalized Laplacian matrix is a real symmetric semi-
positive definite matrix, it can be decomposed into
$\tilde{\mathbf{L}}=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{T}$, where
$\mathbf{U}$ is the eigenvector matrix and $\mathbf{\Lambda}$ is the diagonal
matrix of $[\mathbf{\Lambda}]_{ii}=\lambda_{i}$ with $\lambda_{i}$ being the
eigenvalue. In a graph, each node may have its own attribute information,
which is represented by feature matrix $\mathbf{X}\in\mathbb{R}^{N\times{d}}$,
where $d$ is the dimension of feature vectors. If feature matrix $\mathbf{X}$
changes over time, the graph is defined as a spatial-temporal graph.

### B. Graph Convolutional Neural Networks

Graph convolutional neural networks (GCNs) implement convolutional operation
on graph-structured data. In other words, the convolution operation of GCNs is
transformed from Euclidean space to non-Euclidean space[23]. 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 spectral-based GCNs [24, 25, 26, 27, 28, 29] and spatial-based
GCNs [30, 31, 32, 33]. In the sequel, we simply introduce several classical
models in terms of spectral-based GCNs and spatial-based GCNs.

1) Spectral-based 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 graph-structured data is
converted to the frequency domain. Given the input graph feature matrix
$\mathbf{X}$ and a graph filter $\mathbf{g}\in\mathbb{R}^{N\times d}$, the
graph convolution to $\mathbf{X}$ can be defined as [11]

$\displaystyle\mathbf{X}*_{G}\mathbf{g}$
$\displaystyle=\mathcal{F}^{-1}\left(\mathcal{F}(\mathbf{X})\odot\mathcal{F}(\mathbf{g})\right)$
(1)
$\displaystyle=\mathbf{U}\left(\mathbf{U}^{T}\mathbf{X}\odot\mathbf{U}^{T}\mathbf{g}\right)=\mathbf{U}\hat{\mathbf{g}}\mathbf{U}^{T}\mathbf{X},$

where $*_{G}$ denotes the graph convolution operation,
$\mathcal{F}(\mathbf{X})=\mathbf{U}^{T}\mathbf{X}$ denotes the graph Fourier
transform, $\mathcal{F}^{-1}(\mathbf{X})=\mathbf{U}\mathbf{\hat{\mathbf{X}}}$
denotes the inverse graph Fourier transform, $\odot$ denotes the Hadamard
product, and
$\hat{\mathbf{g}}=\text{diag}\left(\mathbf{U}^{T}\mathbf{g}\right)$.

Different spectral-based GCNs can be defined by changing $\hat{\mathbf{g}}$.
For example, Bruna _et al._ proposed Spectral convolutional neural network
(SpectralCNN) in which $\hat{\mathbf{g}}$ is learnable [24]. However, due to
the existing of the eigen-decomposition of $\tilde{\mathbf{L}}$, 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 [25]. 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.,

$\displaystyle\mathbf{X}*_{G}=\mathbf{W}\left(\mathbf{I}_{N}+\mathbf{D}^{-\frac{1}{2}}\mathbf{AD}^{-\frac{1}{2}}\right)\mathbf{X},$
(2)

where $\mathbf{W}$ is learnable weight matrix. In order to solve the situation
that
$\mathbf{I}_{N}+\mathbf{D}^{-\frac{1}{2}}\mathbf{AD}^{-\frac{1}{2}}\in(0,2)$
may lead to gradient explosion, the authors further transform
$\mathbf{I}_{N}+\mathbf{D}^{-\frac{1}{2}}\mathbf{AD}^{-\frac{1}{2}}$ into
$\widetilde{\mathbf{D}}^{-\frac{1}{2}}\widetilde{\mathbf{A}}\widetilde{\mathbf{D}}^{-\frac{1}{2}}$,
where $\widetilde{\mathbf{A}}=\mathbf{A}+\mathbf{I}_{N},\
\widetilde{\mathbf{D}}_{ii}=\sum\nolimits_{j}\widetilde{\mathbf{A}}_{ij}$.

By comparing Chebnet with GCN, Chebnet has higher computational complexity and
a large number of parameters, but it has stronger expression ability.
Chebnet’s $K$-order convolution operator can cover maximum $K$ steps neighbor
nodes of the central node, while GCN only covers the first-order 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 spectral-based GCNs to solve wireless
network problems, which will be introduced in Section III, the definition of
graph filters for spectral-based GCNs usually combines the adjacency matrix
$\mathbf{A}$ of wireless network topology and the channel state information
$\mathbf{H}$, 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) Spatial-based GCNs. The spatial-based 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 spatial-based 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 spatial-based
convolution operation cannot use a fixed-size convolution kernel like the
image convolution operation. Thus, the key of spatial-based GCNs is to define
convolution operation with different neighborhood numbers and keep local
invariance.

According to the overview result, the most widely used spatial-based GCNs in
wireless network problems are mainly including Message passing neural network
(MPNN) and diffusion-convolutional neural networks (DCNNs). MPNN was proposed
in [31], which is a unified framework of spatial-based GCNs. MPNN decomposes
the spatial-based graph convolution into a message aggregation phase and a
combination phase, i.e.,

$\displaystyle\text{Aggregation:}~{}~{}\mathbf{m}_{v}^{(t)}$
$\displaystyle=\sum\limits_{u\in\mathcal{N}(v)}M_{t}\left(\mathbf{h}_{v}^{(t-1)},\mathbf{h}_{u}^{(t-1)},\mathbf{e}_{vu}\right),$
(3) $\displaystyle\text{Combination:}~{}~{}\mathbf{h}_{v}^{(t)}$
$\displaystyle=U_{t}\left(\mathbf{h}_{v}^{(t-1)},\mathbf{m}_{v}^{(t)}\right),$

where $\mathbf{e}_{vu}$ is the feature representation of the edge between node
$v$ and node $u$, $M_{t}(\cdot)$ and $U_{t}(\cdot)$ are the aggregation
function and the combination function in the $t$-th iteration, respectively.
$\mathbf{m}_{v}^{(t)}$ is the message aggregated from node $v$’s neighbors and
$\mathbf{h}_{v}^{(t)}$ is the new feature representation of node $v$ in the
$t$-th iteration. W. L. Hamilton further proposed GraphSage model via fixing
the number of neighbors for message passing [32], 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

$\displaystyle\mathbf{h}_{v}^{(t)}=\sigma\left(\mathbf{W}^{(t)}g_{t}\left(\mathbf{h}_{v}^{(t-1)},\left\\{\mathbf{h}_{u}^{(t-1)},\forall
u\in S_{\mathcal{N}(v)}\right\\}\right)\right),$ (4)

where $g_{t}(\cdot)$ is an aggregation function in the $t$-th layer,
$S_{\mathcal{N}(v)}$ is a random sample of the node $v$’s neighbors.

The diffusion-convolution 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,
$\mathbf{Z}^{(t)}=\sigma\left(\mathbf{W}^{(t)}\odot\mathbf{P}^{t}\mathbf{X}\right)$,
where $\mathbf{Z}^{(t)}$ is the hidden state in the $t$-th layer,
$\mathbf{P}=\mathbf{D}^{-1}\mathbf{A}$ is the probability transition matrix
and $\mathbf{P}^{t}$ denotes the $t$ power of $\mathbf{P}$.

######  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 spatial-based
GCNs, and there is no work on designing heterogeneous graph convolutions based
on spectral-based GCNs.

### C. Graph Attention Networks

In deep neural networks, attention mechanism [34] 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 graph-structured data and performed the aggregation operation on neighbor
nodes to realize the adaptive allocation of weight to different neighbors
[35], i.e., Graph Attention Network (GAT), which is defined as follows

$\displaystyle\mathbf{h}_{v}^{(t)}=\sigma\left(\sum_{u\in\mathcal{N}(v)}\alpha\left(\mathbf{h}_{v}^{(t-1)},\mathbf{h}_{u}^{(t-1)}\right)\mathbf{W}^{(t-1)}\mathbf{h}^{(t-1)}_{u}\right),$
(5)

where $\alpha(\cdot)$ is the attention mechanism, $\mathbf{h}^{(t)}_{v}$ is
the hidden state of node $v$ at the $t$-th layer. Multi-head attention
mechanism can be implemented to further improve the expression ability of the
attention layer, that is, $K$ independent attention mechanisms can be utilized
and then the output results are concatenated together, i.e.,

$\displaystyle\mathbf{h}_{v}^{(t)}=\|^{K}_{k=1}\sigma\left(\sum_{u\in\mathcal{N}(v)}\alpha^{k}\left(\mathbf{h}_{v}^{(t-1)},\mathbf{h}_{u}^{(t-1)}\right)\mathbf{W}^{(t-1)}\mathbf{h}^{(t-1)}_{u}\right),$
(6)

where $\|$ denotes the concatenation operation and $\alpha^{k}(\cdot)$ is the
$k$-th attention mechanism.

Figure 2: Illustration of the implementation details of 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 Auto-Encoder

Inspired by the conventional auto-encoders, graph auto-encoders (GAEs) utilize
GNNs as encoders to learn low-dimensional 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 [36]. The
general overview of GAEs is shown in Fig. 3. Specifically, the encoder maps
the node $v$ to a low-dimensional vector embedding $\mathbf{z}_{v}$ based on
the node’s structural information, and the decoder extracts the information
that the user interested from the low-dimensional vector embedding.

Figure 3: Overview of GAEs.

GAEs have been used in many fields by virtue of their concise encoder-decoder
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 [37]. The encoder maps each
node to a low-dimensional latent representation using GCN, then a network
embedding $\mathbf{Z}$ can be obtained. The decoder computes the pair-wise
distance given network embedding and applies a non-linear activation. Finally,
the decoder outputs the reconstructed adjacency matrix.

######  Remark 4.

The main advantage of the graph auto-encoder 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 long-term information propagation across the graphs,
there is a trend to utilize the gate mechanism from RNNs, such as long-short
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 long-term 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 [38]. GGNN unrolls the recurrence to a fixed number of
steps $L$ and use backpropagation through time in order to compute gradients.
The basic recurrence paradigm is designed as follows,

$\displaystyle\mathbf{h}_{v}^{(1)}=\left[\mathbf{X}_{v}^{T},\bm{0}\right]^{T},$
(7a)
$\displaystyle\mathbf{a}_{v}^{(t)}=\mathbf{A}_{v:}^{T}\left[\mathbf{h}_{1}^{(t-1)T}\cdots\mathbf{h}_{|\mathcal{V}|}^{(t-1)T}\right]^{T}+\mathbf{b},$
(7b)
$\displaystyle\mathbf{z}_{v}^{(t)}=\sigma\left(\mathbf{W}^{z}\mathbf{a}_{v}^{(t)}+\mathbf{U}^{z}\mathbf{h}_{v}^{(t-1)}\right),$
(7c)
$\displaystyle\mathbf{r}_{v}^{(t)}=\sigma\left(\mathbf{W}^{r}\mathbf{a}_{v}^{(t)}+\mathbf{U}^{r}\mathbf{h}_{v}^{(t-1)}\right),$
(7d)
$\displaystyle\widetilde{\mathbf{h}_{v}^{(t)}}=\mathrm{tanh}\left(\mathbf{W}\mathbf{a}_{v}^{(t)}+\mathbf{U}\left(\mathbf{r}_{v}^{(t)}\odot\mathbf{h}_{v}^{(t-1)}\right)\right),$
(7e)
$\displaystyle\mathbf{h}_{v}^{(t)}=(1-\mathbf{z}_{v}^{(t)})\odot\mathbf{h}_{v}^{(t-1)}+\mathbf{z}_{v}^{(t)}\odot\widetilde{\mathbf{h}_{v}^{(t)}},$
(7f)

where
$\mathbf{A}=[\mathbf{A}^{(out)},\mathbf{A}^{(in)}]\in\mathbb{R}^{d|\mathcal{V}|\times
2d|\mathcal{V}|}$ determines how nodes in the graph communicate with each
other, $\mathbf{A}_{v:}\in\mathbb{R}^{d|\mathcal{V}|\times 2d}$ are the two
columns of blocks in $\mathbf{A}^{(out)}$ and $\mathbf{A}^{(in)}$
corresponding to node $v$, $\mathbf{b}$ is a bias item.
$\mathbf{a}_{v}^{(t)}\in\mathbb{R}^{2d}$ contains activations from edges in
both directions. $\mathbf{W}^{\prime}s$ and $\mathbf{U}^{\prime}s$ are weight
matrices. N. Peng _et al._ also explored a general framework for cross-
sentence $n$-ary relation extraction based on graph LSTMs [39].

######  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, spatial-temporal
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 [40]. 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 non-differentiable objective
function or constraints, while RL can solve this problem by learning from
feedback. In recent years, RL has been gradually applied to graph-structured
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 non-differentiable objective functions and constraints [41]. 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
domain-specific rewards and adversarial rewards. GCPN is trained in an end-to-
end manner via using proximal policy gradient.

Graph attention model (GAM) utilized RL to solve graph classification task
based on random walks [42]. 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 $\mathcal{J}(\theta)=\mathbb{E}_{P(S_{1:L};\theta)}\sum_{l=1}^{L}r_{l}$,
where $r_{l}=1$ if the GAM classified the graph correctly, otherwise,
$r_{l}=-1$. $S_{l}$ is the environment. W. Xiong _et al._ proposed a model,
i.e., DeepPath, for knowledge graph reasoning [43]. 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. Spatial-Temporal Graph Neural Networks

For some wireless network problems, the graph structure and graph input may be
dynamical. Spatial-temporal 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, RNN-based methods and CNN-based
methods.

C. Chen _et al._ utilized residual recurrent graph neural networks (Res-RGNN)
to predict the traffic flow in traffic network [44]. Res-RGNN can utilize the
spatial attributes to capture spatial features using diffusion convolution. In
addition to using graph convolution, Res-RGNN also uses GRU to discover
temporal dependency for each node. Then Res-RGNN can jointly capture the
spatial-temporal dynamics. Specifically, the implementation of RGNN unit at
time $t$ is

$\displaystyle\mathbf{r}^{(t)}=\sigma\left(\mathbf{\Theta}_{r}*_{G}\left[\mathbf{x}^{(t)},\mathbf{e}^{(t)},\mathbf{s}^{(t-1)}\right]+\mathbf{b}_{r}\right),$
(8a)
$\displaystyle\mathbf{u}^{(t)}=\sigma\left(\mathbf{\Theta}_{u}*_{G}\left[\mathbf{x}^{(t)},\mathbf{e}^{(t)},\mathbf{s}^{(t-1)}\right]+\mathbf{b}_{u}\right),$
(8b)
$\displaystyle\mathbf{c}^{(t)}=\mathrm{tanh}\left(\mathbf{\Theta}_{c}*_{G}\left[\mathbf{x}^{(t)},\mathbf{e}^{(t)},(\mathbf{r}^{(t)}\odot\mathbf{s}^{(t-1)})\right]+\mathbf{b}_{c}\right),$
(8c)
$\displaystyle\mathbf{s}^{(t)}=\mathbf{u}^{(t)}\odot\mathbf{s}^{(t-1)}+\left(1-\mathbf{u}^{(t)}\right)\odot\mathbf{c}^{(t)},$
(8d) $\displaystyle\mathbf{y}^{(t+1)}=\mathbf{W}_{o}\mathbf{s}^{(t)},$ (8e)

where $\mathbf{x}^{(t)},\mathbf{e}^{(t)}$ and $\mathbf{s}^{(t)}$ denote graph
signal, external feature and the outputted hidden state at time $t$,
respectively. $\mathbf{r}^{(t)}$ and $\mathbf{u}^{(t)}$ represent the reset
gate and update gate at time $t$, respectively.
$\mathbf{\Theta}_{r},\mathbf{\Theta}_{u}$ and $\mathbf{\Theta}_{c}$ are
learnable graph filters, $\mathbf{W}_{o}$ is the learned weights of the output
layer. $\mathbf{y}^{(t+1)}$ denotes the output at time $t+1$.

S. Guo _et. al_ proposed the attention based spatial-temporal graph
convolutional networks (ASTGCN) based on GCNs and CNNs to predict traffic flow
[45]. Specifically, ASTGCN utilizes spectral-based GCNs, i.e., ChebNet [25] to
capture the spatial dependency among different nodes in traffic network graph.
In addition, 1-D 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.

Figure 4: Illustration of how ASTGCN captures spatial and temporal
dependencies.

######  Remark 7.

Traffic prediction is the most popular application of spatial-temporal 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.

TABLE II: Applications of GNNs in Wireless Networks Area | Year | Application | Algorithm | Scheme
---|---|---|---|---
Power Control | 2019 |  Power control in ad-hoc wireless networks | Spatial-based GCNs | Y. Shen  _et al._ [46]
2021 |  Radio resource management in mmWave networks | Spatial-based GCNs | Y. Shen  _et al._ [47]
2020 | Power allocation in ad-hoc wireless networks | Spectral-based GCNs | M. Eisen  _et al._ [48]
2019 |  Power allocation with a set of transmitter receiver pairs in a large scale wireless network | Spectral-based GCNs | M. Eisen  _et al._ [49]
2020 |  Power control in ad-hoc wireless networks | Spectral-based GCNs | M. Eisen  _et al._ [50]
2021 |  Power control in decentralized wireless networks | Spectral-based GCNs | I. Nikoloska  _et al._ [51]
2020 | Downlink power control in wireless networks | Spectral-based GCNs | N. Naderializadeh  _et al._ [52]
2020 |  Power allocation in a single-hop ad-hoc wireless network | Spectral-based GCNs | A. Chowdhury  _et al._ [53, 54]
2021 | Power control in multi-cell cellular networks | Spatial-based GCNs | J. Guo  _et al._ [55]
2021 |  Joint user association and power allocation in heterogeneous ultra dense network | Spatial-based GCNs | X. Zhang  _et al._ [56]
2021 |  Power control/beamforming (PC/BF) in heterogeneous D2D networks | Spatial-based GCNs | X. Zhang  _et al._ [57]
2020 |  Resource allocation in free space optical (FSO) fronthaul networks | Spectral-based GCNs | Z. Gao  _et al._ [58]
2020 |  Resource allocation problems under asynchronous wireless network setting | Spectral-based GCNs | Z. Wang  _et al._ [59]
Link Scheduling | 2019 |  Link scheduling in D2D networks | Graph embedding | W. M. Lee  _et al._ [60]
2020 |  Schedule transmission for wireless networks in a distributed manner | Spectral-based GCNs | Z. Zhao  _et al._ [61]
2019 |  Temporal link prediction in various network systems | Spectral-based GCNs | K. Lei  _et al._ [62]
2021 |  Joint link scheduling and beam selection in ultra-dense D2D mmWave communication networks | Spatial-based GCNs | S. He  _et al._ [63]
Channel Control | 2020 |  Channel allocation for densely deployed WLANs |  DRL with Spectral- based GCN | K. Nakashima  _et al._ [64]
2020 |  Channel tracking for the massive MIMO networks | Spatial-based GCNs | Y. Yan  _et al._ [65]
2020 |  Channel estimation for wireless networks | GAT | K. Tekbıyık  _et al._ [66]
2020 |  Massive MIMO detection in wireless communication | Spatial-based GCNs | A. Scotti  _et al._ [67]
2021 |  Reflect and Beamform for Intelligent Reflecting Surface (IRS) | Spatial-based GCNs | T. Jiang  _et al._ [68]
Traffic Prediction | 2020 |  Cellular traffic prediction |  Spectral-based GCNs and CNN with GLU | S. Zhao  _et al._ [69]
2020 |  Satellite traffic prediction |  Spectral-based GCNs with graph GRU | L. Yang  _et al._ [70]
Vehicular Communication | 2020 |  Multiagent Cooperative Control for CAV Networks |  Spectral-based GCNs with DRL | J. Dong  _et al._ [71]
2020 |  Active traffic management for CAV networks |  Spectral-based GCNs with DRL | PYJ. Ha  _et al._ [72]
2020 |  Spectrum allocation in vehicle-to-everything (V2X) networks |  Spatial-based GCNs with Multi-agent RL | Z. He  _et al._ [73]
Point Cloud | 2020 |  Efficient point cloud processing | Dynamic GCNs | J. Shao  _et al._ [74]
2020 |  Point cloud delivery | GAE | T. Fujihashi  _et al._ [75]
2021 |  3D Object Detection | 3D GNN | C. S. Jeong  _et al._ [76]
Others | 2019 |  Throughput maximization for UAV assisted ground networks | Spectral-based GCNs | S. Lohani  _et al._ [77]
2021 |  Wireless network localization | Spectral-based GCNs | W. Yan  _et al._ [78]
2021 |  Decentralized control in wireless communication systems | Spectral-based GCNs | M. Lee  _et al._ [79]

### 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
$K$-user interference channels, Y. Shen _et al._ proposed an interference
graph convolutional neural network (IGCNet) [46]. In this scheme, the $K$-user
interference channel is modeled as a complete interference graph with node and
edge labels, which is shown in Fig. 5. The $i$-th transmitter-receiver pair is
viewed as the $i$-th node. The node label contains the state of direct channel
$h_{ii}$ and the weight $w_{i}$ of the $i$-th pair. One edge between two nodes
indicates an interference link, with label as the interference channels
$h_{ij}$ and $h_{ji}$. The aggregation and combination rules of IGCNet are
designed as follows

Figure 5: Illustration of constructing interference graph for 3-user
interference channel.

$\displaystyle\bm{\gamma}_{u,v}^{(t)}$
$\displaystyle=\mathrm{MLP1}\left(h_{uv},h_{vu},w_{v},h_{uu},\bm{\beta}_{u}^{(t-1)}\right),$
(9) $\displaystyle\bm{\alpha}_{v}^{(t)}$
$\displaystyle=\mathrm{CONCAT}\left(\mathrm{MAX}_{u\in\mathcal{N}(v)}\left(\bm{\gamma}_{u,v}\right),\sum_{u\in\mathcal{N}(v)}\bm{\gamma}_{u,v}\right),$
$\displaystyle\bm{\beta}_{v}^{(t)}$
$\displaystyle=\mathrm{MLP2}\left(\bm{\alpha}_{v}^{(t)},h_{vv},\bm{\beta}_{v}^{(t-1)},w_{v}\right),$

where $\mathrm{MAX\left({\cdot}\right)}$ is to take the largest value in a
set, $\mathrm{MLP1}$ and $\mathrm{MLP2}$ denote two different $\mathrm{MLPs}$,
$\mathrm{CONCAT}$ denotes the operation that concatenates two vectors
together. $\bm{\gamma}_{u,v}^{(t)}$ denotes the feature vector of the edge
connecting node $u$ and node $v$ in the $t$-th iteration,
$\bm{\alpha}_{v}^{(t)}$ is the aggregated information from the neighbor nodes
to the central node $v$ in the $t$-the iteration, and $\bm{\beta}_{v}^{(t)}$
is the updated hidden representation of node $v$ in the $t$-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 [47]. Then they proposed a family of neural networks, i.e., message
passing graph neural networks (MPGNNs) to solve the large-scale radio resource
management problem. It demonstrates that MPGNNs satisfy the permutation
equivariance property and have the ability to address the large-scale 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 optimization-based
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
GNN-based scheme, i.e., the random edge graph neural networks (REGNN) [48].
Different from the work in [46], REGNN performs convolutions over random
graphs formed by the fading interference patterns in wireless networks.
Specifically, suppose there are $F_{t}$ filter coefficients
$\bm{\alpha}_{t}=\left[\alpha_{t0};\cdots;\alpha_{t(F_{t}-1)}\right]$ defining
graph filters
$\mathbf{A}^{(t)}(\mathbf{H})=\sum\limits_{k=0}^{F_{t}-1}\alpha_{tk}\mathbf{H}^{k}$
in the $t$-th layer. The intermediate feature $\mathbf{Z}^{(t)}$ in the $t$-th
layer is produced as follows

$\displaystyle\mathbf{Z}^{(t)}=\sigma\left(\mathbf{A}^{(t)}\left(\mathbf{H}\right)\mathbf{X}^{(t-1)}\right).$
(10)

The REGNN operator is defined by
$\mathbf{\Phi}\left(\mathbf{H},\mathbf{Z};\mathbf{A}\right)=\mathbf{Z}^{(L)}$,
where $\mathbf{A}=\\{\alpha_{tk}\\}_{t,k}$. The final problem to be solved by
this work is

$\displaystyle\mathbf{A}^{*}=\operatorname*{argmax}~{}\mathcal{U}_{0}(\mathbf{r}),$
(11) $\displaystyle s.t.$ $\displaystyle\
\mathbf{r}=\mathbb{E}\left[f\left(\mathbf{\Phi}\left(\mathbf{H},\mathbf{X};\mathbf{A}\right);\mathbf{H},\mathbf{X}\right)\right],$
$\displaystyle\ \mathcal{U}(\mathbf{r})\geq
0,~{}\mathbf{\Phi}\left(\mathbf{H},\mathbf{X};\mathbf{A}\right)\in\mathcal{P}\left(\mathbf{H},\mathbf{X}\right).$

where $f(\cdot)$ is a function with respect to the joint space of resource
allocations and states to the space of rewards. $\mathcal{U}_{0}$ and
$\mathcal{U}$ 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 model-free
primal-dual learning algorithm to train the weights of the REGNN [49]. REGNN
is also utilized to solve power control in decentralized wireless networks
[51]. In order to enable the fast adaption of power control policy to time-
varying topologies, the authors applied first-order meta-learning on data from
multiple topologies for a few-shot adaptation to new network configurations.

N. Naderializadeh _et al._ considered the problem of downlink power control in
wireless networks, consisting of multiple transmitter-receiver pairs
communicating with each other over a single shared wireless medium [52]. 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 $5$-th percentile user rates throughout a range of network
configurations. The GNN-based model is also trained based on primal-dual
learning framework.

The previous methods are all based on data-driven neural networks, which have
poor interpretability and scalability. Different from the previous works, A.
Chowdhury _et al._ proposed a hybrid data-model 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 single-hop
ad-hoc wireless network [53]. The optimization problem should be solved is

$\displaystyle\
\min\limits_{\mathbf{w},\mathbf{a},\mathbf{b}}~{}\sum\limits_{i=1}^{M}\left(w_{i}q_{i}-\mathrm{log}w_{i}\right),$
(12) $\displaystyle s.t.{}$ $\displaystyle\
q_{i}=\left(1-a_{i}h_{ii}b_{i}\right)^{2}+\sigma^{2}a_{i}^{2}+\sum\limits_{i\neq
j}a_{i}^{2}h_{ij}^{2}b_{j}^{2},$ $\displaystyle\ b_{i}^{2}\leq p_{max},$

where $\mathbf{w},\mathbf{a}$ and $\mathbf{b}$ are variables should be
optimized. The allocated power is computed by a function of the channel state
matrix $\mathbf{p}=\mathrm{\Phi}\left(\mathbf{H};\mathbf{\Theta}\right)$
through a layered architecture $\Phi$ with trainable weights
$\mathbf{\Theta}$. Precisely, setting
$\mathbf{b}^{(0)}=\sqrt{p_{max}}\mathbf{1}$, the $t$-th layer of UWMMSE is
implemented as follows

$\displaystyle\mathbf{\vartheta}^{(t)}=\Psi\left(\mathbf{H};\bm{\theta}_{\vartheta}^{(t)}\right),~{}~{}\mathbf{\nu}^{(t)}=\Psi\left(\mathbf{H};\bm{\theta}_{\nu}^{(t)}\right),$
(13a) $\displaystyle
a_{i}^{(t)}=\frac{h_{ii}{\nu}_{i}^{(t-1)}}{\sigma^{2}+\sum\limits_{j}h_{ij}^{2}b_{j}^{(t-1)}b_{j}^{(t-1)}},~{}~{}for~{}all~{}i,$
(13b) $\displaystyle
w_{i}^{(t)}=\frac{{\vartheta}_{i}^{(t)}}{1-a_{i}^{(t)}h_{ii}b_{i}^{(t-1)}}+{\nu}_{i}^{(t)},~{}~{}~{}for~{}all~{}i,$
(13c) $\displaystyle
b_{i}^{(t)}=\alpha\left(\frac{a_{i}^{(t)}h_{ii}w_{i}^{(t)}}{\sum\limits_{j}h_{ji}^{2}a_{j}^{(t)}a_{j}^{(t)}w_{j}^{(t)}}\right),~{}~{}~{}for~{}all~{}i,$
(13d)

and the output power is determined as
$\mathbf{p}=\Phi\left(\mathbf{H};\mathbf{\Theta}\right)=\left(\mathbf{b}^{(L)}\right)^{2}$.
$\alpha(z):=[z]_{0}^{\sqrt{p_{max}}}$ simply ensures that
$b_{i}^{(t)}\in{\left[0,\sqrt{p_{max}}\right]}$. The function $\Psi$
parameterized by $\mathbf{\Theta}$ 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 [54]. Compared with neural network models, unfolding
methods have better interpretability and scalability via stacking several
trainable layers that correspond to model-driven iterations.

Figure 6: Illustration of UWMMSE [53].

J. Guo  _et al._ considered the power control problem in multi-cell cellular
networks [55]. 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 multi-cell 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
[80]. For the WIG, the aggregator and the combiner of PGNN are designed as
follows,

BSs aggregating information from UEs

$\displaystyle\ \text{Aggregate:
}\mathbf{a}_{i,\mathrm{BS}}^{(t)}=\sum\limits_{j=1}^{M}\left(\mathbf{V}^{(t)}\mathbf{u}_{j}^{(t-1)}+\mathbf{P}^{(t)}\mathbf{h}_{ij}\right),$
(14) $\displaystyle\ \text{Combine:
}\mathbf{b}_{i}^{(t)}=\sigma\left(\mathbf{S}^{(t)}\mathbf{b}_{i}^{(t-1)}+\mathbf{a}_{i,\mathrm{BS}}^{(t)}\right),$

UEs aggregating information from BSs

$\displaystyle\ \text{Aggregate:
}\mathbf{a}_{i,\mathrm{UE}}^{(t)}=\sum\limits_{j=1}^{M}\left(\mathbf{U}^{(t)}\mathbf{b}_{j}^{(t-1)}+\mathbf{Q}^{(t)}\mathbf{h}_{ji}\right),$
(15) $\displaystyle\ \text{Combine:
}\mathbf{u}_{i}^{(t)}=\sigma\left(\mathbf{T}^{(l)}\mathbf{u}_{i}^{(t-1)}+\mathbf{a}_{i,\mathrm{UE}}^{(t)}\right),$

where $\mathbf{S}^{(t)}$ and $\mathbf{T}^{(t)}$ are model parameters in the
combination function, $\mathbf{V}^{(t)}$ and $\mathbf{U}^{(t)}$ are used to
aggregate the information from UEs and BSs, and $\mathbf{P}^{(t)}$ and
$\mathbf{Q}^{(t)}$ are used to aggregate the information from edges,
respectively. To distinguish the hidden outputs between BSs and UEs,
$\mathbf{b}_{i}^{(t)}$ and $\mathbf{u}_{i}^{(t)}$ are the hidden output of
$\text{BS}_{i}$ and $\text{UE}_{i}$, 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 multi-user multi-cell
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 ultra-dense networks (HUDNs) [56]. The HUDNs are
modeled as a heterogeneous graph, which is shown in Fig. 7. A heterogeneous
GraphSAGE (HGSAGE) that extended from GraphSAGE [81], 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 generalization-representation
learning (GRL) part and the specialization-representation 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 off-line manner. Based on the learned GRL representation, the
SRL finely tune the parameters of GNN online to further improve the
performance for quasi-static user distribution.

Figure 7: Illustration of constructing heterogeneous graph. K denotes the
K-hop neighbor of a node [56].

X. Zhang _et al._ focused on power control/beamforming in heterogeneous
device-to-device (D2D) networks [57]. 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 $n$ to link type $m$ is characterized
as $r=(n,m)$. Take node $i_{m}$ as an example, the update rules in relation
$(n,m)$ is defined as follows

$\displaystyle\mathbf{e}_{j_{n}i_{m}}[l]=\phi_{(n,m)}^{e}\left(\mathbf{v}_{j_{n}}[l-1],\mathbf{e}_{j_{n}i_{m}}[0]\right),$
(16a)
$\displaystyle\mathbf{v}_{i_{m}}^{(n)}=\phi_{(n,m)}^{v}\left(\mathbf{v}_{i_{m}}[l-1],\max\limits_{j\in{\mathcal{N}_{i_{m}}^{(n)}}}\mathbf{e}_{j_{n}i_{m}}[l]\right).$
(16b)

where $\phi_{r}^{e}$ is an edge update function of relation $r$,
$\phi_{r}^{v}$ is a vertex update function of relation $r$. $j_{n}$ denotes
the $j$-th vertex with link type $n$. $\mathbf{v}_{j_{n}}$ is the attribute
vector of vertex $j_{n}$, $\mathbf{e}_{j_{n}i_{m}}$ is the attribute vector of
the edge between vertex $j_{n}$ and $i_{m}$. Then, the aggregation rules is
defined as follow

$\displaystyle\mathbf{v}_{i_{m}}[l]=\rho_{m}^{v\rightarrow{v}}\left(\left\\{\mathbf{v}_{i_{m}}^{(n)}[l]\right\\}_{n}\right)=\frac{1}{c_{i,m}}\sum\limits_{n}\mathbf{v}_{i_{m}}^{(n)}[l],$
(17)

where $c_{i,m}$ is the number of relations causing interference to link
$i_{m}$. $\rho\left(\cdot\right)$ is the aggregation function.

Figure 8: Illustration of constructing heterograph [57].

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 [58]. GNNs are utilized to exploit the FSO
network structure with small-scale training parameters. Then a primal-dual
learning algorithm is developed to train the GNN in a model-free manner.

Z. Wang  _et al._ addressed the asynchronous decentralized wireless resource
allocation problem with a novel unsupervised learning approach [59].
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 (Agg-GNNs) is
designed based on the graph representation of interference and asynchrony. The
Agg-GNN 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 ultra-dense network, power
control alone is not enough to eliminate the strong interference between
short-range 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 [60]. In detail,
this work first models the D2D network as a fully-connected directed graph,
then computes a low-dimensional feature vector based on the distances of both
communication and interference links without requiring the accurate channel
state information for each node. Finally, a multi-layer 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 [61]. The authors first proposed a GCN-based
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 non-linear GCN-GAN model by
leveraging the benefits of GCNs, LSTM as well as the GANs [62]. 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, GCN-GAN 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 high-quality predicted graph snapshot with
an adversarial process, where generative network consists of the GCN and the
LSTM, discriminative network consists of a fully-connected network.

In ultra-dense 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 [63]. The
resulting optimization problem is formulated as an constrained integer
programming problem that is nonconvex and NP-hard. To solve this problem
efficiently, an unsupervised learning model, i.e., GNN-based Beam selection
and Link scheduling (GBLinks), which based on spatial-based 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 near-optimal solution through comparing with the exhaustively search in
small-scale 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 GCN-based deep RL model
for channel allocation [64]. 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 high-movement networks
is a challenging task. Y. Yan  _et al._ proposed a new channel tracking method
based on GNNs [65]. 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 GNN-based channel tracking framework, which includes encoder, core network
and decoder. The numerical results confirmed that the GNN-based scheme
proposed can achieve better performance than the scheme using feed-forward
neural network.

K. Tekbıyık  _et al._ first used GAT for channel estimation [66]. The
performance of the proposed method is studied on the two-way backhaul link of
high-altitude platform stations with reconfigurable intelligent surfaces. The
numerical results show that for the full-duplex 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 full-
duplex 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 small-scale fading characteristics changes. A. Scotti
_et al._ considered the inference task of massive MIMO detection under time-
varying channels and higher-order qadrature amplitude modulation and proposed
a message-passing solution based on GNNs, i.e., MIMO-GNN [67].

### 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 STGCN-HO, which uses the transition probability matrix of the handover
graph to improve the accuracy of traffic prediction [69]. STGCN-HO builds a
stacked residual neural network structure that combines graph convolution and
CNN with gated linear units [82] to capture the spatial and temporal
relationships of traffic. Unlike RNN, STGCN-HO 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, STGCN-HO 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 spatio-temporal characteristics of network traffic,
and then proposed a combination of GCNs and GRU network traffic prediction
model [70]. 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 vehicle-to-everything (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
[71]. 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 decision-making.

P. Y. J. Ha  _et al._ applied RL algorithms to train CAV driving behaviors,
which can be used to relieve highway bottleneck congestion [72]. 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 [73]. The
authors first modeled the V2X network as a graph, where each vehicle-to-
vehicle link is a node in the graph. Then, GNNs are applied to learn the low-
dimensional representations of each node from graph structure information.
According to the learned characteristics, multi-agent 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 Branchy-GNN for efficient point cloud processing [74].
Branchy-GNN 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 three-dimensional data [75]. Specifically, the
proposed scheme combines GNN-based point cloud coding and near-analog
modulation to simultaneously achieve: 1) prevent cliff effect, 2) prevent
leveling effect, 3) high energy compression, and 4) low communication
overhead. In addition, the end-to-end 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
[76].

### 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 [77]. 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 two-
dimensional (2-D) space using GCNs [78]. The adjacent matrix
$\mathbf{A}_{T_{h}}\in{\mathbb{R}^{N\times{N}}}$ of an undirected graph
corresponding a wireless network is constructed as follows

$\displaystyle\left[\mathbf{A}_{T_{h}}\right]_{ij}=\left\\{\begin{array}[]{rcl}0,&&if\quad
x_{ij}\leq T_{h},\\\ 1,&&otherwise,\end{array}\right.$ (18)

where $T_{h}$ is a Euclidean distance threshold to determine whether there is
an edge between two nodes or not. $x_{ij}$ is the distance between node $i$
and $j$. The corresponding augmented adjacency matrix is defined as
$\tilde{\mathbf{A}}_{T_{h}}:=\mathbf{A}_{T_{h}}+\mathbf{I}$. The graph
convolution operation in the $t$-th layer is defined as
$\mathbf{H}^{(t)}\leftarrow\phi\left(\bar{\mathbf{H}}^{(t)}\mathbf{W}^{(t)}\right)$,
where
$\bar{\mathbf{H}}^{(t)}\in{\mathbb{R}^{N\times{D_{t-1}}}}\leftarrow\hat{\mathbf{A}}_{T_{h}}\mathbf{H}^{(t-1)}$
, $\phi$ is a non-linear activation function, and
$\hat{\mathbf{A}}_{T_{h}}\in\mathbb{R}^{N\times{N}}:=\tilde{\mathbf{D}}_{T_{h}}^{-\frac{1}{2}}\tilde{\mathbf{A}}_{T_{h}}\tilde{\mathbf{D}}_{T_{h}}^{-\frac{1}{2}}$.
The initialization is $\mathbf{H}^{(0)}=\hat{\mathbf{X}}$, where
$\hat{\mathbf{X}}=\mathbf{A}_{T_{h}}\odot\mathbf{X}$. 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 [79].

## IV. Key Issues and Future Development

According to the current review results, we summarize some ongoing or future
research directions that are worth exploring:

$\bullet$ 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 small-scale, how to
effectively apply GNNs to solve large-scale wireless network problems is a
topic worth exploring.

$\bullet$ 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.

$\bullet$ 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 long-term topic.

$\bullet$ Deep unfolding is an effective combination of model-driven methods
and data-driven methods, which not only effectively utilizes the
interpretability and scalability of model-driven algorithms, but also uses the
expressive power of data-driven 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.

$\bullet$ As we all know, high dynamics is a major feature of wireless
networks, and currently there are few GNN-based 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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