1. Introduction

2. Related Work

3. Our Method

3.1. T-GCN with Hierarchical Attention Mechanism Combining Dynamic and Static Attention

In this module, we detail the hierarchical attention mechanism introduced in this paper. Our hierarchical attention mechanism is divided into two parts: the static attention mechanism and the dynamic attention mechanism. The static attention mechanism includes information that remains unchanged in space and time within network traffic, such as the relative positions between base stations, population density, and channel models. The dynamic attention mechanism encompasses information that changes over time, such as temporal relationships and the impacts of holidays.

To use graph convolutional neural networks to address wireless communication issues, we first need data structured as a graph. Since the mobile communication wireless network consists of various base stations, if each base station is considered a node and the communication between base stations is considered an edge, we can construct a graph based on this.

The initial adjacency matrix is simple, where connections are represented as 1 and no connections as 0. Thus, the nodes can form a matrix that expresses the spatial relationships between data nodes. Compared to traditional neural networks, the core innovation of graph convolutional neural networks lies in the adjacency matrix. However, values of 0 and 1 can only reflect whether a connection exists, not the strength of the connection. If the connections in the adjacency matrix are designed to adapt to the data itself, rather than being just 0 and 1, then the graph convolutional network transforms into a graph attention mechanism network. Figure 4 illustrates the process of transforming the wireless network traffic prediction problem into a graph and constructing the adjacency matrix.

Figure 2. Figure 2 illustrates the framework of our hierarchical attention mechanism module, which integrates both static and dynamic attention mechanisms.

Figure 2. Figure 2 illustrates the framework of our hierarchical attention mechanism module, which integrates both static and dynamic attention mechanisms.

Graph neural networks have achieved significant research results and practical applications in traffic flow and air quality prediction. However, their research in the mobile communication field remains limited. When processing mobile communication data using graph neural networks, the adjacency matrix of the graph is mostly derived by calculating the correlation between sequences. For instance, if the values of node 1 and node 2 are very close in a time series, their adjacency matrix value will be set higher. Conversely, if the values of node 1 and node 5 differ significantly, the value will be set lower (e.g., the traffic of node 1 in the next 3 hours is 2Gbps, 6Gbps, 10Gbps, and node 2 is 2.5Gbps, 5Gbps, 9Gbps, while node 5 is 11Gbps, 3Gbps, 1Gbps).

Figure 3. Figure 3 illustrates the framework of our heatmap processing module, which primarily consists of two parts. One part transforms raw data into heatmap video data, and the other constructs an LSTM network optimized by a Laplacian-tuned attention mechanism.

Figure 3. Figure 3 illustrates the framework of our heatmap processing module, which primarily consists of two parts. One part transforms raw data into heatmap video data, and the other constructs an LSTM network optimized by a Laplacian-tuned attention mechanism.

This module innovatively combines dynamic data characteristics that change over time with static data characteristics that remain constant over time, proposing a new graph attention mechanism algorithm using a hierarchical attention mechanism. This module includes the following three innovations:

• Introducing a hierarchical attention mechanism that combines both static and dynamic data characteristics.
• Static features include channel characteristics, physical distances, spectrum usage, and other static information.
• The dynamic attention mechanism takes into account the characteristics of real-time data updates, not only incorporating time series correlation calculations but also incremental learning features to meet the needs of different changing scenarios. Additionally, it considers important temporal features such as holidays. Figure 5 shows the overall framework of this module.

3.1.1. Static Attention

In this section, we will detail the construction of our static attention mechanism. By considering various explicit and invariant factors, we construct a weight function for the static attention mechanism and gradually design a mathematical model with learnable parameters. This model aims to comprehensively account for grid population density, physical distance of base stations, radiated power, transmitting antenna gain, system loss factor, electromagnetic wave data (wavelength), environmental occlusion parameters, and the multipath effect on the static attention weights.

First, we define the basic formula for the static attention weight $w_{ij}$, where $w_{ij}$ represents the attention weight of node i to node j:

$$w_{ij}=\mathrm{softmax}\left(f_{\mathrm{att}}(x_i,x_j,E_{ij})\right)$$

(1)

Here, $f_{\mathrm{att}}$ is an attention scoring function, $x_i$ and $x_j$ represent the features of nodes i and j respectively, and $E_{ij}$ represents the edge features between nodes i and j (including distance, radiated power, etc.). The softmax function ensures that the sum of attention weights from node i to all nodes is 1.

Next, we construct the scoring function $f_{\mathrm{att}}$. Here, we consider various spatial factors:

$$f_{\mathrm{att}}(x_i,x_j,E_{ij})=\sigma (a^T[W{X}_i\parallel W{X}_j\parallel \phi \left(E_{ij}\right)\left]\right)$$

(2)

where a is a learnable weight vector, W is a feature transformation matrix that maps node features to a common space, and ∥ denotes the concatenation operation. $\sigma$ is a non-linear activation function such as LeakyReLU. $\phi \left(E_{ij}\right)$ is a function that integrates edge features (e.g., physical distance, radiated power, etc.).

For the edge feature processing function $\varphi \left(E_{ij}\right)$, we have designed the following considering the factors related to wireless network traffic prediction:

$$\varphi \left(E_{ij}\right)=V\left[{\displaystyle \frac{1}{d_{ij}^a}}·{\mathrm{PopDensity}}^{\beta }·P_{rad}^{\gamma }·G_{ant}^{\delta }·L_{sys}^n·{\lambda }^{\theta }·O_{env}^l·M_{path}^k\right]$$

(3)

where $d_{ij}$ represents the physical distance between nodes i and j, ${\mathrm{PopDensity}}^{\beta }$ is the population density of the grid, and $P_{rad}$, $G_{ant}$, $L_{sys}$, $\lambda$ are the radiation power, antenna gain, system loss factor, and wavelength, respectively. $O_{env}$ and $M_{path}$ are the quantifications of the obstruction environment parameter and multipath effect. The parameters $\alpha$, $\beta$, $\gamma$, $\delta$, $\eta$, $\theta$, l, k are learnable parameters that adjust the influence of each factor. V is another learnable weight vector used to transform the comprehensive features into a format required by the attention mechanism.

The function $f_{att}$ is an attention scoring function, and $x_i$ and $x_j$ represent the features of nodes i and j, respectively. $E_{ij}$ represents the edge features between nodes i and j (including distance, radiation power, etc.), and the softmax ensures that the sum of attention weights from node i to all nodes is 1.

During the model training process, the values of population density, radiation power, and antenna gain are fixed. However, we set learnable parameters. By minimizing the prediction error (e.g., using a mean squared error loss function) during model training, we optimize the learnable parameters a, W, V, $\alpha$, $\beta$, $\gamma$, $\delta$, $\eta$, $\theta$, l, k to values suitable for the wireless network traffic prediction scenario. This process will automatically adjust these parameters through backpropagation to comprehensively account for the influence of various spatial factors on static attention weights. After completing the model training, the static attention mechanism remains constant when predicting traffic.

3.1.2. Dynamic Attention

The basic idea is to enhance the dynamic attention mechanism from four distinct aspects: dependency on time series, adaptability to real-time data, incremental learning mechanisms, and adjustment driven by dynamic events. Each aspect contributes uniquely to the robustness of the model, and we adopt a step-by-step construction approach, with specific mathematical formulas detailed below.

First, we construct the basic dynamic attention model. We define the dynamic attention weight function $W_{ij}^t$, which represents the attention weight of node i to node j at time t:

$$W_{ij}^t=\mathrm{softmax}\left(f_{\mathrm{dyn}}^t(H_i^t,H_j^t,\Delta t)\right)$$

(4)

where $f_{\mathrm{dyn}}^t$ is the dynamic attention scoring function. $H_i^t$ and $H_j^t$ are the hidden states of nodes i and j at time t, generated by models such as RNN or LSTM that capture time series features. $\Delta t$ is the time interval, used to capture the dependency on the time series. The softmax ensures that the sum of dynamic attention weights from node i to all nodes is 1.

The time series dependency refers to the model’s ability to recognize and utilize the influence of past data points on future predictions. Capturing this dependency is crucial when processing time series data. ConvLSTM (Convolutional Long Short-Term Memory) is a special LSTM network that incorporates convolution operations into the LSTM structure, allowing the model to handle long-term dependencies in time series data while effectively processing spatial data. ConvLSTM achieves this by adding convolution operations to the input gate, forget gate, and output gate of LSTM, enabling the network to capture dependencies in the time dimension and extract features in the spatial dimension. This is particularly important for spatiotemporal datasets, such as network traffic prediction, where base station traffic is influenced by both time factors and spatial location and surrounding environment. We use ConvLSTM to capture the time series dependency, and the hidden state of the node $H_t$ is updated as follows (where $X^t$ is the input feature at time t):

$$H^{t+1}=\mathrm{convLSTM}(H^t,X^t)$$

(5)

Real-time data adaptability focuses on the model’s ability to adapt to fresh data. Incremental learning mechanisms allow the model to update its knowledge base upon receiving new data rather than retraining from scratch. This mechanism is crucial for handling real-time streaming data as it significantly reduces computational resource consumption and allows the model to dynamically adapt to data changes. In incremental learning, model parameters are updated based on the loss function of the new data batch. By calculating the loss and applying backpropagation, the model can gradually adjust its parameters to better reflect the characteristics and patterns of the new data. This method ensures that the model retains the memory of previously learned knowledge while continuously learning new information. Our incremental learning mechanism is mathematically formulated as follows:

$${\Theta }^{t+1}={\Theta }^t+\eta ·{\nabla }_{\Theta }L(Y_t,{\widehat{Y}}_t)$$

(6)

where $\Theta$ represents the parameters of the LSTM model, $\eta$ is the learning rate, L is the loss function, $Y_t$ is the true value, and ${\widehat{Y}}_t$ is the predicted value. ${\nabla }_{\Theta }L$ is the gradient of the loss function with respect to $\Theta$.

Event-driven dynamic adjustment focuses on the model’s ability to respond to specific events. In applications such as network traffic prediction, specific events (e.g., holidays, large events) may cause significant changes in traffic patterns. To capture such non-periodic changes, the model needs to recognize these events and adjust its behavior accordingly. Our implementation strategy introduces an event recognition component to the model, which identifies the occurrence of specific events based on input data. The model can then adjust the dynamic attention weights to respond to these events, for instance, by increasing the attention to specific time points or areas. This adjustment can be based on predefined rules or a data-driven approach that learns the relationship between events and traffic patterns.

For dynamic adjustment based on specific events, we define an event impact function $E\left(t\right)$ and integrate it into the dynamic attention scoring function:

$$f_{\mathrm{dyn}}^t(H_i^t,H_j^t,\Delta t)=\sigma (a^T\left[H_i^t\right|H_j^t]+b·E\left(t\right))$$

(7)

where $E\left(t\right)$ is the event impact at time t, which can be a model learned from historical data or a rule-based indicator function. a and b are learnable weight vectors, and $\sigma$ is an activation function.

3.1.3. Combining Static Attention and Dynamic Attention

To combine the static and dynamic attention mechanisms through a hierarchical attention mechanism, we use a framework of feature concatenation and interactive learning. This framework not only integrates static and dynamic information but also optimizes this integration through learning to enhance the overall model performance.

First, feature concatenation is a straightforward data fusion method whose core idea is to aggregate features from different sources to form a more comprehensive and rich feature representation. In the hierarchical attention mechanism, feature concatenation is used to combine features generated by static and dynamic attention mechanisms. Static features reflect long-term stable, time-independent attributes, such as geographical location and infrastructure layout. Dynamic features capture information that varies over time, such as traffic fluctuations and event impacts. By concatenating these two types of features, we obtain a comprehensive feature vector that encompasses both spatial static properties and temporal dynamic changes. The advantage of this fusion method lies in its simplicity and directness, allowing the model to access and utilize both aspects of information in subsequent processing, thereby improving the accuracy and reliability of decisions or predictions. The specific implementation of our formula is as follows. Suppose for each node i, we obtain the static feature vector $S_i$ through the static attention mechanism and the dynamic feature vector $D_i^t$ through the dynamic attention mechanism (where t represents time). We can concatenate these two parts of the features to form a comprehensive feature vector $C_i^t$:

$$C_i^t=[S_i\parallel D_i^t]$$

(8)

Here, ‖ denotes the concatenation operation of vectors. This comprehensive feature vector $C_i^t$ will be used as the input for subsequent processing, integrating static spatial information and dynamic temporal information.

Interactive learning further optimizes the interaction between different features on the basis of feature concatenation through learning. The core of this method lies in identifying and utilizing the intrinsic connections between static and dynamic features and their joint influence on the final task (such as prediction). Interactive learning is not just about feature concatenation; it focuses on how to make these features "communicate" and interact more effectively within the model. By introducing one or more fully connected layers (or other types of network structures), the model can learn a nonlinear combination of feature representations that comprehensively considers the interactions and dependencies between features. This learning process enables the model to automatically discover which feature combinations are most important for the prediction task and how to best combine static and dynamic information to improve performance. The advantage of interactive learning lies in its flexibility and adaptability. It can not only automatically adjust the relationships between features but also dynamically adjust the way features are combined according to the needs of the specific task. This means the model can more accurately understand the inherent structure and patterns of the data, thus making more precise judgments or predictions. The formula for our interactive learning is as follows:

$$I_i^t=\sigma (W_C{C}_i^t+b_C)$$

(9)

where $W_C$ is the weight matrix, $b_C$ is the bias term, $\sigma$ is the nonlinear activation function, and $I_i^t$ represents the feature representation after interactive learning.

The feature vector $I_i^t$ obtained through interactive learning can be used for various tasks, and it is particularly suitable for our network traffic prediction. Our goal is to predict a specific output (such as the traffic at node i at the next time step) based on the current static and dynamic information. We can further process $I_i^t$ as follows:

$$Y_i^{(t+1)}=\mathrm{ReLU}(W_Y·I_i^t+b_Y)$$

(10)

Here, $W_Y$ and $b_Y$ are the weights and biases of the output layer, and $Y_i^{(t+1)}$ is the predicted output. ReLU ensures that the output of the task is regressive.

3.2. Tuning Graph Neural Network Aggregation Levels with Laplacian Matrix

We explain the aggregation process of graph neural networks using a simple base station graph we constructed. When dealing with graph-structured data, graph neural networks need to extract spatial information, so it is necessary to aggregate all the nodes first.

For example, for a node v, in the first aggregation, the new node feature $h_v^{\prime }$ is obtained by a weighted average of the features of its neighboring nodes, which can be expressed as:

$$h_v^{\prime }=\mathrm{ReLU}\left(W\ast {\displaystyle \frac{1}{\left|N\right(v\left)\right|}}\sum _{u\in N\left(v\right)}{h}_u\right)$$

(11)

Here, $h_v^{\prime }$ is the new feature of node v after aggregation, and $N\left(v\right)$ is the set of neighboring nodes of v. $h_u$ is the feature of neighboring node u, W is a learnable weight matrix, and ReLU is the activation function. For instance, for node 2 in Figure 5, its new feature $h_2^{\prime }$ will be calculated based on the weighted average of the features of nodes 1, 3, and 4 (adjusted by the weight matrix W and activated by the ReLU function). This aggregation method allows each node to capture information from its neighbors, and with multiple aggregations, it can obtain information from nodes that are further away.

For example, our node 2 (Figure 5), after the first aggregation, not only retains its own information but also incorporates information from nodes 1, 3, and 4. Similarly, after the first aggregation, node 3 has both its own information and information from nodes 2 and 5. Therefore, after the second aggregation, node 2 will have information from node 5 through the aggregation of node 3. Thus, multiple aggregations enable each node to obtain information from more distant areas. Theoretically, the more aggregations, the more comprehensive the spatial information each node can acquire from other nodes. However, aggregation also has a side effect—the over-smoothing problem.

3.2.1. Motivation

In graph neural networks, performing multiple aggregations, as iterated in the aforementioned formula, results in each node continually integrating information from increasingly distant neighbors. Initial aggregations might integrate information from nearby nodes, but as the number of aggregations increases, nodes begin to incorporate features from further away nodes. If the aggregation steps are too many, eventually, all nodes may collect information from almost the entire graph, causing their feature representations to become increasingly similar. This leads to the over-smoothing problem, where the distinguishability between nodes diminishes as their feature representations become identical or similar. Consequently, the model struggles to differentiate between nodes or perform tasks that rely on node-specific characteristics, such as node classification.

3.2.2. Case Study

For example, in our previous case, after two aggregations, the overall model effect remains positive because each node acquires information from other nodes. However, what happens if the model undergoes five, six, or even more aggregations? Take node 2 for instance; after multiple aggregations, its neighboring nodes 3 and 1, due to prior aggregations, already possess information about node 2. Thus, further aggregation results in node 2 redundantly acquiring its own information, and other nodes encounter the same issue. As each node fully integrates its information, they become increasingly similar, leading to over-smoothing.

Conversely, if the number of aggregations is too few, the main issue is insufficient information. Specifically, nodes may fail to receive enough neighborhood information to fully understand their context within the graph. For example, a node may only aggregate information from directly connected neighbors, failing to sense more distant neighbors. In our case, if aggregation occurs only once, node 2 will not be aware of information from node 5. This lack of information restricts the model’s performance, especially in tasks that require understanding broader connectivity patterns, such as graph classification or complex node classification problems. Therefore, determining the appropriate number of aggregation layers is critical when designing graph neural networks.

3.2.3. Our Method

The authors of this paper observed that the graph aggregation process, during practical graph data processing, resembles a high-pass filtering process. During aggregation, high-frequency anomalous components are often lost, while the basic information that most nodes possess tends to be retained. This observation led us to consider frequency domain transformations. Specifically, if we conduct spectral analysis on the graph data during aggregation and ensure that the spectrum retains over 70% (i.e., $\sqrt{2}$) of the information, we can generally achieve a balance between extracting spatial relationships and avoiding over-smoothing. The following is a detailed introduction of our specific implementation:

Graph Theory Background: In graph theory, adjacency matrices, degree matrices, and Laplacian matrices are used to describe the structure of a graph. The adjacency matrix represents whether the nodes in the graph are directly connected (it can be a simple 0/1 matrix or have specific values, as is the case with graph attention mechanisms), while the degree matrix records the number of edges connected to each node (the degree matrix is diagonal, with each value representing the number of edges connected to a specific node). The Laplacian matrix is obtained by subtracting the adjacency matrix from the degree matrix (as shown in the previous figure). It combines the degree information of the nodes with their adjacency relationships. This matrix plays a crucial role in analyzing the structural properties of a graph, especially in the application of graph neural networks, where it helps in understanding and processing graph data.

Eigenvalues and Eigenvectors of the Laplacian Matrix: The eigenvalues and eigenvectors of the Laplacian matrix reveal important structural characteristics of the graph. The eigenvalues reflect the strength of different connectivity patterns in the graph, while the eigenvectors describe the distribution of these patterns across the graph. Each eigenvalue corresponds to an eigenvector, indicating the distribution of a specific connectivity pattern within the graph. In graph neural networks, eigenvalues and eigenvectors are often used to understand the global structure of the graph, helping to optimize the network for more effective graph data processing.

Judgment Strategy: The innovation in our algorithm lies in using the Laplacian matrix of the graph to guide the aggregation process in graph neural networks (GNNs). Specifically, we determine whether to continue aggregation by analyzing the eigenvalues and eigenvectors of the Laplacian matrix. When more than 70% of the eigenvalues and eigenvectors are retained, it indicates that there is still sufficient distinguishability between nodes, and aggregation can continue to obtain more extensive neighborhood information. Once the retained features drop below 70%, aggregation is halted to prevent over-smoothing. This approach innovatively combines the spectral characteristics of the graph with the structural learning in graph neural networks, providing a new perspective for addressing the over-smoothing problem in GNNs.

Figure 4 illustrates our approach to optimizing the number of aggregations through tuning the graph data spectrum.

Figure 4. Tuning Graph Neural Network Aggregation Levels with Laplacian Matrix.

Figure 4. Tuning Graph Neural Network Aggregation Levels with Laplacian Matrix.

4. Experiment

4.4. Prediction Performance Comparison

We compared the performance of each model across different data categories, prediction time spans, and regions.

Table 1 presents the average MAE of each model on the test set for SMS, Call, and Internet traffic data.

The results show that our model achieved the best prediction performance across all three types of communication activities, significantly outperforming the baseline models.

Figure 5. Comparison of MAE for different models across SMS, Call, and Internet traffic data.

Figure 5. Comparison of MAE for different models across SMS, Call, and Internet traffic data.

Table 2 shows the RMSE of each model for tasks predicting 1 hour, 6 hours, and 12 hours ahead.

As the prediction time span increases, the errors of all models increase accordingly. However, our model consistently maintains the lowest RMSE across all time spans, indicating its stability and superiority in both short-term and long-term predictions.

Figure 6. Comparison of RMSE for different models across varying prediction horizons.

Figure 6. Comparison of RMSE for different models across varying prediction horizons.

To verify the generalization ability of our model, we tested it on the Rome mobile communication dataset (a similar data structure to Milan). The results show that our model also achieved the best prediction performance on the Rome dataset, demonstrating its applicability to different cities.

4.5. Ablation Experiments

To understand the contributions of each component of the model, we conducted ablation experiments.

We removed the static attention mechanism and only used the dynamic attention mechanism. The MAE increased from 8.76 to 9.34, and RMSE increased from 11.56 to 12.12. This indicates that the static attention mechanism plays a crucial role in capturing the fixed spatial relationships between base stations.

We then removed the dynamic attention mechanism and only used the static attention mechanism. The MAE increased to 9.56, and RMSE increased to 12.45. This shows that the dynamic attention mechanism helps the model adapt to temporal dynamics in traffic; removing it reduces the model’s ability to capture traffic fluctuations.

Next, we used a fixed number of GCN layers (e.g., 2 layers) without adjusting based on the Laplacian matrix. The MAE increased to 9.12, and RMSE increased to 11.89, verifying the effectiveness of adaptively adjusting the GCN aggregation depth.

Finally, we removed both the hierarchical attention mechanism and the Laplacian adjustment, causing the model to degrade to a standard GCN. The MAE increased to 10.45, and RMSE increased to 13.34, confirming that our proposed innovations significantly contribute to improving model performance.

Figure 7. Ablation study showing the impact of removing different components on model performance.

Figure 7. Ablation study showing the impact of removing different components on model performance.

5. Conclusions

6. Patents

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