Research on topology control method of multi-hop wireless communication network based on graph neural network

With the rapid development of wireless communication technology, multi-hop wireless networks have emerged. Multi-hop wireless network is a kind of wireless communication network topology, in which each node in the network can be used as a relay node for data transmission, so as to realize multiple hop propagation of signals [1–3]. Compared with the traditional point-to-point communication, multi-hop wireless network has a larger transmission range and stronger network scalability. However, this network topology also has some problems that need to be studied and solved.

Firstly, the communication between nodes in multi-hop wireless communication networks is interfered by environmental factors such as geographic location, obstacles and weather, and these interferences can lead to the degradation of signal transmission quality and even cause the loss of data packets [4–6]. Secondly, multi-hop wireless communication networks may lead to unbalanced energy consumption during data transmission due to the different number of times the nodes act as relay nodes in the network, thus affecting the overall network lifetime [7–8]. The lifetime of multi-hop wireless communication networks is a central issue. An energy balancing algorithm can be introduced to dynamically adjust the nodes according to their energy residuals so that the nodes in the network can consume energy in a reasonable and balanced manner [9–11]. In addition, the number of nodes in a multi-hop wireless communication network is huge, and each node may have multiple optional forwarding paths to transmit data, and it is crucial to select the best forwarding path to improve the transmission efficiency of the whole network [12–14]. An intelligent algorithm can be used for routing by modeling the topology and considering parameters such as the number of hops between nodes, hop distance, and signal strength [15–18]. Finally, multi-hop wireless communication networks in which nodes communicate with each other through wireless signals, the security of the network is easily attacked. Encryption algorithms, authentication mechanisms and access control policies can be used to enhance the security of the network [19–21].

All the above issues are related to the efficiency, stability and security of data transmission in multi-hop wireless communication networks. Therefore, the method of topology control of multi-hop wireless communication networks is an important research direction. Through its research, it can provide better support and guarantee body for the development of wireless communication technology.

In this paper, we introduce graph neural network (GCN) to build a network topology intelligent control model (GCNTO), and analyze the topological transformation process of network structure from simple to highly vandalism-resistant through supervised learning of GCN, especially the evolution to “onion-like” structure, in order to optimize the network topology. The key feature information of the network and edges is extracted using GCN, and the relationships and feature representations between nodes are automatically learned while preserving the graph structure. Then, the nodes in the graph are subjected to feature learning and representation learning, and the local and global features of the nodes are extracted by multilayer convolution and pooling operations. Finally, based on the learned feature representations, the probability of existence of each edge in the network topology is judged to optimize topology control for multi-hop wireless communication networks. A simulation experiment scenario under cellular network conditions is set up to conduct simulation performance experiments on GCNTO, focusing on both power control and power allocation performance of multi-hop wireless communication networks.

2

Topology control optimization method for multi-hop wireless communication networks

The emergence of fifth-generation wireless communication systems has resulted in a greater variety of wireless communication networks and a variety of new application scenarios for massive machine-like communication and ultra-reliable and low-latency. These scenarios increasingly demand high concurrency, ultra-reliability, and low latency. In order to effectively achieve these targets, resource allocation and transmission optimization of wireless communication systems will play a key role. Traditional resource provisioning and transmission optimization schemes usually have high computational complexity and can only utilize limited domain knowledge, which cannot meet most of the service requirements in wireless communication systems. Aiming at the challenging problem of how to achieve wireless resource provisioning and transmission optimization with low computational complexity, high interpretability and reliability, and satisfying large-scale complex constraints in multi-hop wireless communication network scenarios, this study aims to optimize the network topology of multi-hop wireless communication networks by fusing graph neural network techniques for supervised learning.

2.1

Modeling Multi-hop Wireless Network Properties and Connectivity

2.1.1

Overview of Multi-Hop Wireless Network Properties

Multi-hop wireless network attributes include two kinds of network intrinsic attributes and network basic attributes [22]. Multi-hop wireless network intrinsic attributes is the network in the application environment, in order to achieve specific functions and have the network, nodes, links and other aspects of the intrinsic characteristics, is the network operation process will not change and is not affected by other intrinsic attributes of the network attributes. Inherent properties mainly include network aspects, such as node density, node distribution, etc.; wireless links, such as transmit power, receive sensitivity, the environment of the channel parameters, the effective communication radius, etc.; nodes, such as node mobility, node movement speed, node energy value, etc.. Network intrinsic attributes are the basis for studying the basic attributes of the network, the factors to be considered in the design of network architecture and network strategy, and the basic parameters for modeling and analyzing network performance.

The study of network basic attributes is the premise of multi-hop wireless network technology research, the basis of network architecture design and network strategy development, and the theoretical guidance of network planning, both in wired and wireless networks, the network basic attributes are the reference basis for the targeted development of network strategy, and the premise of network performance modeling and analysis and the theoretical basis for effective avoidance of network planning limitations.

2.1.2

Network connectivity modeling

In this section, network connectivity is analyzed in depth, measures of network connectivity are defined, and closed-form solutions for the relevant metrics are obtained through modeling solutions. 1)

Definition of Network Connectivity and Measurement Metrics

Network connectivity refers to the interconnection characteristics between any two nodes in the network, which is generally measured by the probability that any two nodes in the network can be interconnected. Generally speaking, the larger the probability is, the better the network connectivity is, and the more stable the network is. Network connectivity is a very important and fundamental attribute in multi-hop wireless networks, and is a prerequisite that must be considered in network strategy design and network planning.

Network connectivity examines the problem of existence, where it is stated that a multi-hop route exists with a certain probability for the nodes in the network. A given route is known, as to whether a node is able to find this multi-hop route is a problem that needs to be solved for network strategy design, and the reliability of this existent route will be studied in the subsequent sections. In this section, the connectivity of the network is studied only for the existence of multi-hop routes.

2)

Research model for network connectivity

First, we assume that the effective communication radius of nodes is r₀ and the effect of channel fading is not considered. The nodes obey Poisson distribution in the network coverage area D, and the node density is ρ. At this time, the distribution of nodes satisfies the following two conditions.

(1)

Within the network area S, the area is s and the number of nodes in the area is N. Then N is satisfied: 1 $P (n n o d e s i n r e g i o n S) = P (N = n) = \frac{λ^{n}}{n!} e^{- λ}$ where λ = E[N] = ρs.

(2)

For any non-overlapping region S_i with area s_i and number of nodes in the region N_i, N_i are independent of each other, i.e.: 2 $P (N_{1} = n_{1}, N_{2} = n_{2} \dots N_{k} = n_{k}) = \prod_{i = 1}^{n} P (N_{i} = n_{i})$

In this paper, we set the study region as D, the size of the region area as D², and the node density as ρ. For a given node density, the number of nodes N_D within the region is only related to the size of the region area. Let the random variable N_D denote the number of nodes in the region D when N_D = n, n > 2, n nodes in the region D satisfy the uniform distribution, which is determined by the nature of the Poisson distribution.

At any point in time, the distance d_ij between node i and node j is less than or equal to the effective communication radius r₀, then we consider these two nodes to be connected.

Therefore, when the number of nodes in the region is N_D = n, n > 2, the probability of connectivity between any node i and node j in the network is: 3 $P_{i j}^{(n)} = P (d_{i j} \leq r_{0} | N_{D} = n)$

In particular, $P_{i t}^{(n)} = 1$ .

When the area of the effective coverage area of a node is much smaller than the network coverage area, i.e., when $D r_{0}^{2} - D^{2}$ is satisfied, the probability of connectivity between node i and node j is: 4 $P_{i t}^{D_{r}} = P (d_{i j} \leq r_{0} | N_{D} = n) = \frac{π r_{0}^{2}}{D^{2}}$

Define the schematic function: 5 $a_{i j}^{(n)} = {\begin{array}{l} 1, p (d_{i j} \leq r_{0} | N_{D} = n) \\ 0, p (d_{i j} > r_{0} | N_{D} = n) \end{array}$

In the case of N_D = n, the network has an adjacency matrix A:

From the definition, it is known that the adjacency matrix A is a Boolean matrix and is symmetric. For any given A, it is known that A^k is a k -hop reachable matrix of the network, where the multiplication operation of matrix A is Boolean, i.e., addition in the matrix corresponds to the or operation and multiplication corresponds to the and operation. Therefore, the k -hop reachable matrix A^k is also a Boolean symmetric matrix. If an element $a_{i j}^{k}$ of A^k is non-zero, it means that nodes can be interconnected by k relays and there may be more than one k relay path between nodes.

In order to obtain the probability of existence of k -hop routes, different from k -hop reachable matrices, we define only k -hop reachable matrices C^k, C^k as Boolean matrices, where an element of 1 at the corresponding position indicates the existence of a k -hop route between two nodes. The relationship between C^k and A^k is as follows: 7 $C^{k} = [A^{k} * \sum_{i = 0}^{k} A^{i}]$ where all matrix operations are numerical matrix operations, i.e., A^k is not treated as a Boolean type operation. The purpose of $\sum_{i = 0}^{k} A^{i}$ is to obtain k -hop intra-reachable matrices where nodes can reach other nodes within k hops, taking into account the reuse of the same relay link. The operator * denotes the hadamard product, which serves to multiply the corresponding bit elements of the matrix and is used between matrices of the same order. The purpose of multiplying $\sum_{i = 0}^{k} A^{i}$ with A^k is to ensure that the node is k hop-accessible to other nodes. The function of operator four is to Booleanize the matrix by setting the non-1 elements within the matrix to zero. In the above equation, if the matrix element within the operator port is greater than 1 then it means that there exists an k - hop route for the node to reach other nodes, but at the same time there exists a route with a smaller number of hops than k. Therefore, an element $c_{i j}^{k}$ of matrix C^k that is a non-zero element indicates that there exists a k hop route between node i and node j and that node i and node j can be reached by only k hops.

Let P(k | N_D = n) denote the probability of the existence of a maximum-hop routing of k hops at any one node in the case where n nodes are distributed in the network area D. Using the statistical parametric model, we are able to C^k find the probability P(k | N_D = n) that the maximum hop-count routing exists at any one node for k hops in the case of N_D = n. Obviously, P(k | N_D = n) is related to the node's effective communication radius r₀, and when k > n, P(k | N_D = n) = 0.

Thus the probability of the existence of k hop routes for any node in the network is obtained under the condition that the node density is ρ and the effective communication radius is r₀: 8 $\begin{array}{l} p_{k c} & = \sum_{n = 2}^{\infty} P (N_{D} = n) P (k | N_{D} = n) \\ = \sum_{n = 2}^{\infty} \frac{λ^{n}}{n!} e^{- λ} P (k | N_{D} = n) \end{array}$ where λ = ρD².

Clearly, p_kc is related to the node density ρ and the effective communication radius r₀ of the nodes. As a measure of network connectivity, the relationship between p_kc and network planning requirements such as maximum network radius and minimum node density will be further analyzed when analyzing network planning constraints in Chapter III.

2.2

Principles of graph neural networks

A graph neural network (GCN) is a type of neural network that performs convolution operations on graph-structured data (non-Euclidean space) [23]. The core idea of GCNs is to learn a mapping function that can combine the information of a vertex's neighboring vertices with its feature information to generate a new representation of the vertex's features. According to the different graph convolution methods, GCNs can be categorized into spectral-domain based GCNs and spatial-domain based GCNs. Next, several classical models of spectral-domain based and spatial-domain based GCNs will be briefly introduced, respectively. 1)

Spectral domain based GCNs

A fixed convolution kernel cannot be used on the graph since the number of neighboring vertices may be different for different vertices. To solve this problem, the graph structure data is usually converted to the frequency domain for processing. Specifically, given an input graph signal x ∈ ℝ ^d and a graph filter g ∈ ℝ ^d, the spectral domain based graph convolution can be defined as: 9 $x * G g = F^{- 1} (F (x) ⊙ F (g)) = U (U^{T} x ⊙ U^{T} g) = U \hat{g} U^{T} x$ where *G denotes the graph convolution operation, F (x) = U^Tx denotes the graph Fourier transform, F⁻¹(F (x)) = UF (x) denotes the Fourier transform. ⊙ denotes the Hardman product, and g = diag(U^Tg). Different spectral domain based GCNs can be defined by varying g. The mode of operation of GCN is defined as: 10 $X * G = W_{G} (I_{N} + D^{- \frac{1}{2}} A D^{- \frac{1}{2}}) x$ where *G is the learnable weight matrix. To avoid the problem that $I_{N} + D^{- \frac{1}{2}} A D^{- \frac{1}{2}} \in (0, 2)$ may lead to gradient explosion, $I_{N} + D^{- \frac{1}{2}} A D^{- \frac{1}{2}}$ is further transformed into ${\tilde{D}}^{- \frac{1}{2}} \tilde{A} {\tilde{D}}^{- \frac{1}{2}}$ , where $\tilde{A} = A + I_{N}$ , ${\tilde{D}}_{i i} = \sum_{j} {\tilde{A}}_{i j}$ . Compared with GCN, Chebnet has higher computational complexity but stronger expressive power. K -order convolution operator of Chebnet can cover the K -order neighbor vertices of the center vertex, while GCN can only cover the 1-order neighbor vertices. In order to perceive higher-order neighbor vertices, GCN can expand its perceptual domain by superimposing multiple graph convolutional layers.

2)

Spatial domain based GCNs

In wireless networks, Message Propagation Neural Networks (MPNNs) and Diffusion Convolutional Neural Networks (DCNNs) are widely used.MPNN networks are a generalized framework for spatial-domain based GCNs and divides spatial-domain based GCNs into two phases of Message Aggregation and Message Combination, viz: 11 $m_{v_{i}}^{(t)} = \sum_{v_{j} \in N (v_{i})} ℳ^{(t)} (x_{i}^{(t - 1)}, X_{j}^{(t - 1)}, e_{i j})$ 12 $x_{i}^{(t)} = U^{(t)} (X_{i}^{(t - 1)}, m_{v_{i}}^{(t)})$ where e_ij is the feature vector representation of the edges between vertex v_i vertex v_j, M^(t)(·) and U^(t)(·) are the aggregation function and combination function of the t th iteration, respectively. $m_{v_{i}}^{(t)}$ is the information obtained from the neighbor aggregation of vertex v_i, and $X_{i}^{(t)}$ is the hidden state information of vertex v_i in the t th iteration. It is not difficult to find that this model of MPNN decreases in computational efficiency as the number of vertices increases.

In order to overcome the defects of MPNN, Hamilton WL et al. further proposed the Graph SAGE model by fixing the number of neighboring vertices of a vertex for message passing. The graph convolution operation of Graph SAGE is defined as: 13 $X_{v_{i}}^{(t)} = σ (W^{(t)} g^{(t)} (X_{i}^{(t - 1)}, {x_{j}^{(t - 1)}, \forall j \in S_{N (v_{i})}}))$ where $W_{G}^{(t)}$ and g^(t)(·) are the learning weight matrix and message aggregation function for layer t, respectively. σ(·) denotes the nonlinear activation function. S_{N(v_i)} is the set of vertices obtained by randomly sampling from the neighboring vertices of vertex v_i. The main difference between MPNN and Graph SAGE is that GraphSAGE randomly samples a fixed number of neighboring vertices for each vertex, whereas MPNN uses all the neighboring vertices of each vertex. Regarding the DCNNs model, the diffusion convolution operation in the DCNNs model obtains a representation of the hidden state of each vertex by scanning the diffusion process of that vertex using a transfer probability matrix P, i.e., $Z^{(t)} = σ (W_{G}^{(t)} ⊙ P^{t} X)$ , where Z^(t) is the hidden state of the t th layer, and P^t denotes the t th power of matrix P.

2.3

Wireless communication graph construction based on wireless communication network

Graph-structured data is the basic prerequisite for applying GNNs. For wireless communication networks, it is necessary to convert the wireless network into graph-structured data suitable for GNNs. Generally, the topology of a wireless network can be constructed as an undirected graph or a directed graph, depending on the specific research objectives. Depending on the types of communication links and communication devices in a wireless network, the topology of a wireless network can be further constructed as a homogeneous or heterogeneous graph. In the next two subsections, the construction methods of wireless communication graphs (WCGs) in various wireless network scenarios, such as Mesh/Ad-hoc networks, Cellular networks/WLANs, will be elaborated. 1)

Wireless Communication Graph Construction Methods in Mesh/Ad-hoc Network Scenarios

First, consider a homogeneous mesh/ad-hoc network scenario consisting of N communication pairs. To construct a WCG for this scenario, the i nd communication pair can be viewed as the i rd vertex v_i in the WCG, and the feature vector of this vertex contains the direct channel state information h_ii and other environmental information (e.g., the weight of the i th vertex w_i). The WCG may be either undirected or directed. For an undirected WCG, the feature vector of an undirected edge between vertices v_i and v_j contains interference channel state information h_ij and h_ji. As for the directed WCG, the feature vectors of the two directed edges between vertices v_i and v_j contain the interference channel state information h_ij and h_ji, respectively. Notably, the dimensional size of the channel state information between vertices v_i and v_j depends on the number of antennas equipped on the communication pair.

For a heterogeneous mesh/ad-hoc network scenario, assume that the scenario has M type of communication link. For the WCG construction of this network scenario, the communication pairs whose communication links are of type m can be regarded as WCG vertices v_{i_m}. The set of neighboring vertices of vertex v_{i_m} whose communication links are of type n is denoted as $N_{i_{m}}^{(n)}$ . The feature vectors of vertex v_{i_m} contain the channel state information h_{i_mi_m} of the direct communication links, as well as other environmental parameters related to the link type m. The design of the feature vectors of the edges between vertices v_{i_m} and v_{j_n} also needs to be considered in terms of both the undirected WCG and the directed WCG. If the WCG is an undirected graph, the feature vectors of the undirected edges between vertices v_{i_m} and v_{j_n} contain the interference channel state information h_{i_mj_n} and h_{j_ni_m}. If the WCG is a directed graph, the feature vectors of the two directed edges between vertices v_{i_m} and v_{j_n} contain h_{i_mj_n} and h_{j_ni_m}, respectively.

2)

Wireless Communication Map Construction Method in Cellular Networks/WLANs Scenario In general, there may be M ≥ 1 access point (AP) and N ≥ 1 users (UEs) in cellular networks/WLANs. First, consider a simple scenario that contains only one AP for resource scheduling and management (e.g., power control, user connectivity, etc.). For this scenario, each UE can be considered as a vertex in the WCG without considering the AP.In this case, the feature vector of the i rd vertex in the WCG contains channel state information h_ii and other environment information. The feature vectors of the edges between vertex v_i and vertex v_j contain channel state information h_ii and h_jj, among others. Considering that all UEs are associated with the same AP, the feature vectors of edges can be disregarded.

Consider further the more complex Cellular Networks/WLANs scenario, where an AP may serve multiple UEs and a UE may also have access to multiple APs.First, consider the scenario where a UE has access to only one AP and one AP serves multiple UEs.

2.4

Intelligent control model for multi-hop wireless communication network topology

In this section, Graph Convolutional Neural Network (GCN), as a deep learning model specifically designed to process graph-structured data, will be introduced to propose a GCN-based Intelligent Topology Control Optimization Strategy (GCNTO) for Multi-hop Wireless Communication Network Topology, which utilizes deep learning techniques to improve the topology of Multi-hop Wireless Communication Networks.

2.4.1

Topology Intelligent Control Strategy

In the process of optimizing network destruction resistance through intelligent algorithms, it is found that the optimized robust network topology often forms an “onion-like” structure. Therefore, more and more researches use the “onion-like” structure for topology destructive enhancement.

By taking advantage of the structural property that nodes with similar degrees tend to be connected, and the property that a large number of edges perpendicular to the center of the network are distributed in the topology, it is possible for nodes with similar degrees to form an “onion-like” structure.

The efficiency and performance of the optimization process can be improved by using GCNs to learn the evolutionary patterns of highly destructive “onion-like” structural topologies. However, this strategy also faces some challenges. First, the optimization of the network topology must take into account the complex relationships between nodes and the dynamic changes of the topology. Second, optimization for highly robust topologies needs to take into account the correlation of degree differences between nodes. Therefore, in this case, the use of graph neural networks to solve these challenges needs to be redesigned. Based on this, this chapter proposes an end-to-end prediction model based on a degree search strategy from the predictive probability model of network connected edges to the prediction of robust topology to improve the optimization and performance.

2.4.2

Topology Intelligent Architecture Design

1)

Graph Convolutional Neural Network Design

In this section the intrinsic mechanism and implementation process of graph convolutional neural network structure in GCNTO will be explored. The basic principle of graph neural network is the aggregation of node information. For Figure G = (V,E), where V is a collection of nodes and E is a collection of edges. Each node v∈V has an initial feature representation.

2)

Node feature embedding

In order to concretely define the process of embedding the adjacency matrix of the initial topology into the node features of the higher dimension h, we can start from the adjacency matrix A of Fig. G = (V,E), where A_ij denotes the connectivity between node i and node j. This process involves converting the adjacency matrix into a high-dimensional feature representation through which each node acquires a kind of embedding that contains information about its relations with other nodes, as defined below: 14 $v_{i}^{0} = W_{0} a_{i} + b_{0}$

The feature vector $v_{i}^{0}$ of each node i is obtained by multiplying the corresponding column a_i ∈ {0,1}^n×1 of its adjacency matrix A with the weight matrix W₀ and adding the bias vector b₀. Where W₀ denotes the weight matrix with dimension h×n and b₀ denotes the bias vector with dimension h×1. 1)

Node Degree Difference Calculation

The key to learning an “onion-like” network topology into a model is to capture the centrality and hierarchy of the nodes in the network, which usually involves the degrees of the nodes and the degree differences between them. The purpose of the new degree difference expression proposed here is to capture the subtle differences in the network topology more finely and efficiently, so that the model can better learn and simulate the evolution from the initial topology to the highly destructive target topology. The details are as follows: 15 $D_{i j} = {\begin{array}{l} \frac{2 \times | d_{i} - d_{j} |}{\max {d_{j}} + \max {d_{j}}} & i \neq j \\ 0 & i = j \end{array}$

2)

Edge feature embedding

A richer representation of edge features is obtained by using a more complex function Multi-Layer Perceptron (MLP) to fuse the node features, the original features of the edges, and the topological features, as defined below: 16 $e_{i j}^{0} = M L P ((W_{2} d_{i j} + b_{2}) ‖ (W_{1} δ_{i j}) ‖ f_{i j} ‖ (v_{i} ‖ v_{j}))$ where f_ij denotes the original feature vector of edge (i, j) ; d_ij represents the new degree difference between nodes i and j ; W₁ ∈ ℝ ^h/2 and W₂ ∈ ℝ ^h/2denote the weight matrices, b₂ ∈ ℝ ^h/2 denotes the bias vector; and v_i and v_j denote the feature vectors of nodes i and j, respectively. Symbol ∥ denotes the connectivity operation, and δ_ij is the connectivity range index. MLP denotes Multilayer Perceptron, a fully connected neural network with a certain depth and a nonlinear activation function for learning a complex mapping from the input features to the edge feature representations.

During the training process, the weight matrix W and bias vector b are randomly initialized to reduce symmetry, and these parameters will be adjusted by back propagation algorithm to minimize the loss function of the model [24].

3)

Graph Convolution Layer

In GCN, the feature vector of a node is updated based on the feature vectors of all the nodes in its neighborhood. This updating process involves aggregating the information of the nodes in the neighborhood and generating new node features through a nonlinear transformation. This operation allows information to flow between nodes in the graph, thus capturing topological features of the graph. The update of the feature vector $v_{i}^{(l + 1)}$ of node i at layer l + 1 can be expressed by the following equation: 17 $v_{i}^{(l + 1)} = σ (W^{(l)} \cdot A G G R E G A T E ({v_{j}^{(l)} : j \in N (i) \cup {i}}) + b^{(l)})$ where $v_{j}^{(l)}$ denotes the feature vector of node j at layer l; N (i) denotes the neighborhood of node i, i.e., the set of all nodes directly connected to node i ; W^(l) is the weight matrix of layer l, which is used to linearly transform the feature vectors within the neighborhood. b^(l) is the bias term in layer l; AGGREGATE is an aggregation function used to aggregate the feature vectors $v_{j}^{(l)}$ of all neighboring nodes j of node i into a single vector, and in this aggregation operation we have chosen to use the mean value; σ is a nonlinear activation function, and here we use the ReLU function, which is used to introduce the nonlinear properties and increase the expressive power of the model [25–26].

As for the way of updating the edge feature vector $e_{i j}^{(l + 1)}$ , it is necessary to consider the dynamic updating mechanism of the edge features, similar to the updating of the node features. This update can be based on the current features of the edges, the features of the nodes connected to the edges, and possibly the global graph features. Since the traditional definition of GCNs focuses on the aggregation and updating of node features, dynamic updating of edge features introduces additional complexity and flexibility for updating edge feature vectors: 18 $e_{i j}^{(l + 1)} = σ (W_{e}^{(l)} \cdot C O N C A T (e_{i j}^{(l)}, v_{i}^{(l)}, v_{j}^{(l)}) + b_{e}^{(l)})$

4)

Output layer

In order to efficiently map the input edge feature vector $e_{i j}^{(l)}$ to the probability of existence of edge ij in the predicted topology, we can adopt a simple yet efficient approach that utilizes a fully connected layer (FC layer) combined with a sigmoid activation function. With this design, we are able to convert the input edge feature vectors into probability values between 0 and 1, thus representing the probability of existence of edge ij in the predicted topology. Given the edge feature vector $e_{i j}^{(l)}$ obtained from the last layer of graph convolution, the output layer can be defined as follows: 19 $p_{i j} = σ (W_{o} \cdot e_{i j}^{(l)} + b_{o})$

5)

Loss function

Before defining the loss function, we first reduce the P_ijk that represents the probability of an edge ij being present in the predicted topology to a n×n matrix where each element represents the probability of the edge being present, ignoring the k dimensionality. This is done to reduce the problem to a traditional binary classification problem and to facilitate model training by applying our binary cross-first loss function. Assuming that P_ij1 denotes the probability that an edge ij exists, we can define the loss function as: 20 $L = - \sum_{i = 1}^{n} \sum_{j = 1}^{n} [w_{1} \cdot Y_{i j} \log (P_{i j 1}) + w_{0} \cdot (1 - Y_{i j}) \log (1 - P_{i j 1})]$

In this setup, Y_ij represents the true label, which is 1 if an edge exists between node i and node j, and 0 otherwise. P_ij1 is the predicted probability that edge ij exists. And w₁ and w₀ are the weights for the positive class (i.e., edge exists) and the negative class (i.e., edge does not exist).

One of the ways to determine the weights w₁ and w₀ is to use an inverse frequency weighting strategy that considers the frequency of occurrence of each category (edge present or absent) in the labeling matrix Y. By calculating the inverse of the frequency of each category, it is possible to balance the contribution of different categories in the loss function by assigning higher weights to less frequent categories and lower weights to more common categories, as defined below: 21 $w_{1} = \frac{1}{c o u n t (Y_{i j} = 1)} A n d w_{0} = \frac{1}{c o u n t (Y_{i j} = 0)}$ where count is the number of counts Y_ij = 1 and Y_ij = 0.

3

Simulation Experiments on Network Topology Control under Cellular Network Conditions

In this chapter, based on the simulation experimental scenario and hardware environment under cellular network conditions, the performance advantages of this paper's intelligent topology optimization strategy (GCNTO) built based on graph neural network will be analyzed from the perspectives of power control and power allocation performance of multi-hop wireless communication networks. The hardware environment parameters of this simulation experiment are specifically shown in Table 1.

Table 1.

Hardware environment

Central processing	12th Gen Intel(R)Core(TM)i7-12700H
CPU main frequency	2.70GHz
Memory	32GB
Operating system	Windows11
Simulation software	PyCharm 2022.2.2

As an example, the cellular network of seven cells is modeled as shown in Fig. 1, with each transmitter distributed in the center of the cell. Figures (a) and (b) represent seven cells and 28 users, and seven cells and 42 users, respectively. The black hexagon represents the boundary of each cell. The green squares indicate the locations of transmitters (base stations) in the cell. The blue circles indicate the random locations of receivers (users).

3.1

Analysis of power control performance in multi-hop wireless communication networks

The power control performance of multi-hop wireless communication networks mainly includes three aspects: convergence and scalability. In the simulation experiments in this section, the FP algorithm is used as the classical optimization algorithm for the power allocation problem to perform a comparative analysis with the GCNTO proposed in this paper.

3.1.1

Convergence analysis

This subsection firstly verifies the convergence of GCNTO through simulation experiments, and secondly compares the performance difference of the model under different hyperparameters by modifying the simulation parameters.The training convergence curve of GCNTO is specifically shown in Fig. 2. The horizontal coordinate training rounds represent the number of times the neural network works on the training set, and the vertical coordinate is the percentage of the weighted sum rate of the system computed by the GCNTO model under each training round to that computed by the FP algorithm.The GCNTO gradually converges from the 70th training round, and the final convergence is 97.58% of the optimization result of the FP algorithm.

Next, we tried to train GCNTO using 1000, 2000, and 3000 training samples in the same network environment respectively, and the training results are specifically shown in Fig. 3. Training GCNTO on datasets of sizes 1000, 2000, and 3000 respectively, the final convergence of the GCNTO model gradually improves as the size of the training set increases, this is because the larger the number of training sets, the more reliable the features extracted by the model, and the better the model's fitness to the optimization problem. When the training set is extended to 3000, the GCNTO model converges to 98.85% of the FP algorithm.

3.1.2

Scalability analysis

In this subsection, the scalability of GCNTO in different network environments will be cross-validated by deflating and deflating the area, as well as increasing and decreasing the number of users. First, 60 users of three communication types are randomly placed in a 1km × 1km range, the trained GCNTO model in this scenario is saved offline, and then the scenario is reset. Based on the unchanged range of 1km×1km, 90, 120, 150, 180, and 300 users using the three communication types are randomly placed, and then the offline models are tested respectively.The optimization performance of GCNTO with respect to the change in the number of devices is specifically shown in Fig. 4. In the new network scenario, the original GCNTO optimization result still achieves more than 96% of the performance of the FP algorithm. This result will allow the GCNTO model to be extended for use in any scenario where the device density is in the range of 5 times that of the original scenario.

Next, keeping the total number of 60 users unchanged, the side length of the square area range is adjusted to 0.2km, 0.3km, 0.5km, 0.8km, and 1.2km, respectively, and the offline model is tested, and the results are specifically shown in Figure 5. Despite the fact that the area range set in the dataset is reduced to 1/25 of the original one, the topology of the communication scenario has not changed relative to the original one, so the power allocation decision with a performance of more than 98.3% of the FP algorithm can still be made using the original GCNTO offline model.

The above experimental results fully validate the scalability of GCNTO, a property which will allow the GCNTO model to be used in any scenario irrespective of the scenario size, with the only qualification being that it does not change the graph topological relationships provided by the original dataset.

3.2

Analysis of power allocation performance in multi-hop wireless communication networks

In this section, the following WMMSE, IWFA, MLP and GCN models are used for comparison experiments. A cellular network environment of 5-cell 20-user network is set up, and the sum rate of equal and random weights weighted by all the models in this environment including the GCNTO in this paper is specifically shown in Table 2. The performance of all the models shows a decreasing trend as the noise power increases. Among them, the gap between WMMSE, GCN and HGNN is gradually narrowing. In addition, the GCNTO proposed in this chapter outperforms the performance of GCN, MLP, and IWFA, and approaches the performance of the WMMSE algorithm. Specifically, GCNTO achieves 94.49% and 95.94% of the performance of the WMMSE algorithm for equal-weighted and random-weighted weighted sum rates, respectively.

Table 2.

Equal weight and Random weight

Index	10 log10 σ2	WMMSE	IWFA	MLP	GCN	GCNTO
Equal weight	10dB	30.747	26.578	21.069	28.362	29.052
	5dB	29.856	25.843	19.991	27.619	28.292
	0dB	28.012	24.375	18.222	26.174	26.72
	-5dB	24.814	21.828	15.754	23.609	23.975
	-10dB	20.313	18.192	12.729	19.814	19.974
Random weight	10dB	17.354	12.671	11.914	15.504	16.585
	5dB	16.81	12.308	11.401	15.019	16.022
	0dB	15.711	11.59	10.518	14.083	14.967
	-5dB	13.815	10.371	9.211	12.536	13.294
	-10dB	11.209	8.636	7.508	10.328	10.754

Further considering the scalability of GCNTO when the number of users varies, the interference channels of 5 cell 20 users and 5 cell 30 users are used for model training, while the interference channel of 5 cell 25 users is used for model testing in the inference phase. The sum rates of GCN and GCNTO during training and validation are specifically shown in Fig. 6. As the number of iterations increases, the sum rates of GCN and GCNTO gradually increase and converge. Specifically, GCNTO obtains a superior performance over GCN in all these different experimental setup scenarios. Moreover, the performance of GCN and GCNTO increases as the number of users increases. Even when dealing with untrained 5-cell 25-user channels, both GCN and GCNTO can adapt well to the variation in the number of users and accurately output the power allocation vectors to ensure the desired data rate.

This section also considers the scalability of GCNTO when the number of cells varies. The model training is carried out using the interference channels of users in cell 5, 20 and cell 7, 28, while the model testing is carried out using the interference channel of users in cell 6, 24 in the inference phase, and the test results are specifically shown in Fig. 7. From the figure, it can be clearly seen that the sum rate of GCN and GCNTO gradually increases and converges. As the number of cells increases, the performance of GCN and GCNTO increases. On the basis of no prior training, when dealing with a 6-cell 24-user channel, they are still able to quickly adapt to the change in the number of cells, dynamically capture the topology and node information, and still maintain the desired data rate.

4

Conclusion

In this paper, a graph neural network (GCN)-based intelligent control model for multi-hop wireless communication network topology (GCNTO) is constructed as a means to achieve intelligent optimization of network topology control. Network topology control simulation experiments are being carried out for GCNTO in cellular network conditions.

The convergence and scalability of GCNTO are examined in network power control performance simulation experiments. After modifying the simulation parameters, GCNTO gradually converges from the 70th training round, and the final convergence result is 97.58% of the optimization result of FP algorithm, and in the case of different number of training samples under the same network environment, GCNTO converges to 98.85% of FP algorithm when the training set is expanded to 3,000, which both show excellent convergence. The scalability of GCNTO can be verified by zooming in and out of the area and increasing and decreasing the number of users. By randomly placing different numbers of users of the three communication types within a range of 1km×1km, the optimization results of the original GCNTO still have more than 96% of the performance of the FP algorithm. On the contrary, by maintaining the number of users and adjusting the range of the square area, the original GCNTO still achieves more than 98.3% of the performance of the FP algorithm. Obviously, GCNTO has excellent performance in power allocation decision making for multi-hop wireless communication networks.

In the same network environment, with increasing noise power, GCNTO outperforms the comparative models of GCN, MLP, and IWFA, achieving 94.49% and 95.94% of the performance of the WMMSE algorithm for the equal-weighted and random-weighted weighted sum rates, respectively. Whether the model is trained using the interference channels of users in cell 5, cell 20 and cell 5, cell 30, and the model is tested using the interference channel of users in cell 5, cell 25 in the inference phase, or the model is trained using the interference channels of users in cell 5, cell 20 and cell 7, cell 28, and the model is tested using the interference channel of users in cell 6, cell 24 in the inference phase, the GCNTO untrained The GCNTO can adapt well to the new changes in the number of users, accurately output the power allocation vector, and maintain the desired data rate.

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Research on topology control method of multi-hop wireless communication network based on graph neural network

Ang Li

Publicado en línea: 21 mar 2025

Recibido: 14 nov 2024

Aceptado: 24 feb 2025

DOI: https://doi.org/10.2478/amns-2025-0566

Palabras claveGraph neural network, Multi-hop wireless communication network, Network topology, Function multilayer perceptron

© 2025 Ang Li, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Palabras clave
Graph neural network, Multi-hop wireless communication network, Network topology, Function multilayer perceptron