Graph Neural Networks for Physical-Layer Security in Multi-User Flexible-Duplex Networks

Complex wireless networks pose a challenge in securing data against intelligent eavesdroppers. Traditional cryptography is often insufficient. Physical layer security (PLS) leverages the dynamic wireless channels to establish secure connections, operating independently of upper-layer integrations. This simplifies network architectures and enhances security [6, 7, 8]. PLS has been extensively investigated for both conventional HD and FD communications, as discussed in [9, 10, 11, 12] and references therein. The literature on FlexD presents diverse algorithms for resource scheduling, including a graph neural network (GNN) approach [2], iterative pattern search algorithms [1], a radio frame selection algorithm for flexible duplex networks [13], and resource management in Narrowband Internet of Things (NB-IoT) [14]. However, the dynamic scheduling demands in FlexD networks pose a computationally complex challenge due to the combinatorial nature, exacerbated by the need for secure communication in the presence of eavesdroppers. Notably, none of the existing approaches specifically address the goal of PLS-targeted resource allocation in FlexD networks.

FlexD-based PLS in multi-user networks necessitates effective power allocation and communication direction. The non-convex nature of power allocation and the combinatorial challenges in dynamic scheduling make classical algorithms less efficient. Recent advancements in wireless communication leverage machine learning to tackle such issues in complex networks, extending to resource allocation [15], wireless sensing [16], and user localization [17]. The intrinsic graph structure of wireless networks aligns with the use of GNNs, tailored for processing non-Euclidean data. Recent GNN applications encompass decentralized inference [18], resource allocation with random edge GNNs [19], and link scheduling [20].

In addressing the research gap between multi-user FlexD networks and PLS, this work pioneers the introduction of a novel multi-user FlexD system model. It further proposes both classical optimization and GNN algorithms as the initial endeavors to maximize network secrecy within the existing literature. This study presents several noteworthy contributions:

1. 1.

   Innovative System Model: Presenting a novel system model that accommodates multiple legitimate users and coordinated eavesdroppers. Assuming optimal conditions for eavesdroppers, we set a lower bound for the system’s sum secrecy rate, initially addressing the optimization problem with classical methods.

2. 2.

   Efficient Graph Representation of FlexD System: Introducing a distinctive graph representation for the FlexD system, consolidating two paired users into a single node to optimize memory and computational resources. Then, integrating passive eavesdropper nodes into user pairs reduces node count and simplifies the network graph.

3. 3.

   GNN-Based Optimization Strategy: Introducing an innovative GNN-based optimization to maximize the sum secrecy rate of FlexD networks. This approach outperforms classical methods in performance and time complexity.

4. 4.

   Enhanced GNN Model: Modifying the GNN model to predict without eavesdropper channel information achieves comparable performance to models with full channel data, enhancing practicality and versatility.

As illustrated in Fig. 1, the FlexD network comprises $2N$ users, denoted as $U_{1},\dots,U_{2N}$, aiming for simultaneous secure communications amid the presence of $K$ coordinating eavesdroppers, labeled as $E_{1},\dots,E_{K}$. In Fig. 1, solid lines represent desired communication links, dashed lines indicate interference links, and dash-dotted lines denote leakage links dedicated to eavesdropping. For clarity, the figure specifically highlights channels involving $U_{2n-1}$, $U_{n}$, and $E_{k}$. Within the FlexD network, without loss of generality, users with adjacent indices form user pairs, denoted as $(1\leftrightarrow 2,\dotsc,(2n-1)\leftrightarrow 2n,\dotsc,(2N-1)\leftrightarrow 2N)$. In a resource block, each user pair designates one node as the transmitter and the other as the receiver. FlexD users managing interference as receivers face $(N-1)$ transmitter nodes. As transmitters, they minimize leakage to $K$ eavesdroppers and interference to other users. FlexD user pairs dynamically schedule communication directions, capitalizing on wireless channel variations to maximize sum secrecy rates.

Eavesdroppers within the network engage in passive monitoring of FlexD user pair communications. Our analysis assumes adaptive coordinated eavesdroppers, strategically adjusting their positions to optimize the signal-to-interference plus noise ratio (SINR), thereby maximizing the potential for secure communication leakage. Eavesdroppers employ a Minimum Mean Square Error (MMSE) receiver, treating interference as noise, representing the best-case scenario with Gaussian codebook sequences [21]. For simplicity, we assume single-antenna configurations for FlexD users and eavesdroppers, though the analysis can extend to multi-antenna scenarios.

Consider a FlexD user pair denoted as $U_{m}-U_{n}$, where $n\in{1,\cdots,2N}$ and $m=2(n\bmod 2)+n-1$. For a given resource block, the SINR of the $n$th FlexD node is expressed as

where $p_{m}$ denotes the transmit power of the $m$th node, $h_{n,m}\in\mathbb{C}$ denotes the wireless channel from $U_{m}$ to $U_{n}$, and $\sigma^{2}$ denotes the complex Gaussian noise power. The term $\sum_{j\neq m,n}^{2N}t_{j}p_{j}|h_{nj}|^{2}$ represents the total interference from $(N-1)$ transmitter nodes. The binary variable $t_{m}\in\{0,1\}$ indicates whether $U_{m}$ is a transmitter or a receiver with $t_{m}=1$ and $t_{m}=0$, respectively. This variable ensures that transmitter nodes have zero received SINR and receiver node channels are excluded from the interference expression. Additionally, to analyze the more challenging generalized scenario, we assume non-reciprocal channels, i.e., $h_{nm}\neq h_{mn}$.

Coordinated eavesdroppers attempt to intercept FlexD communications by treating individual links as desired signals and considering others as interference. This process can be parallelized to decode all the communications from FlexD user pairs. Then, the received SINR of FlexD user $m$ at the coordinated eavesdroppers’ end, assuming MMSE receivers, is given by

where $\bm{g}_{m}\in\mathbb{C}^{K\times 1}$ and $\bm{I}_{K}$ denotes the channel vector from the $m$th FlexD user to $K$ eavesdroppers and the identity matrix, respectively.

Using (1) and (2), the instantaneous secrecy rate achieved by the communication from $m$th user to $n$th user is expressed as [22]

This expresses the achievable secrecy rate by selecting communication from $m$ to $n$, with a similar expression for $n$ to $m$. One expression evaluates to zero based on the communication direction. The instantaneous secrecy rate expression can be evaluated for all user pairs and communication directions to obtain the sum secrecy rate of the FlexD network. This rate is assured over a fixed channel state in a given resource block. In the next subsection, we formulate the objective function to maximize the sum secrecy rate for all user pairs in the network.

The optimization goal is to maximize the sum secrecy rate of the FlexD network, as shown in (3). This entails optimizing power and transmission direction for all user pairs. The objective is mathematically represented as

where $P_{\max}$ is the maximum transmit power of a FlexD node.

Notably, Addressing this challenge involves grappling with various complexities: i) the problem’s inherent nature is marked by non-convexity, non-differentiability, and a combinatorial structure, making it inherently intricate; ii) the computational burden is heightened due to the inclusion of high-dimensional vector spaces; iii) the absence of readily available optimal solutions hampers seamless integration with supervised learning methods; and iv) real-world scenarios present the added hurdle of lacking eavesdroppers’ channel data, compelling legitimate users to rely on location/distance data for estimating channel conditions. Addressing the challenges of this problem demands innovative solutions. In Section III, we tackle complexities i), ii), and iii) by using classical methods and GNN approaches. Subsequently, in Section IV, we extend our GNN approach to address complexity iv) related to the unavailability of full eavesdroppers’ channel knowledge.

We introduce an iterative algorithm designed to address a relaxed version of the optimization objective presented in (4). Due to the absence of differentiability, we eliminate the non-negative operator in (3). Consequently, the optimization problem can be projected into a higher-dimensional space using the MMSE-SINR equivalence. The resulting relaxed problem is then solved iteratively for $\bm{p}$ and $\bm{t}$, employing algorithms proposed in [23] and [1], respectively.

This block-coordinate descent mechanism is repeated until the sum secrecy rate converges to a user-defined threshold. However, it is worth noting that this algorithm exhibits diminished performance due to the challenges posed by the high-dimensional optimization landscape. Specifically, in scenarios with an increased number of users in the network, the algorithm optimizing $\bm{t}$ becomes predominant, resulting in a time complexity of $\mathcal{O}(N^{4})$ under the assumption of linear time complexity for determining the sum secrecy rate. Conversely, in situations with more eavesdroppers in the network, the algorithm optimizing $\bm{p}$ takes precedence, leading to a time complexity of $\mathcal{O}(NK^{3})$. This elevated time complexity can pose a computational bottleneck, especially in dense real-world scenarios.

The classical approach to (4) is computationally complex and suboptimal due to coordinate descent’s susceptibility to converge to local optima. we propose a novel GNN-based unsupervised learning strategy for maximizing sum secrecy rate in the FlexD network. Unlike traditional methods, our approach concurrently optimizes $\bm{t}$ and $\bm{p}$, reducing computational complexity to $\mathcal{O}(N^{2})$ with our proposed graph structure as input.

We consider a network consisting of $N$ user pairs and $K$ eavesdroppers which are spanned over $1\times 1\;\text{km}^{2}$ area. User pairs and the eavesdroppers are Poisson disk distributed over the given area. Users are paired randomly to form user pairs in the network. Channel realizations are generated by assuming Rayleigh fading with free space path loss at $1\;\text{GHz}$ frequency and $8\;\text{dB}$ log-normal shadowing. Peak power of $30\;\text{dBm}$ is assumed along with a noise level of $-100\;\text{dBm}$.

Further, simulation results are presented for two additional baseline strategies: HD, the classical approach without direction selection, and Max Power, where the direction with the strongest channel of the desired link is selected, and $P_{\max}$ is used as the transmit power. All the algorithms are implemented in Python language and tested on CPU for time comparisons. The GNN model consists of 3 layers and the MLPs used to realize (10) and (11) are kept consistent throughout the experiments. GNN models are trained with 10000 network realizations with a batch size of 128 samples. The AdamW optimizer is used with the learning rate of $0.002$.

Fig. 3 presents a comparison of the average sum secrecy rate (ASSR) performance across varying user and eavesdropper counts. The inherent limitations imposed by the coordinated nature of eavesdroppers are evident in the observed spectral efficiency, thereby providing valuable insights into the network’s capabilities under challenging conditions.

Notably, the GNN approach outperforms the classical approach, Max Power, and HD communications by a substantial margin. For instance, with $4$ users and $2$ eavesdroppers, the GNN achieves a performance gain of $5$-fold and $10$-fold compared to classical and HD scenarios, respectively. However, this performance discrepancy diminishes with an increasing number of users and decreasing number of eavesdroppers due to the diversity introduced by more desired channels and the reduction in leakage links. A noteworthy observation is that the GNN model maintains a constant learnable parameter count, excluding the linear projection matrix, throughout the experiments. This highlights the scalability of the GNN, demonstrating minimal increases in learnable parameters and subsequently reducing computational costs, particularly in larger networks.

Across all scenarios, the classical algorithm with direction selection consistently outperforms the HD communications. This substantiates the claim that direction selection in FlexD networks significantly contributes to performance gains. For instance, in a $16$-user network with $2$ and $4$ eavesdroppers, the performance gain from FlexD or direction selection is $1.6$ times and $2$ times, respectively. Additionally, the superior performance of the Max Power strategy with a better direction selection criterion validates this claim.

Fig.4a and Fig.4b show heuristic time complexity comparisons of algorithms. Each plot indicates single CPU time for algorithm execution, revealing insights into underlying mathematical operation time complexity. Max Power approach is computationally efficient than the GNN model but suffers from low performance due to the lack of proper power allocation. The GNN model consistently outperforms classical and HD communications with varying user and eavesdropper counts. This advantage becomes more pronounced with an increasing number of eavesdroppers while maintaining a fixed number of users. In Fig.4b with $12$ eavesdroppers, the time gap is on the order of $10^{-3}$, while in Fig.4a with $12$ users, it is on the order of $10^{-4}$. With an increasing number of eavesdroppers, the GNN approach maintains low time complexity growth due to dimensionality reduction in linear projection (6). As a result, the number of neurons needed for eavesdropper information processing remains nearly constant despite the growing number of eavesdroppers.

In Fig. 4c, we evaluate two GNN models: one using distance knowledge from users to eavesdroppers and the other incorporating full CSI. Distance knowledge proves effective as a proxy for channel information, yielding substantial performance. However, with an increasing number of users, a growing performance gap between the two approaches appears. This implies neural networks may have limitations in fully extracting diversity information solely from distance knowledge.