# Heterogeneous Graph Neural Network for Power Allocation in Multicarrier-
Division Duplex Cell-Free Massive MIMO Systems

Bohan Li, Lie-Liang Yang, Fellow, IEEE, Robert G. Maunder, Senior Member,
IEEE, Songlin Sun, Senior Member, IEEE, Pei Xiao, Senior Member, IEEE B. Li,
L.-L. Yang and R. Maunder are with the School of Electronics and Computer
Science, University of Southampton, SO17 1BJ, UK. (E-mail: bl2n18, lly,
<EMAIL_ADDRESS>http://www-mobile.ecs.soton.ac.uk/lly). S. Sun is with the
School of Information and Communication Engineering, Beijing University of
Posts and Telecommunications (BUPT). P. Xiao is with 5GIC & 6GIC, Institute
for Communication Systems (ICS), University of Surrey, Guildford GU2 7XH, UK.
(Email<EMAIL_ADDRESS>The project was supported in part by the EPSRC,
UK, under Project EP/P034284/1 and EP/P03456X/1, in part by the Innovate UK
project, and in part by the China Scholarship Council (CSC).

###### Abstract

In-band full duplex cell-free (CF) systems suffer from severe self-
interference and cross-link interference, especially when CF systems are
operated in distributed way. To this end, we propose the multicarrier-division
duplex as an enabler for achieving full-duplex operation in the distributed CF
massive MIMO systems, where downlink and uplink transmissions occur
simultaneously in the same frequency band but on the mutually orthogonal
subcarriers. To maximize the spectral-efficiency (SE), we introduce a
heterogeneous graph neural network (HGNN) specific for CF systems, referred to
as CF-HGNN, to optimize the power-allocation (PA). We design the adaptive node
embedding layer for CF-HGNN to be scalable to the various numbers of access
points (APs), mobile stations (MSs) and subcarriers. The attention mechanism
of CF-HGNN enables individual AP/MS nodes to aggregate information from the
interfering and communication paths with different priorities. For comparison,
we propose a quadratic transform and successive convex approximation (QT-SCA)
algorithm to solve the PA problem in classic way. Numerical results show that
CF-HGNN is capable of achieving 99% of the SE achievable by QT-SCA but using
only $10^{-4}$ times of its operation time. CF-HGNN significantly outperforms
the traditional greedy unfair method in terms of SE performance. Furthermore,
CF-HGNN exhibits good scalability to the CF networks with various numbers of
nodes and subcarriers, and also to the large-scale CF networks when assisted
by user clustering.

###### Index Terms:

Multicarrier-division duplex, cell-free, massive MIMO, deep learning,
heterogeneous graph neural network, power allocation, spectral efficiency.

##  I Introduction

With the unprecedented increase of data-hungry devices, wireless communication
communities have been continuing to thrive in the last few of years to seek
the cutting-edge techniques for further improving spectral efficiency (SE) and
quality of services (QoSs). In this regard, cell-free massive MIMO (CF-mMIMO)
and in-band full-duplex (IBFD) have been envisioned as the main pillars of
future wireless communication systems. Specifically, by integrating the
concepts of boundaryless cells and distributed antenna mMIMO system, CF-mMIMO
is capable of providing the users within a coverage area with not only higher
data rates but also more consistent service than the conventional small-cell
or co-located mMIMO systems [1]. On the other side, IBFD has the potential to
double the SE with off-the-shelf system resource and hence, can resolve the
critical defects of the traditional half-duplex (HD) modes, such as the
inefficient usage of time-frequency resource and the high latency generated by
both physical and network layer operations [2].

However, to date, although there exist a large number of works on CF-mMIMO [3]
and IBFD systems [4], the combination of these two techniques has rarely been
addressed. One of the underlying reasons is that IBFD-based CF (IBFD-CF)
systems suffer from the problems of severe self-interference (SI) and cross-
link interference (CLI). Here, SI is referred to the interference imposed by
the transmitting signals of access point (AP) or mobile station (MS) on its
receiving signals. SI may be mitigated by propagation-, analog- and digital-
domain self-interference cancellation (SIC) methods [5, 6, 7, 8, 9]. However,
the implementation of SIC imposes large complexity and resource burden on the
small-scale APs in CF networks [10]. Moreover, none of the existing methods
can fully suppress SI, and unavoidably, the residual SI causes the performance
degradation of desired transmission. CLI, known as the inter-AP interference
(IAI) or inter-MS interference (IMI), has been shown to be the most serious
problem in multi-cell IBFD and dynamic time-division duplex (TDD) systems
[11]. The situation is even exacerbated in the CF networks, where the IAI
received by an AP may come from various directions. Noticeably, compared with
IAI, IMI is more difficult to manage, as the baseband processors of MSs are
usually not as powerful as those of APs. Fortunately, the transmit power of
MSs is relatively low and in this case, the IMI can be greatly mitigated in
the analog domain using the power control and/or user scheduling approaches
[12].

###  I-A Related Works

In [13], authors studied the IBFD-CF in the context of the centralized CF-
mMIMO systems, where a small amount of IAI is mitigated in digital domain by
the coordinated precoding and successive interference cancellation operated at
APs’ receiver. With this method, each AP has to reduce its transmit power to
decrease the effect of IAI on the neighboring APs, which can result in
performance degradation. On top of [13], the authors of [14, 15] proposed a
so-called network-assisted full duplex (NAFD) mode for the centralized CF
networks, where digital-domain IAI and IMI are mitigated by the central
processing unit (CPU) and user scheduling, respectively, while the analog-
domain CLI is not fully considered. In this case, the digital-domain IAI
mitigation relies on both the relatively precise estimation of IAI channels
and the undistorted fronthaul transmissions, which significantly increase
system overhead. Possibly due to the above-mentioned challenges, to the best
of our knowledge, there is no research article in the open literature having
considered the full-duplex (FD)-style distributed CF systems, as the CLI
problem in distributed CF systems is much severer than that in the centralized
CF systems.

To fill the research gap, in [16], we proposed a distributed CF-mMIMO system
driven by the multicarrier-division duplex (MDD), which enables FD operation
in the same time slot and the same frequency band but on different
subcarriers. In our MDD-assisted CF (MDD-CF) systems, the subcarriers of one
band are divided into two mutually exclusive subsets, namely DL and UL
subcarrier subsets, to support DL and UL transmissions, respectively. With the
aid of these arrangements, despite the MDD-CF system suffers from the problems
of SI and CLI in analog domain similarly as the IBFD-CF system, it can be free
from the digital-domain SI and CLI. This can be readily achieved by the fast
Fourier transform (FFT) operation at receiver, as the SI/CLI and desired
signals in the MDD-CF system are transmitted on orthogonal subcarriers [10].
Our studies in [16] demonstrated that the MDD-CF scheme can significantly
enhance the system performance, when compared with the IBFD-CF and TDD-CF
systems.

However, the PA problem in the distributed MDD-CF system is typically non-
convex and computationally challenging due to the large size of the
optimization variables involved. In literature, there are various optimization
algorithms proposed for resource allocation, including the Dinkelbach’s
transform [17], quadratic transform (QT) [18, 19, 16], dual decomposition [20]
and the successive convex approximation (SCA) [21, 22]. These approaches can
in general converge to the local optimum within several iterations, but their
computational complexity grows exponentially with the increase of network
size. Moreover, once network changes, these algorithms have to rerun for
obtaining the updated solutions, which introduces extra computational cost and
latency. As the result, these traditional methods can hardly be operated in a
timely manner for the PA in the CF-mMIMO systems. Needless to say, the
situation is more complicated in the FD-style scenarios.

Recently, machine learning (ML) has shown its potential for solving the PA
problem in wireless communications. For instance, in [23], a convolutional
neural network (CNN)-based PA algorithm was proposed and shown to be superior
to the conventional weighted minimum mean square error (WMMSE) method in terms
of the achievable SE and computation time. In [24], the authors proposed an
ensemble PA network (ePCNet) consisting of multiple independent and fully-
connected multi-layer neural networks, which was demonstrated to outperform
the WMMSE and greedy-relied PA. Furthermore, in [25] and [26], the deep
reinforcement learning-based PA was leveraged to solve respectively the max-
min and SE maximization problems in the CF-mMIMO systems. In [27], the authors
resorted to the deep CNN for maximizing SE in the CF-mMIMO systems, and
demonstrated that the learning-based method can outperform the well-known use-
and-then-forget-based PA method. In [28], a clustered DNN model exploiting the
large-scale fading information was introduced to implement PA in the CF-mMIMO
systems, showing that its performance is comparable to that of the WMMSE-
assisted alternating direction method.

The aforementioned learning-based PA methods have been shown to prevail over
the traditional methods. However, none of them have considered to exploit the
structures of wireless networks in the optimization process. Hence, they can
not be generalized to the unseen scenarios, such as the networks with varying
sizes or AP/MS densities. To this end, the graph neural network (GNN) reaping
the advantages of scalability, generalization and parallel execution has
attracted significant research interests in wireless communications.
Specifically, in [29], a wireless channel graph convolution network (WCGCN)
was proposed for solving the PA and bearmforming problem in device-to-device
(D2D) communication systems. It was shown that the WCGCN trained on the basis
of small size systems can be generalized to use for optimization in the large
systems with higher density of MSs and larger cell size. Additionally, the
authors in [30] and [31] studied the GNN-assisted PA in the ad hoc networks.
However, the above-mentioned GNN-related models for ad hoc or D2D networks are
all homogeneous, where only one type of nodes exist. Hence, they are not
suitable for the operation in the more complicated scenarios like the MDD-CF
networks, where APs and MSs are different types of nodes having different node
features. Furthermore, in the MDD-CF networks, each AP/MS node can be
connected with different types of nodes either via the communication path or
via the interference path.

Against the background, in this paper, we propose a CF-heterogeneous graph
neural network (HGNN), to solve the PA problem in the distributed MDD-CF
systems. To the best of our knowledge, this is the first HGNN-based network
proposed for handling the PA problem in the FD-style CF-mMIMO systems.

###  I-B Contributions

The major contributions are summarized as follows:

  * •

The PA problem for maximizing SE in the distributed MDD-CF systems is
formulated under the constraints of QoS requirements, where the effect of
residual SI and CLI are practically modeled. In order to solve the
optimization problem, we propose a quadratic transform and successive convex
approximation (QT-SCA) optimization algorithm. In this QT-SCA algorithm, QT is
used to first transform the optimization function, which belongs to the
multiple-ratio concave-convex fractional programming (MRCCFP) problem, into an
iterative convex problem. Then, SCA is employed to substitute the non-convex
constraints imposed by QoS requirements with their approximated convex forms.
Furthermore, the feasible initialization, convergence behavior and
computational complexity of the QT-SCA algorithm are analyzed.

  * •

To achieve the scalability and make the optimization general, we propose the
CF-HGNN to solve the PA problem in the distributed MDD-CF systems. In our CF-
HGNN, an adaptive node embedding layer and an adaptive output layer are
implemented so that the CF-HGNN can adapt to the MDD-CF networks with various
numbers of APs, MSs and subcarriers. Two types of meta-paths are defined,
namely the data transmission path and interfering path, to connect the
involved AP/MS nodes. Furthermore, to learn the importance factors of the
information received at the AP/MS nodes via two meta-paths during message
passing, the meta-path attention mechanism is implemented with the CF-HGNN.

  * •

We conduct a range of experiments to evaluate the effectiveness of the CF-HGNN
for the PA problem in the distributed MDD-CF systems, where the CF-HGNN is
trained in an unsupervised way using unlabeled data. Simulation results show
that, without exploiting any prior information, the CF-HGNN is capable of
learning nearly the same PA strategy as the QT-SCA algorithm, and it can
significantly outperform the traditional greedy unfair method. Our studies
also demonstrate that the CF-HGNN has the advantage over the QT-SCA in terms
of computational complexity and operation time. Furthermore, the CF-HGNN
exhibits the scalability to the MDD-CF networks with various numbers of nodes
and subcarriers. Specifically, assisted by user clustering, the CF-HGNN
exhibits the scalability to the large-scale MDD-CF networks, which cover large
area, have a big number of subcarriers and simultaneously support a large
number of nodes.

The rest of the paper is organized as follows. In Section II, we describe the
distributed MDD-CF systems, and formulate the optimization problem for PA.
Then, the QT-SCA algorithm for PA is analyzed. In Section III, the CF-HGNN for
PA in the distributed MDD-CF systems is detailed. Simulation results are
presented and analyzed in Section IV. Finally, conclusions are summarized in
Section V.

##  II System Model

An MDD-CF system as shown in Fig. 1 is studied. We assume the set
$\mathcal{D}=\left\\{1,...,d,...,D\right\\}$ of single-antenna MSs and the set
$\mathcal{L}=\left\\{1,...,l,...,L\right\\}$ of APs of each with $N$ antennas.
The MSs and APs are operated in the MDD mode that relies on the mutually
orthogonal subcarrier sets [32], denoted as
$\mathcal{M}=\left\\{1,...,m,...,M\right\\}$ with $\left|\mathcal{M}\right|=M$
and $\mathcal{\bar{M}}=\left\\{1,...,\bar{m},...,\bar{M}\right\\}$ with
$\left|\mathcal{\bar{M}}\right|=\bar{M}$, for supporting DL and UL data
transmissions, respectively. The total number of subcarriers is
$M_{\text{sum}}=M+\bar{M}$. Furthermore, we assume that the CF system is
operated in the distributed manner, where CPU offloads the majority of tasks
to APs to relieve its computation burden, and only sends the coded data to APs
for DL transmissions or integrates the received UL data from APs via fronthaul
links without requiring any channel knowledge.

Figure 1: Illustration of MDD-CF networks.

###  II-A Channel Model

For the convenience of notation, we denote the SI channels at the $l$-th AP
and $d$-th MS by $\boldsymbol{H}_{ll}\in\mathbb{C}^{N\times N}$ and $h_{dd}$,
respectively. These two SI channels are modeled as

$\left(\boldsymbol{H}_{ll}\right)_{i,j}=\sqrt{\xi_{l}^{\text{SI}}}\
\alpha_{s},\ h_{dd}=\sqrt{\xi_{d}^{\text{SI}}}\ \alpha_{s},$ (1)

where $\alpha_{s}\sim\mathcal{CN}(0,1)$ is the small-scale fading, while
$\xi_{l}^{\text{SI}}\in(0,1]$ and $\xi_{d}^{\text{SI}}\in(0,1]$ denote the
levels of the residual SI at AP and MS receivers, respectively.

Furthermore, we denote the time-domain channel impulse responses (CIRs) of the
communication channels between the $d$-th MS and the $n$-th antenna at the
$l$-th AP, the IAI channels between the $n$-th antenna at the $l$-th AP and
the $n^{\prime}$-th antenna at the $l^{\prime}$-th AP, and the IMI channels
between the $d$-th MS and $d^{\prime}$-th MS by
$\boldsymbol{g}_{ld}^{n}\in\mathbb{C}^{U_{ld}\times 1}$,
$\boldsymbol{g}_{ll^{\prime}}^{nn^{\prime}}\in\mathbb{C}^{U_{ll^{\prime}}\times
1}$ and $\boldsymbol{g}_{dd^{\prime}}\in\mathbb{C}^{U_{dd^{\prime}}\times 1}$,
respectively, where
$U\in\left\\{U_{ld},U_{ll^{\prime}},U_{dd^{\prime}}\right\\}$ is the number of
taps of multipath channels. Specifically, the $u$-th tap of these channels is
generally modeled as $\left(\boldsymbol{g}\right)_{u}=\sqrt{\beta/U}\
\alpha_{s}$ with
$\boldsymbol{g}\in\left\\{\boldsymbol{g}_{ld}^{n},\boldsymbol{g}_{ll^{\prime}}^{nn^{\prime}},\boldsymbol{g}_{dd^{\prime}}\right\\}$,
where
$\beta\in\left\\{\beta_{ld},\beta_{ll^{\prime}},\beta_{dd^{\prime}}\right\\}$
accounts for the large-scale fading of path loss and shadowing. Different
channel taps are assumed to be independent. Given the time-domain CIRs, the
frequency-domain subcarrier channels can be obtained as
$\boldsymbol{h}=\boldsymbol{F}\boldsymbol{\Psi}\boldsymbol{g}$, where
$\boldsymbol{h}\in\left\\{\boldsymbol{h}_{ld}^{n},\boldsymbol{h}_{ll^{\prime}}^{nn^{\prime}},\boldsymbol{h}_{dd^{\prime}}\right\\}$,
$\boldsymbol{F}\in\mathbb{C}^{M_{\text{sum}}\times M_{\text{sum}}}$ is the FFT
matrix, and $\boldsymbol{\varPsi}\in\mathbb{C}^{{M_{\text{sum}}\times U}}$ is
constructed by the first $U$ columns of $\boldsymbol{I}_{M_{\text{sum}}}$.
Moreover, the single DL/UL subcarrier channel can be expressed as
$h[m]=\boldsymbol{\phi}_{\text{DL}}^{T}\boldsymbol{h}$ and
$h[\bar{m}]=\boldsymbol{\phi}_{\text{UL}}^{T}\boldsymbol{h}$, respectively,
where $\boldsymbol{\phi}_{\text{DL}}=\boldsymbol{I}_{M_{\text{sum}}}^{(:,m)}$
and
$\boldsymbol{\phi}_{\text{UL}}=\boldsymbol{I}_{M_{\text{sum}}}^{(:,\bar{m})}$
are mapping vectors.

###  II-B Downlink Transmission

The data transmitted on the $m$-th DL subcarrier for the $d$-th MS is denoted
by $x_{d}[m]$, which satisfies
$\mathbb{E}\left\\{\left|x_{d}[m]\right|^{2}\right\\}=1$. The transmitted
signal on the $m$-th DL subcarrier by the $l$-th AP is given by

$\boldsymbol{s}_{l}[m]=\sum_{d\in\mathcal{D}}\sqrt{p_{ldm}}\boldsymbol{f}_{ld}[m]x_{d}[m],$
(2)

where $\boldsymbol{f}_{ld}[m]\in\mathbb{C}^{N\times 1}$ denotes the precoding
vector with $\left\|\boldsymbol{f}_{ld}[m]\right\|_{2}^{2}=1$, and $p_{ldm}$
is the power allocated to the $d$-th MS on the $m$-th subcarrier. The total
power budget at the $l$-th AP is expressed as $P_{l}$, satisfying
$\sum_{m\in\mathcal{M}}\sum_{d\in\mathcal{D}}p_{ldm}\leq P_{l}$.

The signal received from the $m$-th DL subcarrier by the $d$-th MS can be
expressed as

$\displaystyle
y_{d}[m]=\sum_{l\in\mathcal{L}}\boldsymbol{h}_{ld}^{H}[m]\boldsymbol{s}_{l}[m]+z^{\text{SI}}_{d}+z^{\text{IMI}}_{d}+n_{d},$
(3)

where $n_{d}\sim\mathcal{CN}(0,\sigma^{2})$ is the additive white Gaussian
noise (AWGN). According to [33, 34], the residual interference in the digital
domain arising from SI and IMI (i.e., $z^{\text{SI}}_{d}$ and
$z^{\text{IMI}}_{d}$ in (3)) can be modeled as Gaussian noise. Specifically,
we express
$z^{\text{SI}}_{d}\sim\mathcal{CN}(0,\mathbb{E}\left[\bar{z}^{\text{SI}}_{d}\left(\bar{z}^{\text{SI}}_{d}\right)^{\ast}\right])$
with
$\bar{z}^{\text{SI}}_{d}=h_{dd}\sum_{\bar{m}\in\bar{\mathcal{M}}}\sqrt{p_{d\bar{m}}}x_{d}[\bar{m}]$,
where $x_{d}[\bar{m}]$ denotes the data transmitted on the $\bar{m}$-th UL
subcarrier by the $d$-th MS, $p_{d\bar{m}}$ denotes the transmitted power.
$z^{\text{IMI}}_{d}\sim\mathcal{CN}(0,\xi_{d}^{\text{IMI}}\mathbb{E}\left[\bar{z}^{\text{IMI}}_{d}\left(\bar{z}^{\text{IMI}}_{d}\right)^{\ast}\right])$
with
$\bar{z}^{\text{IMI}}_{d}=\sum_{d^{\prime}\in\mathcal{D}\backslash\left\\{d\right\\}}\sum_{\bar{m}\in\bar{\mathcal{M}}}\sqrt{p_{d^{\prime}\bar{m}}}h_{dd^{\prime}}[\bar{m}]x_{d^{\prime}}[\bar{m}]$,
where $\xi_{d}^{\text{IMI}}$ denotes the residual IMI level at MS $d$.

Based on (3), the received SINR on the $m$-th DL subcarrier at the $d$-th MS
is given by

$\text{SINR}_{d,m}=\frac{\left|\sum_{l\in\mathcal{L}}\sqrt{p_{ldm}}\boldsymbol{h}_{ld}^{H}[m]\boldsymbol{f}_{ld}[m]\right|^{2}}{\text{MUI}_{d,m}+\text{var}\left\\{z^{\text{SI}}_{d}\right\\}+\text{var}\left\\{z^{\text{IMI}}_{d}\right\\}+\sigma^{2}}\
,$ (4)

where
$\text{MUI}_{d,m}=\sum_{l\in\mathcal{L}}\sum_{d^{\prime}\in\mathcal{D}\backslash\left\\{d\right\\}}p_{ld^{\prime}m}|\boldsymbol{h}_{ld}^{H}[m]\boldsymbol{f}_{ld^{\prime}}[m]|^{2}$
is the component of multiuser interference (MUI).

###  II-C Uplink Transmission

The received UL signal by the $l$-th AP from the $\bar{m}$-th subcarrier can
be expressed as

$\boldsymbol{y}_{l}[\bar{m}]=\sum_{d\in\mathcal{D}}\sqrt{p_{d\bar{m}}}\boldsymbol{h}_{ld}[\bar{m}]x_{d}[\bar{m}]+\boldsymbol{z}^{\text{SI}}_{l}+\boldsymbol{z}^{\text{IAI}}_{l}+\boldsymbol{n}_{l},$
(5)

where $p_{d\bar{m}}$ denotes the power allocated by MS $d$ to the $\bar{m}$-th
UL subcarrier, which satisfies
$\sum_{\bar{m}\in\bar{\mathcal{M}}}p_{d\bar{m}}\leq P_{d}$. Similar to the
received signals at MSs, the residual interferences due to the SI and IAI are
modeled as Gaussian noise, where
$\boldsymbol{z}^{\text{SI}}_{l}\sim\mathcal{CN}(0,\text{diag}(\mathbb{E}[\bar{\boldsymbol{z}}^{\text{SI}}_{l}(\bar{\boldsymbol{z}}^{\text{SI}}_{l})^{H}]))$
with
$\bar{\boldsymbol{z}}^{\text{SI}}_{l}=\boldsymbol{H}_{ll}\sum_{m\in\mathcal{M}}\boldsymbol{s}_{l}[m]$,
and
$\boldsymbol{z}^{\text{IAI}}_{l}\sim\mathcal{CN}(0,\xi_{l}^{\text{IAI}}\text{diag}(\mathbb{E}[\bar{\boldsymbol{z}}^{\text{IAI}}_{l}(\bar{\boldsymbol{z}}^{\text{IAI}}_{l})^{H}]))$
with
$\bar{\boldsymbol{z}}^{\text{IAI}}_{l}=\sum_{l^{\prime}\in\mathcal{L}\backslash\left\\{l\right\\}}\sum_{m\in\mathcal{M}}\boldsymbol{H}_{ll^{\prime}}[m]\boldsymbol{s}_{l^{\prime}}[m]$,
where $\xi_{l}^{\text{IAI}}$ denotes the residual IAI level at the AP side.

Due to the distributed processing nature in our proposed system, each AP
firstly processes the signal received from MSs using the local combining
vectors, expressed as
$\tilde{y}_{l}[\bar{m}]=\boldsymbol{w}_{ld}^{H}[\bar{m}]\boldsymbol{y}_{l}[\bar{m}]$,
where $\boldsymbol{w}_{ld}[\bar{m}]$ denotes the local combining vector of AP
$l$ for detecting MS $d$. Then, the locally estimated data by all APs are
forwarded to the CPU for final processing, which can be expressed as
$y_{\text{cpu}}[\bar{m}]=\sum_{l\in\mathcal{L}}\tilde{y}_{l}[\bar{m}]$. It can
be shown that the SINR at the CPU for detecting the data transmitted on the
$\bar{m}$-th UL subcarrier of MS $d$ can be expressed as

$\text{SINR}_{d,\bar{m}}=\frac{p_{d\bar{m}}|\tilde{\boldsymbol{w}}_{d}[\bar{m}]\tilde{\boldsymbol{h}}_{d}[\bar{m}]|^{2}}{\text{MUI}_{d,\bar{m}}+\text{SI}_{d,\bar{m}}+\text{IAI}_{d,\bar{m}}+\sigma^{2}\left\|\tilde{\boldsymbol{w}}_{d}[\bar{m}]\right\|^{2}}\
,$ (6)

where
$\tilde{\boldsymbol{w}}_{d}[\bar{m}]=[\boldsymbol{w}_{1d}^{H}[\bar{m}],...,\boldsymbol{w}_{Ld}^{H}[\bar{m}]]$,
$\tilde{\boldsymbol{h}}_{d}[\bar{m}]=[\boldsymbol{h}_{1d}^{H}[\bar{m}],...,\boldsymbol{h}_{Ld}^{H}[\bar{m}]]^{H}$,
$\text{MUI}_{d,\bar{m}}=\sum_{d^{\prime}\in\mathcal{D}\backslash\left\\{d\right\\}}p_{d^{\prime}\bar{m}}|\tilde{\boldsymbol{w}}_{d}[\bar{m}]\tilde{\boldsymbol{h}}_{d^{\prime}}[\bar{m}]|^{2}$,
$\text{SI}_{d,\bar{m}}=\sum_{l\in\mathcal{L}}\mathbb{E}[\left\|\boldsymbol{w}_{ld}^{H}[\bar{m}]\boldsymbol{z}^{\text{SI}}_{l}\right\|^{2}]$
and
$\text{IAI}_{d,\bar{m}}=\sum_{l\in\mathcal{L}}\mathbb{E}[\left\|\boldsymbol{w}_{ld}^{H}[\bar{m}]\boldsymbol{z}^{\text{IAI}}_{l}\right\|^{2}]$.

###  II-D Beamforming Design

In this paper, the ZF beamforming strategy is employed for both transmitting
and receiving at APs. It is well-known that the MMSE beamforming slightly
outperforms the ZF beamforming in mMIMO, when perfect CSI is available and
especially when SINR is low [35]. However, considering the multiuser
interference suppression, computation complexity as well as the concise
formulation, ZF is considered in the following analysis.

According to the ZF principle [36], the precoder/combiner at the $l$-th AP,
expressed as
$\boldsymbol{F}^{\text{ZF}}_{l}[m]=[\boldsymbol{f}_{l1}^{\text{ZF}}[m],...,\boldsymbol{f}_{lD}^{\text{ZF}}[m]]$
and
$\boldsymbol{W}^{\text{ZF}}_{l}[\bar{m}]=[\boldsymbol{w}_{l1}^{\text{ZF}}[\bar{m}],...,\boldsymbol{w}_{lD}^{\text{ZF}}[\bar{m}]]$,
can be derived as
$\boldsymbol{F}^{\text{ZF}}_{l}[m]=\boldsymbol{H}_{l}^{H}[m](\boldsymbol{H}_{l}[m]\boldsymbol{H}_{l}^{H}[m])^{-1}$
and
$\boldsymbol{W}^{\text{ZF}}_{l}[\bar{m}]=\boldsymbol{H}_{l}[\bar{m}](\boldsymbol{H}_{l}^{H}[\bar{m}]\boldsymbol{H}_{l}[\bar{m}])^{-1}$,
respectively, where
$\boldsymbol{H}_{l}[m]=[\boldsymbol{h}_{l1}[m],...,\boldsymbol{h}_{lD}[m]]^{H}$
and
$\boldsymbol{H}_{l}[\bar{m}]=[\boldsymbol{h}_{l1}[\bar{m}],...,\boldsymbol{h}_{lD}[\bar{m}]]$.
Note that, in order to ensure that the MUI is fully suppressed, the
implementation of ZF beamforming should adhere to the constraint of $N\geq D$
111We will discuss the case of $D\gg N$ in Section IV-D along with the user-
centric clustering algorithm.. In this case, the MUI terms in (4) and (6) are
equal to zero. Therefore, the $\text{SINR}_{d,m}$ and
$\text{SINR}_{d,\bar{m}}$ can be rewritten as

$\displaystyle\text{SINR}_{d,m}$
$\displaystyle=\frac{|\sum_{l\in\mathcal{L}}\sqrt{p_{ldm}}\omega_{ldm}|^{2}}{\xi_{d}^{\text{SI}}\Theta_{\text{DL}}+\sigma^{2}},$
$\displaystyle\text{SINR}_{d,\bar{m}}$
$\displaystyle=\frac{p_{d\bar{m}}L^{2}}{\sum_{l\in\mathcal{L}}\upsilon_{ld\bar{m}}(\xi_{l}^{\text{SI}}\Theta_{\text{UL}}+\sigma^{2})},$
(7)

where $\omega_{ldm}=1/{\|\boldsymbol{f}_{ld}^{\text{ZF}}[m]\|_{2}}$,
$\upsilon_{ld\bar{m}}=\|\boldsymbol{w}_{ld}^{\text{ZF}}[\bar{m}]\|_{2}^{2}$,
$\Theta_{\text{DL}}=\sum_{\bar{m}\in\bar{\mathcal{M}}}p_{d\bar{m}}+\sum_{d^{\prime}\in\mathcal{D}\backslash\left\\{d\right\\}}\sum_{\bar{m}\in\bar{\mathcal{M}}}(\xi_{d}^{\text{IMI}}\beta_{dd^{\prime}}p_{d^{\prime}\bar{m}}/{\xi_{d}^{\text{SI}}}{M_{\text{sum}}})$
and
$\Theta_{\text{UL}}=\sum_{m\in\mathcal{M}}\sum_{d\in\mathcal{D}}p_{ldm}+\sum_{l^{\prime}\in\mathcal{L}\backslash\left\\{l\right\\}}\sum_{m\in\mathcal{M}}\sum_{d\in\mathcal{D}}(\xi_{l}^{\text{IAI}}\beta_{ll^{\prime}}p_{l^{\prime}dm}/{\xi_{l}^{\text{SI}}}{M_{\text{sum}}})$.

###  II-E Problem Formulation

Given the $\text{SINR}_{d,m}$ and $\text{SINR}_{d,\bar{m}}$ as shown in
(II-D), the optimization problem can be formulated as

$\displaystyle\max_{p_{ldm},p_{d\bar{m}}}\Lambda_{\text{SE}}$ (8a) s.t.
$\displaystyle~{}~{}\sum_{m\in\mathcal{M}}\sum_{d\in\mathcal{D}}p_{ldm}\leq
P_{l},\ \forall l\in\mathcal{L},$ (8b)
$\displaystyle~{}~{}\sum_{\bar{m}\in\bar{\mathcal{M}}}p_{d\bar{m}}\leq P_{d},\
\forall d\in\mathcal{D},$ (8c)
$\displaystyle~{}~{}\sum_{m\in\mathcal{M}}\ln(1+\text{SINR}_{d,m})\geq\chi_{\text{DL}},\
\forall d\in\mathcal{D},$ (8d)
$\displaystyle~{}~{}\sum_{\bar{m}\in\mathcal{\bar{M}}}\ln(1+\text{SINR}_{d,\bar{m}})\geq\chi_{\text{UL}},\
\forall d\in\mathcal{D},$ (8e)

where
$\Lambda_{\text{SE}}=\frac{1}{M_{\text{sum}}}\sum_{d\in\mathcal{D}}\big{(}\sum_{m\in\mathcal{M}}\ln(1+\text{SINR}_{d,m})+\sum_{\bar{m}\in\mathcal{\bar{M}}}\ln(1+\text{SINR}_{d,\bar{m}})\big{)}$,
$\chi_{\text{DL}}$ and $\chi_{\text{UL}}$ denote the DL and UL QoS
requirements, respectively.

It can be observed that (8) is a NP-hard nonconvex problem with the nonconvex
constraints as shown in (8d) and (8e). Generally, to solve this kind of
complicated problems, we have to transform the main optimization function as
well as the constraints into the approximated convex ones, and then solve them
in an iterative way. With this consideration, in this paper, we propose a QT-
SCA method, which can be found in detail in Appendix A. Unfortunately, the
complexity of QT-SCA increases significantly as the size of network becomes
large, especially when the multicarrier FD-like systems, such as our proposed
MDD-CF system, are considered. To this end, we resort to the GNN to solve this
optimization problem, which will be demonstrated to exhibit low complexity and
high efficiency.

##  III Graph Learning in MDD-CF networks

In order to leverage the structural information of MDD-CF network to solve the
complicated problem, as that formulated in (8), the heterogeneous graph
learning based PA scheme is introduced. We aim to learn a scalable and
transferable HGNN to efficiently distribute both APs’ and MSs’ transmit power
to maximize the SE of the distributed MDD-CF systems.

###  III-A Definition of Heterogeneous Graph

A heterogeneous graph can be represented as $G=(\mathcal{V},\mathcal{E})$,
where $\mathcal{V}$ is the set of nodes, and $\mathcal{E}$ is the set of
edges. The heterogeneous graph has a node type mapping function
$\phi:\mathcal{V}\longrightarrow\mathcal{Q}$ and an edge type mapping function
$\psi:\mathcal{E}\longrightarrow\mathcal{P}$, where $\mathcal{Q}$ and
$\mathcal{P}$ denote the sets of predefined node types and link types,
$\left|\mathcal{Q}\right|+\left|\mathcal{P}\right|>2$ [37]. Explicitly, we can
write $\mathcal{Q}=\\{Q_{1},Q_{2},...\\}$ and
$\mathcal{P}=\\{P_{1},P_{2},...\\}$, where $Q_{i}$ and $P_{j}$ are the $i$-th
node type and $j$-th link type. Let
$\boldsymbol{v}_{i}\in\mathbb{R}^{F_{v}\times 1}$ denote a node with
$F_{v}$-dimensional features and
$\boldsymbol{e}_{i,j}\in\mathbb{R}^{F_{e}\times 1}$ denote an edge pointing
from $\boldsymbol{v}_{j}$ to $\boldsymbol{v}_{i}$, which has
$F_{e}$-dimensional features. Given the mapping function $\phi$ and $\psi$,
each node belongs to a particular node type of
$\phi(\boldsymbol{v})\in\mathcal{Q}$ and each edge belongs to a specific
relation of $\psi(\boldsymbol{e})\in\mathcal{P}$. The neighborhood of a node
$\boldsymbol{v}_{i}$ is defined as
$\mathcal{N}_{i}=\left\\{\boldsymbol{v}_{j}\in\mathcal{V}|\boldsymbol{e}_{i,j}\in\mathcal{E}\right\\}$.
Furthermore, in a heterogeneous graph, two nodes may be connected via
different semantic paths. For example, an AP and an MS can be connected via
two paths, namely the AP-DL-MS and MS-UL-AP links. Therefore, we introduce the
concept of meta-path as in [38]. In detail, a meta-path $\Phi$ is defined as a
path in the form of $Q_{1}\stackrel{{\scriptstyle
P_{1}}}{{\longrightarrow}}Q_{2}\stackrel{{\scriptstyle
P_{2}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
P_{n}}}{{\longrightarrow}}Q_{n+1}$, which defines a composite relation
$P=P_{1}\circ P_{2}\circ\ldots P_{n}$ from node type $Q_{1}$ to node type
$Q_{n+1}$, where $\circ$ denotes the composition operator on relations. Once a
meta-path $\Phi$ is given, the specific neighbors $\mathcal{N}_{i}^{\Phi}$ of
node $\boldsymbol{v}_{i}$ can be obtained, which are given by the set of nodes
connected with $\boldsymbol{v}_{i}$ via meta-path $\Phi$.

Figure 2: An exemplified graph of CF network.

###  III-B Heterogeneous Graph for MDD-CF Networks

Intuitively, as shown in Fig. 2, it is straightforward to model an MDD-CF
network as a heterogeneous graph. It can be observed from the figure that
there are two types of nodes, namely AP and MS, each of which is associated
with two meta-paths. Specifically, for AP $l$, all MSs are connected with it
via the meta-path $\Phi_{1}^{\text{I}}$ (MS-UL-AP), and the remaining APs are
connected with it via the meta-path $\Phi_{2}^{\text{I}}$ (AP-IAI-AP). Note
that, the SI caused by AP $l$ itself is classified into $\Phi_{2}^{\text{I}}$
by adding a self-loop. Similarly, the meta-paths $\Phi_{1}^{\text{II}}$ and
$\Phi_{2}^{\text{II}}$ associated with MS $d$ are AP-DL-MS and MS-IMI-MS,
respectively.

In the heterogeneous graph of MDD-CF, the node feature vectors of AP $l$ and
MS $d$ are defined as

$\displaystyle\boldsymbol{v}_{l}\in\mathbb{R}^{(DM+3)\times 1}$
$\displaystyle=\left[\boldsymbol{\omega}_{l1}^{T}...\boldsymbol{\omega}_{lM}^{T},P_{l},\xi_{l}^{\text{SI}},\xi_{l}^{\text{IAI}}\right]^{T},$
$\displaystyle\boldsymbol{v}_{d}\in\mathbb{R}^{(L\bar{M}+3)\times 1}$
$\displaystyle=\left[\boldsymbol{\upsilon}_{1d}^{T}...\boldsymbol{\upsilon}_{\bar{M}d}^{T},P_{d},\xi_{d}^{\text{SI}},\xi_{d}^{\text{IMI}}\right]^{T},$
(9)

respectively, where
$\boldsymbol{\omega}_{lm}=\left[{\omega}_{l1m},...,{\omega}_{lDm}\right]^{T}$
is the $m$-th DL equivalent subchannel gains between AP $l$ and all the $D$
MSs, and
$\boldsymbol{\upsilon}_{\bar{m}d}=\left[{\upsilon}_{1d\bar{m}},...,{\upsilon}_{Ld\bar{m}}\right]^{T}$
is the $\bar{m}$-th UL equivalent subchannel gains between MS $d$ and all the
$L$ APs, when the ZF beamforming as in equation (II-D) is applied. Moreover,
to simplify the model for the sake of reducing complexity, the attribute of
edges is assumed to be the Euclidean distance between any two nodes, expressed
as $e_{i,j}=d_{i,j},\forall i,j\in\mathcal{V}$. If $i=j,e_{i,j}=0$ denotes the
edge feature of self-loop.

######  Remark 1.

In the distributed MDD-CF systems, each AP equipped with a baseband processor
is able to independently implement DL/UL beamforming, and the CPU only needs
to collect the processed signals from APs and then accomplish the final data
detection. Hence, in order to fully exploit the APs’ computational potentials
while avoiding using long-stacked channel vectors as node features, we assume
that, each AP first computes the equivalent DL/UL subchannel gains based on
the estimated CSI. Then, the equivalent DL/UL subchannel gains are transmitted
to the CPU as the AP/MS node features. All these are done during the offline
graph training222In principle, apart from ZF, the beamforming methods, such as
MMSE or matched filtering, may also be employed. If the number of antennas at
individual AP and the numbers of MSs and subcarriers are small, it is possible
to represent the AP and MS node features by their channel vectors, which have
the size of $(2DNM+3)$ and $(2LN\bar{M}+3)$, respectively, when both the real
and imaginary parts of the complex vectors are considered. Correspondingly,
the beamforming vectors associated with PA can be directly learned at the
final output layer, rather than choosing one of the known beamforming schemes
to generate the network inputs. However, if the numbers of antennas at each
AP, MSs and subcarriers are large, the huge dimensions of node features will
make the model extremely hard to train.. Note that in some references, such as
[39], the authors used the geographic location information (GLI) as the input
of the learning-based network to reduce the training overhead. This approach
has the advantage that the low-dimensional coordinate values of AP/MS can be
directly used as node features without requiring to computing beamformers.
However, in MDD-based systems, power needs to be allocated among different
subcarriers, while the GLI lacks not only the small-scale information of the
involved communication channels but also the multi-antenna characteristics of
MIMO systems. Explicitly, the GLI is unable to provide the required
information and hence, is infeasible in MDD-based systems.

###  III-C Heterogeneous Graph Learning Assisted Power Allocation in MDD-CF
Networks

In this subsection, we formally present the CF-HGNN to solve the PA problem in
MDD-CF networks. The architecture of CF-HGNN consists of four components: 1)
adaptive node embedding; 2) meta-path based message passing; 3) meta-path
based attention; 4) downstream PA learning. The overall CF-HGNN is type-
specific and the parameters for processing AP and MS nodes are not shared.

Figure 3: Adaptive node embedding layer.

####  III-C1 Adaptive Node Embedding

In general, in order to guarantee the scalability of GNN, the size of node
feature should be irrelevant to the number of nodes involved. As we can see in
(III-B), the feature dimensions of input nodes are related to $L$ and $D$,
which vary with respect to the scales of CF networks. Hence, they are not
feasible to exhibit the scalability and transferability. To tackle this
problem, we propose an adaptive node embedding layer, which can handle graph
nodes with varying input feature dimensions through adaptively activating or
deactivating the neurons in the multi-layer perceptron (MLP), i.e.,
$\Xi_{\text{ada}}(\boldsymbol{v}_{l})$ and
$\Xi_{\text{ada}}(\boldsymbol{v}_{d})$, as depicted in Fig. 3. Specifically,
before the normalization layer, AP and MS nodes are transformed by two
embedding matrices, which are expressed as

$\displaystyle\boldsymbol{v}_{l}^{{}^{\prime}}$
$\displaystyle=\boldsymbol{W}_{\text{AP}}^{(:,1:DM+3)}\boldsymbol{v}_{l},$
$\displaystyle\boldsymbol{v}_{d}^{{}^{\prime}}$
$\displaystyle=\boldsymbol{W}_{\text{MS}}^{(:,1:L\bar{M}+3)}\boldsymbol{v}_{d},$
(10)

where
$\boldsymbol{W}_{\text{AP}}\in\mathbb{R}^{F_{\text{AP}}^{{}^{\prime}}\times
F_{\text{AP}}}$ and
$\boldsymbol{W}_{\text{MS}}\in\mathbb{R}^{F_{\text{MS}}^{{}^{\prime}}\times
F_{\text{MS}}}$ map different AP and MS input nodes into two feature domains
with predefined sizes of $F_{\text{AP}}^{{}^{\prime}}$ and
$F_{\text{MS}}^{{}^{\prime}}$. Note that, as the ZF beamforming is applied,
the maximum input size of the embedding layer for AP nodes is subject to the
number of antennas employed at individual AP, having the relationship of
$F_{\text{AP}}=NM+3$. In the context of MS, we assume
$F_{\text{MS}}=L^{{}^{\prime}}\bar{M}+3$, where $L^{{}^{\prime}}$ denotes the
maximum number of APs that can be deployed in a CF network within a certain
area.

######  Remark 2.

In Section II-D, we assumed that, in our proposed distributed MDD-CF system,
each AP having $N\geq D$ antennas individually communicates with all the $D$
MSs using ZF beamforming. In this case, the above-mentioned embedding layer
imposes strict limitation on the maximum number of MSs, which should not
exceed the number of antennas configured at each AP. However, if MSs are
densely distributed or each AP is only equipped a small number of antennas,
the AP node features can no longer be set as (III-B), as the ZF beamforming
cannot be achieved in the case of $D>N$ 333Although the MMSE beamforming can
be applied in the case of $D>N$, the increasing number of MSs leads to large
multiuser interference and high dimensions of AP node features.. Moreover, in
the above model, the number of APs is restricted to $L^{{}^{\prime}}$, meaning
that the CF-HGNN cannot be generalized to the CF networks with more than
$L^{{}^{\prime}}$ APs 444In principle, $L^{{}^{\prime}}$ can be predefined to
a sufficiently large number so as to make CF-HGNN scalable to the network with
densely distributed APs. However, in doing so, it will significantly increase
the training overhead of the CF-HGNN.. In these cases, to guarantee the
scalability of CF-HGNN and reduce the computational complexity, one possible
approach is to transform the dense graph into the sparse graph with the aid of
user-centric clustering [1], where each AP only serves a certain number of
MSs. For example, if ZF beamforming is used, each AP can serve up to $N$ MSs
depending on the channel conditions. Given this constraint, the dimensions of
the AP node features are only related to the number of antennas, while that of
the MSs node features are only relied on a fraction of APs. Hence, our node
feature definition of (III-B) is still applicable. In Section IV-D, we will
present an example accounting for the user-centric clustering.

Figure 4: The overall architecture of CF-HGNN network.

####  III-C2 Meta-Path Based Message Passing

According to the example shown in Fig. 2, each AP/MS node connects with its
neighboring nodes via two meta-paths, and all the meta-paths are mutually
independent. Hence, within each meta-path $\Phi$, the process of message
passing from neighboring nodes to itself is formulated as

$\boldsymbol{z}_{i}^{\Phi}=\text{Mean}\Big{(}\Xi_{2}\Big{(}\boldsymbol{v}_{i}^{{}^{\prime}}\oplus\frac{1}{\left|\mathcal{N}_{i}^{\Phi}\right|}\Xi_{1}(\sum_{j\in\mathcal{N}_{i}^{\Phi}}\boldsymbol{v}_{j}^{{}^{\prime}}\cdot
e_{i,j})\Big{)}+\underbrace{\boldsymbol{v}_{i}^{{}^{\prime}}}_{\text{Res}}\Big{)},$
(11)

where $\Xi_{1}$ and $\Xi_{2}$ represent two different MLPs, with each
including the fully-connected, activation and normalization layers. The Res
$\boldsymbol{v}_{i}^{{}^{\prime}}$ term denotes the residual connection, which
can maintain the original node information after the multi-layer message
passing. $\oplus$ is the concatenation operation.

####  III-C3 Meta-Path Based Attention

In general cases, the information update of node $i$ is the summation of the
message collected from all the meta-paths with the same coefficients,
expressed as $\boldsymbol{z}_{i}=\sum_{{\Phi}}\boldsymbol{z}_{i}^{\Phi}$.
However, in MDD-CF networks, since AP/MS nodes receive information via both
the interfering path and data transmission path, it is intuitive that these
two meth-paths may has quite different impact on the information integration.
For instance, if an AP node is closely surrounded by MS nodes but the other AP
nodes are far away from it, the meta-path $\Phi_{1}^{\text{I}}$ should be more
important than $\Phi_{2}^{\text{I}}$. Based on this observation, we propose
the meta-based attention in CF-HGNN to enable the GNN to automatically learn
the importance of the two meta-paths. An example of calculating the attention
vector of an AP node is as follows [40]:

$\displaystyle\alpha_{\Phi_{1}^{\text{I}}}$
$\displaystyle=\frac{1}{L}\sum_{l\in\mathcal{L}}\boldsymbol{q}^{T}\Xi_{\text{att}}(\boldsymbol{z}_{l}^{\Phi_{1}^{\text{I}}}),$
$\displaystyle\beta_{\Phi_{1}^{\text{I}}}$
$\displaystyle=\frac{\text{exp}(\alpha_{\Phi_{1}^{\text{I}}})}{\text{exp}(\alpha_{\Phi_{1}^{\text{I}}})+\text{exp}(\alpha_{\Phi_{2}^{\text{I}}})},$
(12)

where $\Xi_{\text{att}}$ is the MLP layer for attention, $\boldsymbol{q}$ is
the learnable attention vector, $\boldsymbol{z}_{l}^{\Phi_{1}^{\text{I}}}$
denotes the aggregated information via meta-path $\Phi_{1}^{\text{I}}$. Then,
the final node representation of AP $l$ is
$\boldsymbol{z}_{l}=\beta_{\Phi_{1}^{\text{I}}}\boldsymbol{z}_{l}^{\Phi_{1}^{\text{I}}}+\beta_{\Phi_{2}^{\text{I}}}\boldsymbol{z}_{l}^{\Phi_{2}^{\text{I}}}$,
where $\boldsymbol{z}_{l}\in\mathbb{R}^{F_{\text{AP}}^{{}^{\prime}}\times 1}$.
Similarly, $\boldsymbol{z}_{d}\in\mathbb{R}^{F_{\text{MS}}^{{}^{\prime}}\times
1}$ at MS $d$ can be obtained. It is noteworthy that the process of message
passing plus attention can be iteratively implemented for $K$ times by
initializing $\boldsymbol{z}_{l}^{(0)}=\boldsymbol{v}_{l}^{\prime}$ and
$\boldsymbol{z}_{d}^{(0)}=\boldsymbol{v}_{d}^{\prime}$, so as to collect the
high-hop neighbors. Owing to this, our proposed model can also be termed as
the K-layer CF-HGNN.

####  III-C4 Downstream Power Allocation Learning

After the $K$-th iteration, the final representation of AP and MS nodes, i.e.,
$\boldsymbol{z}_{l}^{(K)}$ and $\boldsymbol{z}_{d}^{(K)}$, are used for the
downstream PA learning, which can be expressed as:

$\begin{split}\boldsymbol{p}_{l}\in\mathbb{R}^{DM\times
1}&=\text{Relu}\Big{(}\Xi_{\text{PA}}(\boldsymbol{z}_{l}^{(K)})^{(1:DM)}\Big{)},\\\
\boldsymbol{p}_{d}\in\mathbb{R}^{\bar{M}\times
1}&=\text{Relu}\Big{(}\Xi_{\text{PA}}(\boldsymbol{z}_{d}^{(K)})\Big{)},\end{split}$
(13)

where $\Xi_{\text{PA}}$ denotes the MLP for PA learning, and Relu($\cdot$) is
used to constrain the power to be positive. Corresponding to the adaptive node
embedding considered in part 1), in order to make it feasible for the network
with different number of MSs, the adaptive output layer can support the PA for
up to $NM$ elements by activating or deactivating the neurons. By contrast,
for UL PA, provided that the number of available UL subcarriers is fixed, the
size of the output layer for MS node remains to $\bar{M}$, as each MS only
needs to allocate UL power.

The overall architecture of the CF-HGNN for an AP node is shown in Fig. 4,
which is the same for MS nodes, except the different size of output layer as
above-mentioned. Note that all the MLPs shown in Fig. 4 are AP-specific, which
means that the MLPs for MSs are different from that for APs.

Finally, to perform the CF-HGNN training in an unsupervised way, we define the
loss function as

$\displaystyle\mathcal{L}(\boldsymbol{\theta})$
$\displaystyle=\mathbb{E}\bigg{[}-\Lambda_{\text{SE}}+\sum_{d=1}^{D}\bigg{(}\kappa_{1}\text{Relu}\big{(}\chi_{\text{DL}}-\sum_{m\in\mathcal{M}}\ln(1+\text{SINR}_{d,m})\big{)}$
$\displaystyle+\kappa_{2}\text{Relu}\big{(}\chi_{\text{UL}}-\sum_{\bar{m}\in\mathcal{\bar{M}}}\ln(1+\text{SINR}_{d,\bar{m}})\big{)}$
$\displaystyle+\kappa_{3}\text{Relu}\big{(}\sum_{\bar{m}\in\bar{\mathcal{M}}}p_{d\bar{m}}-P_{d}\big{)}\bigg{)}+\sum_{l=1}^{L}\kappa_{4}\text{Relu}\big{(}\sum_{m\in\mathcal{M}}\sum_{d\in\mathcal{D}}p_{ldm}-P_{l}\big{)}\bigg{]},$
(14)

where $\boldsymbol{\theta}$ denotes all the parameters of the neural network,
$\kappa_{i}$ denotes the weighted factor, and the expectation is taken with
respect to the channel realizations. In (III-C4), each ReLU penalty term has a
positive value, only if the DL/UL QoS requirements and transmit power budgets
are not satisfied. This will enforce the training process towards satisfying
the given requirements. Additionally, in (III-C4), the positive parameters
$\kappa_{i}$ give different priorities to the penalty terms.

##  IV Simulation Results and Analysis

Let us now demonstrate the achievable SE performance of the distributed MDD-CF
systems supported by our proposed CF-HGNN.

###  IV-A Simulation Setup

The large-scale fading model is given by [1]:

$\beta[\text{dB}]=-30.5-36.7\log_{10}{(d)}+\sigma_{\text{sh}}z,$ (15)

where $d$ denotes the distance between any two nodes, $\sigma_{\text{sh}}z$ is
the shadowing fading with a standard deviation of $\sigma_{\text{sh}}=4$ dB
and $z\sim\mathcal{N}(0,1)$. Furthermore, we assume that APs are capable of
providing 30 dB of IAI suppression in the propagation/analog domain by
employing the existing approaches [4]. Then, assume that the 12-bit ADCs are
applied, the MDD system can suppress IAI up to 72 dB (i.e.,
$\xi_{l}^{\text{MDD-IAI}}=-72$ dB, $\forall l$, in (5)), of which 42 dB is
attributed to the digital-domain cancellation by FFT 555The 12-bit ADC has a
maximum dynamic range of 42dB, which means it can accommodate up to 42 dB of
power of IAI, and transform it into digital signal without extra quantization
noise. Then, in the digital domain, since the interference signal is
transmitted over DL subcarriers, which is mutually orthogonal to the desired
UL signal, MDD systems supported by the FFT operation can then remove the
remaining IAI.. By contrast, as MSs are of lightweight equipments with single
antenna and can hardly share channel knowledge with other MSs, they can not
actively suppress IMI. However, similar to APs, MSs are able to cancel 42 dB
IMI in the digital domain (i.e., $\xi_{d}^{\text{MDD-IMI}}=-42$ dB, $\forall
d$, in (3)) with the aid of the FFT operation [33]. Unless otherwise noted,
the simulation parameters of the MDD-CF network are listed in Table I.

As for the settings of the neural network, we adopt a 2-layer CF-HGNN based on
Pytorch Geometric [41]. The general MLPs $\Xi_{1}$ and $\Xi_{2}$ in (11)
during the message passing stage contains multiple fully-connected linear
layers followed by the LeakyRelu activation layer and batch normalization
layer. By contrast, the $\Xi_{\text{PA}}$ in (13) is employed with multiple
fully-connected linear layers without batch normalization layer. Moreover, the
$\Xi_{\text{ada}}$ in (III-C1) is a single-layer MLP with one fully-connected
linear layer followed by one batch normalization layer, while
$\Xi_{\text{att}}$ in (III-C3) is a single-layer MLP with only one fully-
connected linear layer. As formulated in (III-C4), the overall learning is
unsupervised without any ground truth. To optimize the CF-HGNN, we adopt the
Adam optimizer with a learning rate of 0.001 [42]. Furthermore, we empirically
set $\kappa_{i}$ in (III-C4) as $\\{0.1,1,0.1,0.1\\}$ during training. For the
training data, we randomly generate 10000 and 1000 CF network layout samples
for training and testing, respectively, under the assumption that APs and MSs
are uniformly distributed within a square area of $(S_{D}\times
S_{D})\text{m}^{2}$ . The batch size for training is 64, and the network
parameters are only updated during training, which stay constant during
testing. The CF-HGNN is run on a GeForce GTX laptop 3080Ti, while the other
algorithms are implemented on the 12th Gen Intel(R) Core(TM) i7-12700H 2.70
GHz.

For comparison, we introduce QT-SCA described in Appendix A as a benchmark,
which is implemented using the CVX tool during simulations [43]. In addition,
our proposed CF-HGNN is also compared with the greedy unfair allocation method
[44], where the water-filling algorithm is assumed by each AP/MS to distribute
their power over the DL/UL subcarriers, regardless of the QoS constraints.

TABLE I: Simulation parameters Default parameters | Value
---|---
Cell area ($S_{\text{D}}\times S_{\text{D}}$) | $(400\times 400)\text{m}^{2}$
Number of antennas per AP ($N$) | 8
Number of DL/UL subcarriers ($M,\bar{M}$) | (4, 2)
AP and MS power ($P_{l},P_{d},\forall l,d$) | $(40,30)\ \text{dBm}$
QoS requirements ($\chi_{\text{DL}},\chi_{\text{UL}}$) | (0.5, 0.1) nats/s/Hz
Noise power ($\sigma^{2}$) | -94 dBm
Delay taps ($U$) | 4
Residual SI level at AP ($\xi_{l}^{\text{SI}},\forall l$) | -120 dB
Residual SI level at MS ($\xi_{d}^{\text{SI}},\forall d$) | -110 dB

###  IV-B Performance Comparison

In this section, we make a comprehensive performance comparison between
different PA methods in MDD-CF networks, where the numbers of MSs and APs are
fixed during training and testing, and are set to $L=24,D=6$. In this case,
the embedding matrices in $\Xi_{\text{ada}}$ and $\Xi_{\text{PA}}$ act as
identical matrices and are not learnable.

Firstly, we compare the QT-SCA, CF-HGNN and the greedy unfair methods in terms
of the SE distribution obtained from 1000 testing CF network layouts. As shown
in Fig. 5, the CF-HGNN achieves nearly the same performance as the QT-SCA in
terms of the 95%-likely SE. In more detail, the SE performance gaps between
the QT-SCA and CF-HGNN with regard to these 1000 network layouts are depicted
in Fig. 6. It can be observed that the absolute SE gaps between these two
methods are lower than 1.5 nats/s/Hz. Furthermore, there are several layouts,
where the CF-HGNN outperforms the QT-SCA, which reflects that the CF-HGNN is
capable of learning the near-optimal solutions for PA in MDD-CF networks. As
shown in Fig. 5, both the QT-SCA and CF-HGNN significantly outperform the
greedy unfair method. The rationale behind is that although the greedy unfair
method can maximize the SE of the classic HD-based multiuser OFDM systems
[44], it lacks the capability to manage the complicated interference in the CF
FD-like systems, hence leading to poor performance at both APs and MSs.

Figure 5: Cumulative distribution of SE, when the MDD-CF network has $L=24$
APs and $D=6$ MSs. Figure 6: SE performance gap between QT-SCA and CF-HGNN,
when the MDD-CF network uses $L=24$ APs to support $D=6$ MSs.

Secondly, we randomly select one of the CF network layouts used in the testing
samples, as shown in Fig. 7, to deeply investigate the PA attained by the QT-
SCA and CF-HGNN. To make the drawing clear in Fig. 7, the DL connections with
transmit power less than $2W$ are omitted, and different colored lines are
used to denote the DL connections obtained by either methods or by both. It is
not surprise that, as Fig. 7 shows, except MS 2, both the CF-HGNN and QT-SCA
yield the same subset of the major serving APs for each of MSs, and also
obtain the similar results of UL PA. As for MS 2, since it locates relatively
far away from APs, more APs are required to transmit signal to it in order to
meet the demand of DL’s QoS. Although different DL connections are obtained by
the two methods, there is in fact no difference between them. For example,
apart from the common connections, the CF-HGNN picks AP 3 and 9 to serve MS 2,
while the QT-SCA chooses AP 8 located between AP 3 and 9 to serve MS 2. Also,
AP 20 and 17 with the similar distances from MS 2 are selected by the CF-HGNN
and QT-SCA, respectively.

Figure 7: An MDD-CF network topology with $L=24$ APs and $D=6$ MSs.

It is worth mentioning that the DL power (and UL power) used for Fig. 7 is the
sum of power allocated to DL subcarriers (and UL subcarriers). In Table II, an
example showing the detailed PA to subcarriers at AP 1, MS 2 and MS 6 are
presented. It can be observed that the CF-HGNN and greedy unfair methods yield
the more consistent allocation among the subcarreirs, which means that the
power distribution is largely relied on large-scale fading, while the QT-SCA
has better capability of exploiting the small-scale fading. Moreover, from the
results of the CF-HGNN and QT-SCA, we can see that MS 6 has a lower UL
transmit power than MS 2. The reason behind is that MS 6 is in the close
proximity of MS 1, the increase of its UL transmit power may cause not only
more SI on the DL reception, but also larger IMI on MS 1, leading to the
degradation of SE. By contrast, the greedy unfair scheme allocates power
depending only on the quality of communication channels regardless of the
effect of interference.

TABLE II: PA among Subcarriers at AP 1, MS 2 and MS 6. Connections | Method |  DL subcarriers’ power ($W$) |  DL power ($W$) |  UL subcarriers’ power ($W$) |  UL power ($W$)
---|---|---|---|---|---
AP 1$\rightarrow$MS 6, | QT-SCA | 1.89 | 0.81 | 1.90 | 1.89 | 6.49 | 0.006 | 0.008 | 0.014
CF-HGNN | 1.77 | 1.74 | 1.78 | 1.79 | 7.08 | 0.003 | 0.010 | 0.013
MS 6$\rightarrow$APs | Greedy unfair | 1.26 | 1.19 | 1.25 | 1.25 | 4.95 | 0.499 | 0.501 | 1.000
AP 1$\rightarrow$MS 2, | QT-SCA | 0.75 | 0.33 | 0.49 | 0.31 | 1.88 | 0.026 | 0.038 | 0.064
CF-HGNN | 0.25 | 0.23 | 0.22 | 0.27 | 0.97 | 0.000 | 0.117 | 0.117
MS 2$\rightarrow$APs | Greedy unfair | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.000 | 1.000 | 1.000
Figure 8: SE convergence behavior and operation time of different methods,
where $L=24,D=6$.

The performance of different methods in terms of SE, convergence behavior and
operation time is plotted in Fig. 8. As expected, the QT-SCA and CF-HGNN
achieve the comparable SE, with a performance gap of only 0.35 nats/s/Hz.
However, the QT-SCA is an iterative algorithm with very high computational
complexity, which converges within 6 iterations and takes 7.03s operation time
in total. On the contrary, with the aid of the parallel computation of GPU,
the CF-HGNN is capable of reaching 99% of the SE achieved by the QT-SCA but
using only $10^{-4}$ times of its operation time to. As to the greedy unfair
method, although it has a similar operation time as the CF-HGNN, due to the
fact that only the water-filling algorithm is applied at both APs and MSs, its
SE performance is significantly worse than that of the other two methods.

Next, we evaluate the generalization performance of the proposed CF-HGNN in
MDD-CF networks with different cell sizes. As shown in Table III, the results
under the Generalized CF-HGNN (GCF-HGNN) are obtained by the CF-HGNN model
trained on the specific network with $S_{\text{D}}=400$, while the results
under the Dedicated CF-HGNN (DCF-HGNN) are obtained by the CF-HGNN models
trained correspondingly using different cell sizes. It can be observed from
the table that as the cell size reduces, the performance of the GCF-HGNN
degrades slightly, but is still in the acceptable range. Note that, the DCF-
HGNN finally outperforms the QT-SCA when $S_{\text{D}}=150$, which means that
the DCF-HGNN is capable of obtaining a better PA strategy in dense networks.

TABLE III: Generalization to Different Cell Sizes. The SE of the CF-HGNN are normalized by that of the QT-SCA. $S_{\text{D}}$ |  QT-SCA (nats/s/Hz) | DCF-HGNN | GCF-HGNN
---|---|---|---
350 | 37.85 | 96.64% | 95.19%
300 | 40.64 | 98.08% | 93.13%
250 | 43.76 | 98.42% | 91.84%
200 | 46.84 | 99.91% | 91.78%
150 | 50.10 | 101.42% | 92.79%

The comparison between QT-SCA and CF-HGNN in terms of computational complexity
and operation time are then analyzed. According to [45], the approximated
computational complexity of the QT-SCA method is
$\mathcal{O}\big{(}(LDM+2DM+3D\bar{M})^{2}(L+D+3DM_{\text{sum}})^{2.5}+(L+D+3DM_{\text{sum}})^{3.5}\big{)}$
per iteration. As to the CF-HGNN, the computational complexity is mainly due
to the matrix computation, as shown in (11)-(13). Table IV summarizes the
results for the different values of $L,D,M$ and $\bar{M}$, showing that when
doubling the numbers of APs, MSs and subcarriers, the computational complexity
of the QT-SCA increases much more quickly than that of the CF-HGNN, which is
about three orders for the QT-SCA versus one order for the CF-HGNN. The
operation time cost by the QT-SCA and CF-HGNN is plotted in Fig. 9.
Explicitly, the CF-HGNN spends much less time than the QT-SCA to accomplish
the PA. Moreover, thanks to the parallel computation of GPU, the CF-HGNN
trained on GPU has the lowest operation time.

TABLE IV: Computational complexity comparison $L$ | $D$ | $M/\bar{M}$ | Methods
---|---|---|---
QT-SCA | CF-HGNN
6 | 2 | 4/2 | $4.48\times 10^{8}$ | $6.83\times 10^{6}$
12 | 4 | 8/4 | $4.78\times 10^{11}$ | $2.75\times 10^{7}$
24 | 8 | 16/8 | $6.79\times 10^{14}$ | $1.48\times 10^{8}$
Figure 9: Comparison of operation time between the QT-SCA and CF-HGNN methods.

###  IV-C Scalability: Case 1

The simulation results in Section IV-B demonstrated that the CF-HGNN is able
to achieve the similar performance as the QT-SCA. However, the CF-HGNN
considered so far can only be applied to the specific MDD-CF networks with the
fixed numbers of nodes and subcarriers, as the fully-connected liner layers in
$\Xi_{\text{ada}}$ and $\Xi_{\text{PA}}$ act as identical matrices. With this
regard, to enable the CF-HGNN to be adaptive to the varying MDD-CF networks,
we now invoke the adaptive node embedding layer and adaptive output layer. In
particular, we consider two cases regarding to the scalability of CF-HGNN. In
Case 1, we still assume that $L<L^{\prime}=24$ and $N=8$, and all MSs are
served by all APs using the ZF beamforming under the constraint of $N\geq D$.
Specifically, 10000 MDD-CF network samples with the specific numbers of nodes
and subcarriers are leveraged to train the DCF-HGNN. For fair comparison, to
train the Adaptive CF-HGNN (ACF-HGNN), 10000 MDD-CF network samples consisting
of various numbers of nodes and subcarriers are used.

Table V shows that by employing the adaptive layers, the ACF-HGNN can attain
the relatively stable performance, when dealing with the various networks.
Hence, it is feasible for implementing dynamic PA in MDD-CF networks. On the
contrary, the QT-SCA and DCF-HGNN have to re-solve the complicated
optimization problem or re-train the CF-HGNN, once the CF network has some
changes in terms of the numbers of APs, MSs and subcarriers. Note that, in the
table, the SE decreases with the increase of the number of subcarriers. This
is because the total power allocated to subcarriers is constrained, while the
SE is normalized by $M_{\text{sum}}$, as shown in (8). Apparently, the total
SE obtained by multiplying $M_{\text{sum}}$ with the value in the table always
increases with the increase of the number of subcarriers, which is attributed
to the subcarrier diversity.

TABLE V: Generalization to the different numbers of APs, MSs and subcarriers. The SE of the CF-HGNN and greedy unfair algorithm are normalized by that of the QT-SCA. $L$ | $D$ | $M/\bar{M}$ |  QT-SCA (nats/s/Hz) | DCF-HGNN | ACF-HGNN | Greedy unfair
---|---|---|---|---|---|---
$24$ | 4 | $4/2$ | 26.42 | 98.07% | 94.14% | 74.45%
5 | 31.34 | 98.02% | 94.51% | 74.06%
6 | 34.23 | 97.78% | 94.77% | 73.06%
7 | 36.24 | 98.18% | 95.45% | 73.68%
20 | $6$ | $4/2$ | 33.04 | 98.09% | 97.22% | 72.79%
16 | 29.99 | 97.70% | 95.07% | 71.16%
12 | 25.69 | 98.52% | 94.08% | 69.95%
8 | 21.13 | 97.40% | 94.46% | 68.67%
16 | 6 | 32/16 | 9.26 | 98.81% | 91.58% | 72.35%
16/8 | 15.53 | 99.03% | 91.50% | 68.45%

###  IV-D Scalability: Case 2

Although the proposed ACF-HGNN can handle the PA in various MDD-CF networks,
it can hardly cut the mustard when the numbers of APs and MSs become too
large, i.e., when $L\gg L^{\prime}$ and $D\gg N$, as explained in Remark 2.
Therefore, in Case 2, we resort to the user-centric clustering strategy to
transform the dense graph into the sparse graph, thereby maintaining the
scalability of the CF-HGNN for operation in the large-scale MDD-CF networks.
For the sake of explanation, here we consider an extreme scenario, where each
AP is only equipped with one antenna and serves only one MS using the ZF
beamforming. Correspondingly, the clustering can be achieved in two
steps666Note that the considered two-stage clustering approach is suboptimal,
and becomes less efficient in the case of $N>1$, where each AP can
simultaneously serve more than one MS using the increased spatial degrees of
freedom. To this end, our future work motivates to propose a GNN-assisted deep
Q-learning network so as to obtain the optimal user grouping and PA at the
same time.: 1) Initialize the set of APs serving MS $d$ as
$\mathcal{L}_{d}=\emptyset,\forall d=1,...,D$, and the set of MSs assigned to
AP $l$ as $\mathcal{D}_{l}=\emptyset,\forall l=1,...,L$. To guarantee a non-
zero SE, each MS $d$ is firstly assigned to a master AP by following the
optimization of

$l=\operatorname*{argmax}_{l\in\\{1,...,L\\},m\in\\{1,...,M\\}}\omega_{ldm},\
\ \mathcal{L}_{d}=\\{l\\},\ \mathcal{D}_{l}=\\{d\\}\ .$ (16)

After this optimization, we have $\left|\mathcal{L}_{d}\right|=1,\forall
d=1,...,D$. 2) If there are still idle APs, which satisfy
$\\{l^{\prime}|\mathcal{D}_{l^{\prime}}=\emptyset\\}$, allocate them to MSs
according to the optimization of

$d=\operatorname*{argmax}_{d\in\\{1,...,D\\},m\in\\{1,...,M\\}}\omega_{l^{\prime}dm},\
\ \mathcal{D}_{l^{\prime}}=\\{d\\},\
\mathcal{L}_{d}=\mathcal{L}_{d}\cup\\{l^{\prime}\\}\ .$ (17)

To train the Large-Scale ACF-HGNN (LSACF-HGNN), we have 10000 training samples
collected from the networks with $L=12$ single-antenna APs and $D=6$ MSs
uniformly distributed within an area of ($200\sqrt{2}\times
200\sqrt{2})\text{m}^{2}$. We then increase the number of APs and MSs in
testing samples while the densities of AP and MSs are fixed. Note that all the
involved training and testing samples have been pre-processed by the proposed
clustering algorithm. As shown in Table VI, although the performance of the
LSACF-HGNN is much better than that of the greedy unfair method, the gap
between the QT-SCA and the LSACF-HGNN becomes larger as the network size
increases. The rationale behind is possibly because the larger CF network has
much more complicated CLI problem than the smaller one. Hence, the LSACF-HGNN
may require extra training or an improved model structure to manage the CLI.
The detailed analysis is out of the scope of this paper and is left for the
future research. However, we should mention that, as shown in Table IV, the
QT-SCA relied optimization is extremely time-consuming, when it is applied to
the large-scale CF networks, while the LSACF-HGNN is capable of accomplishing
PA with high-efficiency.

TABLE VI: Generalization to the Large-scale MDD-CF networks. The SE of the CF-HGNN and greedy unfair algorithm are normalized by that of the QT-SCA. $S_{\text{D}}$ | $L$ | $D$ |  QT-SCA (nats/s/Hz) | LSACF-HGNN | Greedy unfair
---|---|---|---|---|---
400 | 24 | 12 | 22.93 | 93.68% | 65.68%
450 | 30 | 15 | 28.28 | 92.33% | 65.38%
500 | 38 | 19 | 35.18 | 91.42% | 65.66%
400$\sqrt{2}$ | 48 | 24 | 43.93 | 88.05% | 65.65%
400$\sqrt{3}$ | 72 | 36 | 63.92 | 82.09% | 65.35%
800 | 96 | 48 | 82.10 | 80.58% | 67.23%

##  V Conclusions

In this paper, we proposed a distributed MDD-CF system, and investigated the
PA at both AP and MS sides for SE maximization under the constraints of QoS
requirements. In order to solve the non-convex and NP-hard PA problem, we
first proposed the QT-SCA algorithm, which achieves the optimization in the
classic way. Then, the CF-HGNN was proposed to solve the optimization from a
learning perspective. The CF-HGNN was trained in an unsupervised fashion with
unlabeled data, which significantly reduces the system overhead. Our studies
and numerical results show that the CF-HGNN is capable of achieving comparable
SE performance to the QT-SCA but demanding much less operation time and
computational complexity. The CF-HGNN significantly outperforms the greedy
unfair method in terms of SE performance. Furthermore, with the aid of
adaptive node embedding layer and adaptive output layer, the CF-HGNN can
implement PA in the MDD-CF networks with various numbers of APs, MSs and
subcarriers. Additionally, with the aid of user clustering, the CF-HGNN
trained based on a relatively small-scale MDD-CF network can be generalized
for operation in the large-scale MDD-CF networks.

##  Appendix A QT-SCA Optimization

As the main function of (8) belongs to the general multiple-ratio concave-
convex fractional programming problem, it can be firstly reformulated by the
QT method [18] as

$\displaystyle\max_{\boldsymbol{p},\boldsymbol{z}}\frac{1}{M_{\text{sum}}}\sum_{d=1}^{D}\bigg{(}\sum_{m=1}^{M}\ln\big{(}1+2z_{dm}\sqrt{A_{d,m}(\boldsymbol{p})}-z_{dm}^{2}B_{d,m}(\boldsymbol{p})\big{)}$
$\displaystyle+\sum_{\bar{m}=1}^{\bar{M}}\ln(1+2z_{d\bar{m}}\sqrt{A_{d,\bar{m}}(\boldsymbol{p})}-z_{d\bar{m}}^{2}B_{d,\bar{m}}(\boldsymbol{p})\bigg{)},$
(18)

where
$\text{SINR}_{d,m}=\frac{A_{d,m}(\boldsymbol{p})}{B_{d,m}(\boldsymbol{p})}$,
$\text{SINR}_{d,\bar{m}}=\frac{A_{d,\bar{m}}(\boldsymbol{p})}{B_{d,\bar{m}}(\boldsymbol{p})}$,
$\boldsymbol{p}=(\left\\{p_{ldm}\right\\},\left\\{p_{d\bar{m}}\right\\})$. In
(A), $\ln(1+x)$ is a non-decreasing and concave function. For a given
$\boldsymbol{p}$, both $\text{SINR}_{d,m}$ and $\text{SINR}_{d,\bar{m}}$ are
also concave-convex. Hence, when $\boldsymbol{p}$ is fixed, the optimal
$\boldsymbol{z}=\left(\left\\{z_{dm}\right\\},\left\\{z_{d\bar{m}}\right\\}\right)$
maximizing (A) can be obtained as
$\boldsymbol{z}^{\ast}=\frac{\sqrt{A(\boldsymbol{p})}}{B(\boldsymbol{p})}$.
Then, given a fixed value of $\boldsymbol{z}$, the problem (A) is a concave
maximization problem over $\boldsymbol{p}$.

However, the constraints of (8d) and (8e) are still nonconvex, which need to
be approximated by the convex ones using the SCA method. As shown in (8d) and
(8e), these original constraints are extremely complicated, which contain the
sum of $M$ and $\bar{M}$ nonconvex components, respectively. Hence, we firstly
simplify them to the following ones:

$\displaystyle~{}~{}\text{SINR}_{d,m}\geq e^{\frac{\chi_{\text{DL}}}{M}}-1,\
\forall d\in\mathcal{D},m\in\mathcal{M},$ (19)
$\displaystyle~{}~{}\text{SINR}_{d,\bar{m}}\geq
e^{\frac{\chi_{\text{UL}}}{\bar{M}}}-1,\ \forall
d\in\mathcal{D},\bar{m}\in\mathcal{\bar{M}}.$ (20)

Then, based on (II-D), the simplified constraint (19) can be equivalently
written as

(21c)

where $\varpi_{d,m}$ and $\psi_{d,m}$ are new variables, while (A) and (21c)
are linear constraints. For (A), since the function
$f_{\text{ca}}(\varpi_{d,m},\psi_{d,m})\triangleq\varpi^{2}_{d,m}/\psi_{d,m}$
with $(\varpi_{d,m},\psi_{d,m})\in\mathbb{R}_{++}^{2}$ is convex, it can be
approximated with the aid of the SCA properties as [46]

$\displaystyle f_{\text{ca}}(\varpi_{d,m},\psi_{d,m})$
$\displaystyle\geq\frac{2\varpi_{d,m}^{(t)}}{\psi_{d,m}^{(t)}}\varpi_{d,m}-\frac{(\varpi_{d,m}^{(t)})^{2}}{(\psi_{d,m}^{(t)})^{2}}\psi_{d,m}:=f_{\text{ca}}^{(t)}(\varpi_{d,m},\psi_{d,m}),$
(21v)

where $(\varpi_{d,m}^{(t)},\psi_{d,m}^{(t)})$ is the feasible point obtained
at the $t$-th iteration. Consequently, (A) can be substituted by the new
constraint given by

$f_{\text{ca}}^{(t)}(\varpi_{d,m},\psi_{d,m})\geq
e^{\frac{\chi_{\text{DL}}}{M}}-1.$ (21w)

Following the same spirit, (20) can be equivalently expressed by the following
convex constraints:

(21xc)

1 Initialization:

2 Compute
$\left\\{\omega_{ldm}\right\\},\left\\{\upsilon_{ld\bar{m}}\right\\},\forall
l,d,m,\bar{m}$;

3 Set $t=0$,
$\varpi_{d,m}^{(0)}=1,\varpi_{d,\bar{m}}^{(0)}=1,\psi_{d,m}^{(0)}=1,\psi_{d,\bar{m}}^{(0)}=1,\forall
d,m,\bar{m}$;

4 Compute $\boldsymbol{p}^{(0)}$ by solving optimization problem (A);

5 [QT Iteration]  __

6 repeat

7 Compute $\boldsymbol{z}^{(t)}$ using
$\boldsymbol{z}^{(t)}=\frac{\sqrt{A(\boldsymbol{p}^{(t)})}}{B(\boldsymbol{p}^{(t)})}$
for a fixed $\boldsymbol{p}^{(t)}$ ;

8 Update $\boldsymbol{p}^{(t+1)}$ via (A), for a fixed $\boldsymbol{z}^{(t)}$
;

9 Update
$\varpi_{d,m}^{(t+1)},\varpi_{d,\bar{m}}^{(t+1)},\psi_{d,m}^{(t+1)},\psi_{d,\bar{m}}^{(t+1)}$;

10 Set $t=t+1$;

11 until  _Convergence_ ;

12 end

13

Output : $\boldsymbol{p}$

Algorithm 1 QT-SCA Algorithm for SE maximization in MDD-CF

To this point, the optimization problem (8) can be rewritten as

$\displaystyle\max_{\boldsymbol{p},\boldsymbol{z},\boldsymbol{\varpi},\boldsymbol{\psi}}$
$\displaystyle\frac{1}{M_{\text{sum}}}\sum_{d=1}^{D}\big{(}\sum_{m=1}^{M}\ln\big{(}1+2z_{dm}\sqrt{A_{d,m}(\boldsymbol{p})}-z_{dm}^{2}B_{d,m}(\boldsymbol{p})\big{)}$
$\displaystyle+\sum_{\bar{m}=1}^{\bar{M}}\ln(1+2z_{d\bar{m}}\sqrt{A_{d,\bar{m}}(\boldsymbol{p})}-z_{d\bar{m}}^{2}B_{d,\bar{m}}(\boldsymbol{p})\big{)}$
s.t. $\displaystyle z_{dm}\in\mathbb{R},\ \forall
d\in\mathcal{D},m\in\mathcal{M},$ $\displaystyle z_{d\bar{m}}\in\mathbb{R},\
\forall d\in\mathcal{D},\bar{m}\in\mathcal{\bar{M}},$
$\displaystyle(\ref{eq:MDDGNN:SE_formulation}\text{b}),(\ref{eq:MDDGNN:SE_formulation}\text{c}),\eqref{eq:MDDGNN:SINRdmEqu:b},\eqref{eq:MDDGNN:SINRdmEqu:c},\eqref{eq:MDDGNN:SINRCA},\eqref{eq:MDDGNN:SINRdbarmEqu}.$
(21xy)

During the optimization of (A), $\boldsymbol{p}$ and $\boldsymbol{z}$ are
iteratively optimized. Hence, we first need to choose the feasible values of
$\boldsymbol{p}$. Specifically, $(\varpi_{d,m}^{(0)},\psi_{d,m}^{(0)})$ and
$(\varpi_{d,\bar{m}}^{(0)},\psi_{d,\bar{m}}^{(0)})$ can be set to 1, for
$\forall d\in\mathcal{D},m\in\mathcal{M},\bar{m}\in\mathcal{\bar{M}}$. With
this regard, we resort to the optimization problem described as

$\displaystyle\operatorname*{argmax}_{\boldsymbol{p},\boldsymbol{\varpi},\boldsymbol{\psi}}\sum_{d=1}^{D}\left(\sum_{m=1}^{M}\alpha_{d,m}+\sum_{\bar{m}=1}^{\bar{M}}\alpha_{d,\bar{m}}\right)$
s.t. $\displaystyle\alpha_{d,m}\leq 0,\alpha_{d,\bar{m}}\leq 0,\forall
d\in\mathcal{D},m\in\mathcal{M},\bar{m}\in\mathcal{\bar{M}},$ $\displaystyle\
2\varpi_{d,m}-\psi_{d,m}\geq
e^{\frac{\chi_{\text{DL}}}{M}}-1+\alpha_{d,m},\forall
d\in\mathcal{D},m\in\mathcal{M},$ $\displaystyle\
2\varpi_{d,\bar{m}}-\psi_{d,\bar{m}}\geq
e^{\frac{\chi_{\text{UL}}}{\bar{M}}}-1+\alpha_{d,\bar{m}},\forall
d\in\mathcal{D},\bar{m}\in\mathcal{\bar{M}},$
$\displaystyle~{}~{}(\ref{eq:MDDGNN:SE_formulation}\text{b}),(\ref{eq:MDDGNN:SE_formulation}\text{c}),\eqref{eq:MDDGNN:SINRdmEqu:b},\eqref{eq:MDDGNN:SINRdmEqu:c},\eqref{eq:MDDGNN:SINRdbarmEqu:b},\eqref{eq:MDDGNN:SINRdbarmEqu:c},$
(21xz)

where $(\left\\{\alpha_{d,m}\right\\},\left\\{\alpha_{d,\bar{m}}\right\\})$
are new variables, and when they are zeros, the initial $\boldsymbol{p}$ can
be appropriately obtained.

The overall QT-SCA algorithm is summarized as Algorithm 1. Note that, as the
proposed QT-SCA algorithm mainly relies on the QT and SCA methods, the
detailed proof of their convergence can be found in [18] and [46],
respectively. Additionally, the convergence behavior and computational
complexity of the algorithm are evaluated in Section IV of this paper.

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