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Article

LHGCN: A Laminated Heterogeneous Graph Convolutional Network for Modeling User–Item Interaction in E-Commerce

by
Kang Liu
,
Mengtao Kang
*,
Xinyu Li
and
Wenqing Dai
School of Artificial Intelligence, China University of Mining and Technology-Beijing, Beijing 100083, China
*
Author to whom correspondence should be addressed.
Symmetry 2024, 16(12), 1695; https://doi.org/10.3390/sym16121695
Submission received: 26 November 2024 / Revised: 16 December 2024 / Accepted: 18 December 2024 / Published: 21 December 2024
(This article belongs to the Topic Advances in Computational Materials Sciences)
Browse Figures
Figure 1
<p>Alibaba dataset structure pattern (<b>left</b>) and feature distribution (<b>right</b>).</p> "> Figure 2
<p>The overall architecture of LHGCN.</p> "> Figure 3
<p>Laminate generation module.</p> "> Figure 4
<p>Adaptive convolution module (gate outputs z and r and controls 2 types of laminates enjoying different receptive fields); the detailed implementation of attention and gate is shown in <a href="#symmetry-16-01695-f005" class="html-fig">Figure 5</a>.</p> "> Figure 5
<p>Information flow in convolutional layers, exemplified by laminate U.</p> "> Figure 6
<p>Laminate fusion module (the AC Layer is the adaptive convolution layer).</p> "> Figure 7
<p>Effects of the module design: <b>full</b> indicates the full model, <b>-laminate</b> excludes the LGM, <b>-gate</b> excludes the ACM, and <b>base</b> omits both modules mentioned above.</p> "> Figure 8
<p>Comparison of experimental results between GRU and LSTM.</p> "> Figure 9
<p>Multiplex effectiveness analysis.</p> "> Figure 10
<p>Parameter sensitivity analysis.</p> ">
Versions Notes

Abstract

:
The e-commerce data structure is a typical multiplex graph network structure, which allows multiple types of edges between node pairs. However, existing methods that rely on message-passing frameworks are not sufficient to fully exploit the rich information in multiplex graphs. To improve the performance of link prediction, we propose a novel laminated heterogeneous graph convolutional network (LHGCN) consisting of three core modules: a laminate generation module (LGM), an adaptive convolution module (ACM), and a laminate fusion module (LFM). More specifically, the LGM generates symmetric laminates that cover diverse semantics to create rich node representations. Then, the ACM dynamically adjusts the node receptive field and flexibly captures local information, thereby enhancing the representation ability of the node. Through symmetric information propagation across laminates, the LFM combines multiple laminated features to optimize the global representation, which enables our model to accurately predict links. Moreover, an elaborate loss function, consisting of positive sample loss, negative sample loss, and L2 regularization loss, drives the network to preserve critical information. Extensive experiments on various benchmarks demonstrate the superiority of our method over state-of-the-art alternatives in terms of link prediction.

1. Introduction

In recent years, with the rapid development of e-commerce, the number of online shoppers around the world has increased significantly, and e-commerce platforms have gradually become the mainstream method of daily consumption. The behavior of users on these platforms is undergoing complex changes, including browsing, clicking, adding items to shopping carts, purchasing, and rating. On these platforms, the relationship between users and products is inherently symmetrical. Users can freely select and purchase products, while the same product can be recommended to multiple users, establishing a bidirectional, equal relationship between the two entities.
Traditional methods, such as collaborative filtering, clustering algorithms, ranking models, and probabilistic approaches, have been widely applied to model user–product interactions. However, these approaches face certain challenges when applied to heterogeneous and dynamic e-commerce platforms. For example, collaborative filtering often struggles with the sparsity problem in large-scale datasets, making it difficult to provide accurate recommendations for users with limited interaction history. Clustering methods like Louvain [1] focus on global modularity optimization, which limits their ability to differentiate multiple types of edges. Similarly, ranking models such as PageRank [2] prioritize global node importance but lack the granularity needed to capture subtle local interaction patterns essential for edge prediction. Probabilistic methods, including stochastic block models (SBM [3]) and Bayesian networks [4], while theoretically powerful, are constrained by strong assumptions that reduce their ability to handle heterogeneity and multi-relational structures, with high computational complexity, further impacting their scalability in large-scale scenarios.
To address these challenges and leverage the strengths of graph-based approaches, researchers have increasingly turned to graph neural networks (GNNs). Since the relationship between users and products in e-commerce platforms can be constructed as a graph structure, graph neural networks (GNNs) can effectively express the multi-level interactive relationship between users and products. Moreover, by leveraging symmetrical relationships inherent in graph structures, such as bidirectional influence between nodes (users and products) or repeated patterns across similar user groups, GNNs can utilize neighborhood information more effectively to improve the accuracy of recommendations. Therefore, to better capture the symmetrical and hierarchical relationships between users and products, researchers began to introduce GNNs to analyze user behavior in e-commerce platforms.
Early GNNs mainly relied on recurrent neural networks and graph embedding techniques [5,6], which are very effective for small-scale graph datasets. Furthermore, the graph convolutional network [7,8,9] marked a key breakthrough in large-scale graph data processing. In recent years, more sophisticated GNN variants, such as graph attention networks (GAT) [10] and graph isomorphism networks (GIN) [11], have further improved the performance of the model. For example, GNNs perform well in community detection in social network analysis and can accurately analyze the interaction between users and items in recommendation systems [12,13]. In addition, in the field of drug development, GNNs help predict molecular properties, thereby accelerating the development of new drugs [14,15,16].
According to the type characteristics between nodes and edges in the graph network, GNNs can be roughly divided into homogeneous methods [17,18] and heterogeneous methods [19,20,21]. Homogeneous methods, including GCN [8], GAT [10], and graph autoencoder [22], mainly address the case where there is only one type of node and one type of relationship in the graph. However, homogeneous methods have difficulty in effectively capturing the diverse structures and relationships between nodes in complex networks, which limits their applicability in complex scenarios. In contrast, heterogeneous methods, including heterogeneous GCNs [23] and heterogeneous graph attention networks (HANs) [24], can effectively handle graph networks with more than one node type or relationship type in the graph. These methods exploit various forms of information from nodes and edges to model and analyze complex graph networks, for example, in the field of bioinformatics [25,26]. At present, heterogeneous GNNs are mainly used for simple graph network analysis, but there are many complex graph networks in the real world, such as user–product interaction in e-commerce, which is beyond the scope of simple graphs and needs to be modeled through multiplex graphs.
A multiplex graph is a special graph structure that allows multiple edges between the same pair of nodes, and each edge represents a different relationship or connection pattern. An illustrative example of a multiplex graph is the Alibaba dataset, depicted in Figure 1. There are three types of edges connecting users and products in the graph such as clicks, inquiries, and contacts. The analysis of multiplex graphs faces many challenges, mainly in the following aspects. (a) Node feature fusion: the features of nodes in a multiplex graph can represent different semantic information. When homogeneous methods are applied to them indiscriminately, message passing will lead to semantic confusion. Therefore, an innovative message-passing method tailored for multiplex graphs is needed. (b) Complex relationship modeling: a multiplex graph contains multiple edges, representing different types of relationships between the same node pairs. Therefore, it is crucial to design a novel edge aggregation mechanism to merge the information conveyed by different relationships.
In this paper, we propose a laminated heterogeneous graph convolutional network (LHGCN) framework to effectively aggregate vital information for link prediction. We introduce a laminate generation module (LGM), which partitions the original nodes into symmetric laminates containing different semantic information according to the node semantic attributes. Then, an adaptive convolution module (ACM) is designed for each laminate, which can dynamically adjust the neighborhood receptive field of each node in each laminate, thereby flexibly capturing the local information of the node. Through symmetric information propagation across laminates, the laminate fusion module (LFM) combines node features from multiple laminates to optimize node global feature information, which enables our model to accurately predict links. The main contributions in this paper are as follows:
  • We design a laminated heterogeneous graph convolutional network (LHGCN) framework that can effectively aggregate the multiple semantic information contained in nodes in multiplex graphs and improve the accuracy of link prediction between nodes. The laminate generation module can effectively alleviate the semantic confusion caused by node features in the message propagation process, thereby achieving efficient extraction of different semantic features of nodes in the multiplex graph and improving the performance of the model;
  • To address the challenges associated with modeling complex relations, we propose an adaptive convolution module using a gating mechanism. Message passing within the framework is carefully designed to flexibly control the influence range of near and far neighbors, thereby adaptively determining the neighborhood influence of each node;
  • Extensive experiments are implemented on the DTI, Amazon, Alibaba-s, Alibaba, and Douban datasets, demonstrating that the proposed network obtains competitive performances with ten state-of-the-art approaches in terms of quantitative comparison. Ablation studies show the effectiveness of the module design.
The remainder of this paper is organized as follows. Section 2 introduces previous related works. In Section 3, we provide a detailed description of the proposed LHGCN framework for link prediction. Extensive experiments compared with ten methods on the five datasets are illustrated in Section 4. Finally, Section 5 shows the conclusion.

2. Related Works

Graph Attention Networks. Graph attention networks (GATs) [10] use an attention mechanism to dynamically assign attention weights to enhance the aggregation process of node embeddings. By incorporating attention weights, the model is able to selectively aggregate information from neighboring nodes based on the importance of node embeddings, thereby enhancing its expressiveness and generalization capabilities. Based on GATs, the use of a multi-head attention mechanism enables the model to capture more complex relational patterns. GATv2 [27] proposes an enhanced dynamic attention mechanism. In addition, some researchers are also committed to extending the application of GATs to more complex heterogeneous graphs and dynamic graphs. A heterogeneous attention network (HAN) [24] introduces a specialized attention mechanism that can adapt to various types of nodes and edges to capture complex relationships in heterogeneous graphs.
Temporal Graph Neural Networks. Gating mechanisms, such as long short-term memory (LSTM) [28] networks and gated recurrent units (GRU) [29], have been integrated into GNNs to process temporal graphs. Using LSTM or GRU in graph neural networks can not only enhance the model’s ability to represent complex graph data but also alleviate the inherent limitations of traditional GNNs in processing dynamic or temporal information. Therefore, these methods provide innovative solutions for spatio-temporal prediction in graph-structured datasets, such as traffic forecasting and financial temporal analysis. A temporal graph attention network (TGAT) [30] can effectively process graph data characterized by the continuous evolution of nodes and edges by combining temporal encoding and temporal attention mechanisms.
Heterogeneous Graph Neural Network. Heterogeneous graphs allow the coexistence of different types of entities and relationships, enabling the modeling of various complex real-world scenarios, such as citation networks [31], social networks [32], and knowledge graphs [33]. They can be roughly divided into two categories: hand-crafted-based methods and deep neural network-based methods. The hand-crafted-based methods randomly initialize node embeddings and utilize hand-crafted features and rules to learn the embeddings. To address the inherent complexity of heterogeneous graphs, decomposition-based methods [34,35,36] divide heterogeneous graphs into subgraphs with simpler structures, retain the node similarity within each subgraph, and, finally, integrate the node information in the subgraphs to ensure the comprehensiveness and accuracy of information processing, while random walk-based methods [37,38] generate node sequences guided by meta-paths within the graph, effectively capturing local patterns by sampling neighboring nodes. Deep neural network-based methods focus on the properties of nodes and their interaction patterns. Many methods based on homogeneous neural networks use message-passing mechanisms to effectively capture complex semantic information in heterogeneous graphs and integrate information from neighboring nodes, such as HAN [24], MAGNN [39], HetSANN [40], and other methods [41,42].
However, the research on multiplex heterogeneous graphs remains in its nascent stages. Multiplex heterogeneous graphs extend the concept of heterogeneous graphs by allowing multiple types of edges to exist between each pair of nodes, thereby being able to represent complex interactions between multiple types of nodes in a more detailed manner. To more fully elucidate the multi-dimensional and multi-level interaction dynamics between various entities in complex systems, HGT [43] proposed a framework based on the graph transformer architecture, which combines relative temporal encoding techniques to effectively manage dynamic heterogeneous graphs. DualHGCN [44] effectively transforms multiple bipartite structures into two sets of homogeneous hypergraphs and then utilizes spectral hypergraph convolution to enhance inter-domain and intra-domain feature interaction. BPHGNN [45] performs adaptive learning in multiplex heterogeneous networks while collaboratively inferring node representations in various structural relationships between nodes.

3. Proposed Method

In this section, we provide a detailed explanation of our laminated heterogeneous graph convolutional network. First, we provide the problem formulation of the multiplex bipartite graph. Then, we give the network architecture of the proposed LHGCN. Finally, we describe the designed loss function in detail.

3.1. Problem Formulation

Definition 1
(Heterogeneous Graph). A heterogeneous graph is denoted as G = ( V , E ) , where V = V 1 ∪ V 2 ⋯ ∪ V t presents the set of nodes, with V i indicating nodes of Class i; E = E 1 ∪ E 2 ⋯ ∪ E s denotes the set of edges, with E i representing edges of Class i. Additionally, t and s signify the number of distinct node types and edge types, respectively, and t + s > 2 .
Definition 2
(Multiplex Bipartite Graph). A multiplex bipartite graph represents a specific instance of a heterogeneous graph, defined as G = ( U , V , { E 1 ∪ E 2 ⋯ ∪ E s } ) , where U and V are disjoint sets of nodes representing two distinct types of entities. The set { E 1 ∪ E 2 ⋯ ∪ E s } contains multiple layers of edges, with each layer E i describing a distinct type of interaction or relationship between nodes in U and V.
Dealing with multiplex bipartite graphs is challenging because there are two types of nodes, and different types of edges represent different interactions between nodes. The different interactions between nodes in the graph make it difficult for the model to capture the structure of the graph. However, the interactions between nodes in the graph, also known as link prediction, are crucial for many practical applications, such as recommending related products to users in e-commerce platforms.

3.2. Network Architecture

The overall architecture of the proposed architecture is shown in Figure 2. First, the laminate generation module (LGM) separates the latent representation of each node and creates node representations under different semantic contexts. Then, the adaptive convolution module (ACM) dynamically adjusts the receptive field of the node to ensure that the convolution process adapts to the local features of the graph. Finally, the laminate fusion module (LFM) combines multiple laminated features to optimize the global representation, which enables our model to accurately handle link prediction tasks. In addition, the elaborate loss function, which is composed of positive sample loss, negative sample loss, and L2 regularization loss, drives the network to retain key information.

3.2.1. Laminate Generation Module

For the multiplex bipartite graph, different types of nodes are located in different semantic spaces. Taking user behavior on a shopping platform as an example, there are two different types of nodes: users and products. Semantic attributes related to users may include age, gender, purchase preferences, browsing history, etc. Similarly, semantic attributes related to products may include electronics or clothing, price, brand, rating, etc. Therefore, these two types of nodes cannot be directly compared. To address the above problem, we develop a laminate generation module (LGM) that partitions the multiplex bipartite graph into symmetric laminates based on node semantic information. Each laminate represents a unique semantic perspective of a node, so the original node can be represented by the combination of laminates.
Given the heterogeneity of the multiplex bipartite graph, there are two different node types. The input multiplex bipartite graph is divided into two laminates, as shown in Figure 3. If the category of the node conforms to the laminate semantics, its features are linearly transformed; otherwise, it is initialized as a pseudo node. In each laminate, all nodes are initialized using Equation (1):
h v t = W t x v ∈ R d 1 , if t = ϕ ( v ) , 0 ∈ R d 1 , if t ≠ ϕ ( v ) ,
where x v represents the initial representation of node v, and W t ∈ R d 1 × d ϕ ( v ) denotes the transformation matrix corresponding to the class t laminate. d ϕ ( v ) refers to the initial dimension of representations for class ϕ ( v ) , and d 1 indicates the dimension of the transformed node representations.
If laminate t is not compatible with type ϕ ( v ) , node v is initialized to the zero vector, aligned with the other nodes; otherwise, it is initialized using the initial representation of v through a linear transformation W t specific to the class of laminate t.

3.2.2. Adaptive Convolution Module

As the complexity of the graph structure increases, the local feature information of the nodes will also change significantly. For example, when predicting the next purchase of a user in an e-commerce recommendation system, the association between the purchase behavior of different users and the product may vary greatly. Therefore, we develop an adaptive convolution module, as shown in Figure 4. For each laminate, this module convolves neighborhood messages for each center node while dynamically adjusting the convolution receptive field and adopting a gating mechanism to update the center node representation.
For the given node v and all its neighboring nodes u ∈ N ( v ) , the attention mechanism is employed to calculate a first-order neighborhood message m v , l t :
e u , l , v t = a l t · L e a k y R e L U ( h u t ‖ h v t ) , α u , l , v t = s o f t m a x ( e u , l , v t ∣ l ∈ E ) , α ^ u , l , v = a v e r a g e ( α u , l , v t ) , m v , l t = ∑ u ∈ N ( v ) α ^ u , l , v h u t ,
where e u , l , v t denotes the intermediate quantity utilized for computing α u , l , v t , a l t represents the attention learning parameter, l indicates the edge type connecting node u and node v, ‖ represents the horizontal concatenation of vectors, and E ∈ U × V signifies the set of edges.
The number of convolutional layers is denoted by k, and the neighborhood message associated with edge type l in the k-th layer is computed as follows:
m v , l t , ( k ) = ∑ u ∈ N ( v ) α ^ u , l , v ( k ) h u t , ( k − 1 ) .
A gating mechanism is used to integrate the node neighborhood feature information and the output of the previous convolutional layer to update the representation of each node, as shown in Figure 5. When node v is convolved at layer k, it is denoted as h v t , ( k ) , and the propagation dynamics of the gating mechanism are articulated as follows:
z v , l t , ( k ) = σ ( W z m v , l t , ( k ) + B z h v t , ( k − 1 ) ) , r v , l t , ( k ) = σ ( W r m v , l t , ( k ) + B r h v t , ( k − 1 ) ) , h v , l t , ( k ) ˜ = t a n h ( W m v , l t , ( k ) + B ( r v , l t , ( k ) ⊙ h v t , ( k − 1 ) ) ) , h v , l t , ( k ) = ( 1 − z v , l t , ( k ) ) ⊙ h v t , ( k − 1 ) + z v , l t , ( k ) ⊙ h v , l t , ( k ) ˜ ,
where m v , l t , ( k ) represents the aggregation of neighborhood messages, while W and B denote the corresponding learnable transformation matrices. The variables z and r function as update gates and reset gates, respectively, constrained within the value range of [ 0 , 1 ] . ⊙ signifies element-wise multiplication.
The node representation at level k is obtained by averaging various edge types as shown below.
h v t , ( k ) = σ ( 1 s ∑ l = 1 s h v , l ( t , ( k ) ) .

3.2.3. Laminate Fusion Module

In link prediction of multiplex bipartite graphs, the interactions between nodes usually show cross-type characteristics. Relying solely on the node feature information in a single laminate is not enough to fully reveal the link pattern between nodes. Therefore, we propose a laminate fusion module to integrate the node features in multiple laminates to form the final node feature.
The graph convolution operation expands the perception range of node features while the gating mechanism controls the degree of update. In order to reduce the complexity of the model, nonlinear functions are omitted between convolutional layers.
H t , ( k ) = 1 s ∑ l = 1 s G R U ( A l ( α l ) H t , ( k − 1 ) , H t , ( k − 1 ) ) ,
where A l ( α l ) is the adjacency matrix with attention.
After K layers of convolution operations, as shown in Figure 6, the two laminates are integrated to form the final output representation.
Z = 1 2 ( H 1 , ( K ) + H 2 , ( k ) ) .
This module is able to aggregate node features from different laminate semantic spaces, enabling the model to capture complex interactions between nodes more effectively, thereby improving link prediction accuracy.

3.3. Loss Function

We designed a loss function to drive the model to update its parameters, ensuring that the model can effectively extract important node information from multiplex bipartite graphs and improve the accuracy of link prediction. The loss function contains three components: positive sample loss, negative sample loss, and L2 regularization loss.

3.3.1. Positive Sample Loss and Negative Sample Loss

For a node pair u v with an edge, the positive sample loss is calculated to evaluate the accuracy of the model in predicting the connected nodes. For each positive edge, negative edges are randomly sampled from nodes u and v, and the negative sample loss is calculated to ensure that the model can effectively distinguish between real connections and random noise.
L u v + = l o g σ ( Z u T Z v ) , L u v − = ∑ i = 1 n ( E u i ∼ P ( u ) l o g ( 1 − σ ( Z u T Z u i ) ) + E v i ∼ P ( v ) l o g ( 1 − σ ( Z v T Z v i ) ) ,
where σ is the sigmoid activation function, P ( u ) is the negative sample node distribution of u, and n is the negative sample number.

3.3.2. L2 Regularization Loss

In order to enhance the generalization ability of the model and alleviate the overfitting problem on complex datasets, L2 regularization loss is incorporated as a robust regularization technique.
L 2 = η ∑ i = 1 N ‖ θ i ‖ 2 2 ,
where η is the regularization coefficient, θ i denotes the weight of each parameter of the model, and ‖ · ‖ 2 is the norm of 2. L2 regularization improves the generalization ability of the model by imposing constraints on the weights to prevent the parameters from being too large.

3.3.3. Total Loss

The final loss function is the sum of the above three loss functions, assigning weight coefficients to positive and negative losses to balance the importance of positive and negative samples.
L t o t a l = ∑ u v ∈ E [ λ · L u v + + ( 1 − λ ) · L u v − ] + L 2 .
Through the above design, we can effectively control the complexity of the model while balancing the importance of positive and negative samples and improving the performance of the model on the test set.

4. Experiment

4.1. Datasets

The experimental evaluation uses three e-commerce datasets, Amazon [46], Alibaba-s [44], and Alibaba [44], and two generalization validation datasets, DTI [47] and Douban [48]. Among them, Amazon and Alibaba datasets are representative of typical e-commerce platforms, encompassing diverse product categories and user behaviors, providing a strong foundation for evaluating the model. The inclusion of DTI and Douban further tests the model’s ability to generalize to other domains, ensuring robustness across different data characteristics. The details are given in Table 1.
  • Amazon: This dataset comes from the e-commerce platform Amazon and is widely used in recommender system research. Nodes represent users or products, and edges represent users’ behaviors;
  • Alibaba: This dataset is a large-scale dataset extracted from the Alibaba e-commerce platform for recommendation systems and advertising research. The dataset contains a large amount of user behavior data. Nodes represent users or products, and edges represent various user behaviors such as clicks and inquiries;
  • Alibaba-s: This dataset is a small-scale version of Alibaba, mainly used to quickly verify the performance of the model in large-scale recommendation tasks;
  • DTI: This dataset is used to study the interaction between drugs and targets. The nodes in the dataset represent drugs or targets, and the edges represent the interactions between them;
  • Douban: This dataset comes from Douban, which is a social network dataset where nodes represent users and edges represent social relationships between users, such as friends or common interest groups.
The datasets used in this study naturally form heterogeneous graphs where nodes represent users and products, and edges represent interactions such as clicks or enquiries. These graphs inherently contain rich information in their structure, such as the relationships between different product categories, user behaviors, and product preferences. By leveraging the internal characteristics of these graphs, such as the connections between users and products, as well as the features of the nodes themselves, the model can uncover patterns and associations that are crucial for making accurate predictions.

4.2. Experimental Configurations

To evaluate the performance of our LHGCN, we compare our model with 11 state-of-the-art approaches, including one cluster method (Louvain [1]), one Graph embedding-based method (Node2vec [49]), three Homogeneous graph neural network-based methods (GCN [8], GAT [10], GraphSAGE [30]), four Heterogeneous graph neural network-based methods (FiLM [50], HGT [43], HAN [24], PSHGCN [51]), and two Multiple heterogeneous graph networks-based methods (DualHGCN [44], BPHGNN [45]).
Two commonly used statistical metrics are selected to quantify the evaluation, including AUROC (the area under the ROC curve) and AUPRC (the area under the PR curve). AUROC measures the ability of a model to distinguish between positive and negative classes. It evaluates the true positive rate versus the false positive rate at different thresholds. AUPRC evaluates the trade-off between precision and recall. Both AUROC and AUPRC are designed to quantify model performance, with higher values indicating better performance. For both metrics, values closer to 1.0 are preferred and represent ideal classification ability.

4.3. Implementation Details

To ensure a fair comparison, we use the same optimizer and training strategy in all models. We uniformly sample 50% of the edges from the dataset as the training set, assign 25% as the validation set, and reserve 25% as the test set. The model consists of three layers with a hidden layer dimension of 32. The number of negatively sampled edges, denoted as n, is set to 2. The learning rate is set to 0.01, dropout is applied at a rate of 0.5, and the total number of iterations is 4000. To explore the impact of hyperparameter optimization, we conducted additional experiments by tuning key parameters. For example, the learning rate was varied in the range [0.001, 0.005, 0.01, 0.05], and the dropout rate was adjusted within [0.3, 0.5, 0.7]. These optimizations were performed using a grid search strategy. Results showed that a learning rate of 0.01 and a dropout rate of 0.5 provided the best trade-off between model performance and training stability. Furthermore, the dimension of the hidden layer was also tested with values in [16, 32, 64], demonstrating that 32 achieved optimal predictive capacity.

4.4. Quantitative Comparative Experiments for Link Prediction

The performance of the LHGCN model on all test datasets is competitive with state-of-the-art methods in terms of node features initialized using adjacency matrices, and the experimental results are shown in Table 2. The best results are highlighted in bold, and the second-best are underlined. For the Amazon dataset, LHGCN improves the AUROC and AUPRC metrics by 2.16% and 2.38%, respectively, over the suboptimal method. On the Alibaba-s dataset, BPHGNN improves the AUROC by more than 6% compared to DualHGCN while maintaining the same AUROC as BPHGNN, and the AUPRC value of LHGCN increases by 5.3%. For the Alibaba dataset, LHGCN achieves 0.9% and 6.3% higher AUROC and AUPRC than the suboptimal method, respectively. On the DTI dataset, LHGCN’s AUROC surpasses all methods and is 1.02% higher than the suboptimal method, while AUPRC is only 0.07% lower than DualHGCN. For the Douban dataset, the LHGCN model is relatively poor at AUROC but improves 3.63% on AUPRC compared with the suboptimal method.
In addition, since the nodes in the Amazon and Alibaba datasets contain attribute information, the attributes can be used to generate the initial features of the nodes and their link prediction results are shown in Table 3. Our model improves AUROC and AUPRC indicators by 3.9% and 2.7%, respectively, on the Amazon dataset, while on the Alibaba dataset, both indicators improve by 3.4% and 4.2%.

4.5. Ablation Experiment

4.5.1. Effects of the Module Design

To evaluate the effectiveness of each module in the model, three variants were created: -laminate excludes the laminate generation module, -gate excludes the adaptive convolution module, and base omits both the laminate generation and adaptive convolution modules. The link prediction results are shown in Figure 7.
After removing the laminate generation module, the link prediction accuracy of the model on the Amazon, Alibaba-s, Alibaba, DTI, and Douban datasets decreased by 1.51%, 1.74%, 0.58%, 1.26%, and 2.26%, respectively. Experimental results show that the laminate generation module establishes an independent representation domain for each semantic space of the node, enabling the model to capture fine-grained heterogeneous information more effectively. The use of laminates can better retain the feature of the original nodes in different semantic spaces while minimizing interference between different semantics, allowing the model to more accurately extract key information across multiple semantic contexts, thereby significantly improving the overall performance.
The adaptive convolution module can dynamically adjust the node convolution receptive field according to the context information, which helps to accurately obtain the local features of the node. Removing the adaptive convolution module causes the model’s link prediction accuracy to drop by 1.88%, 0.66%, 1.73%, 2.12%, and 0.65% on the Amazon, Alibaba-s, Alibaba, DTI, and Douban datasets, respectively. When the adaptive module is removed from the model, all nodes will be restricted to the unified convolution receptive field and thus cannot adapt to changes in the neighborhood of the nodes, which leads to feature aliasing, and different semantic nodes are overly confined to the shared feature space, resulting in information interference. Therefore, compared with fixed convolution kernels, adaptive convolution can better adapt to the heterogeneity of nodes.
Removing the laminate generation module and the adaptive convolution module resulted in an average drop in link prediction accuracy of 4.6% across all datasets, with the Alibaba dataset experiencing the most significant drop of 7.3%. The collaboration of the two modules enables the model to simultaneously optimize the global and local feature extraction of nodes and effectively manage heterogeneous information in complex graph structures.

4.5.2. Analyses of the Attention Strategy

The attention mechanism is used in the adaptive convolution module, and three attention strategies are compared: static single-head attention (abbreviated as static-one), static multi-head attention (abbreviated as static-multi), and dynamic attention. Different attention strategies have little effect on the link prediction results, as shown in Table 4. This subtle difference in performance suggests that static multi-head attention is sufficient for the model to capture complex relationships, while dynamic adjustment provides limited additional benefits.

4.5.3. Analyses of the Aggregation Across Edges

After capturing neighborhood messages by node type, cross-edge aggregation is performed, and three aggregation operator implementations are considered: aggregation according to the proportion of each edge type, semantic level attention implementation in HAN, and average aggregation. The experimental results are shown in Table 5.
The average aggregation method effectively alleviates the impact of uneven data distribution between edge types by averaging the neighbor node messages corresponding to each edge type. Therefore, the average aggregation method shows relatively stable performance on all datasets. However, the proportional aggregation method drops by 0.84% on each dataset, and the semantic aggregation method performs slightly better but still drops by 0.57% compared with the average aggregation method.

4.5.4. Analyses of the Gating Mechanism

The gating mechanism enables the model to capture long-term dependencies while effectively filtering out irrelevant information when processing sequential data. Our goal is to experimentally determine a gating mechanism suitable for the link prediction task in graph neural networks. The experimental results are shown in Figure 8.
Although LSTM has the ability to manage more complex dependencies in theory, its experimental performance in this paper is not as good as GRU. Because the limited information propagation paths in graph neural networks weaken the long-term dependency advantage of LSTM. Across all datasets, the link prediction accuracy using LSTM dropped by 5.9% while the number of parameters increased by 25%. In the link prediction task of graph neural networks, GRU has been shown to be better at aggregating neighborhood information of nodes, thus effectively balancing computational complexity and model performance.

4.5.5. Analyses of the Activation Function

To verify the effectiveness of removing the activation function in the laminate fusion module, we use two variants of the model: (1) active, combining convolutional layers with nonlinear activation functions, and (2) non-active, removing the activation function between convolutional layers.
Both models are trained on the same dataset while keeping the same optimization algorithm and hyperparameters. According to the experimental results in Table 6, it can be seen that the AUROC and AUPRC of the model without activation function on all datasets significantly surpass the model with activation function. This means that the non-active model can not only improve the link prediction accuracy but also reduce the computational complexity.

4.6. Multiplex Effectiveness Analysis

To further verify the performance of the proposed model on multiplex graphs, we conducted a study on the effectiveness of multiple graphs (see Figure 9).
Specifically, we input the first, second, and third types of edges into the model sequentially and verify their effects on the link prediction accuracy. Experimental results show that the link prediction performance of the model does not reach the optimal level when a single type of edge is used as input. In the Alibaba dataset, when node attributes are used as node features, the prediction results are between 70% and 86% when only one type of edge is considered. However, when three types of edges are input into the model at the same time, its prediction accuracy exceeds 90%. Therefore, relying on a single edge type may impose certain limitations on effectively capturing the structural characteristics of the graph. In contrast, combining edges across types can yield richer and complementary structural information, thereby significantly improving the link prediction ability of the model.
This experimental result demonstrates the adaptability and robustness of our model in dealing with multiplex graphs. By integrating different edge types, it can better illustrate the complex relationships between nodes and ultimately improve link prediction accuracy.

4.7. Parameter Sensitivity Analysis

4.7.1. Effect of the Number of Convolution Layers

In order to analyze the impact of different numbers of model layers on the link prediction accuracy, the number of convolutional layers of the model is set to 1, 2, 3, 4, and 5, respectively. The experimental results are shown in Figure 10a.
Taking the Alibaba dataset as an example, we observed that as the number of convolutional layers increased from 1 to 3, the prediction accuracy of the model gradually improved, reaching 85%, 90.47%, and 91.36%, respectively. In a single-layer convolution scenario, each node only uses its first-order neighborhood information, which limits the understanding of local structure. As the number of convolutional layers increases, the model can gradually aggregate information from a larger neighborhood, thereby capturing more complex graph structures and relationships between nodes. However, as the number of layers increases to 4 and 5, the accuracy drops to 88% and 89%, respectively. It is worth noting that the performance of the model with 5 layers is even worse than that with 2 layers, which can be attributed to factors such as overfitting caused by the increase in the number of layers and the amplification of noise during the propagation process. Therefore, the model uses 3 convolutional layers in all experiments.

4.7.2. Effect of the Weight Coefficient of the Loss Function

For the hyperparameter experiment of the loss function, different values ranging from 0.1 to 0.9 were selected, and the experimental results are shown in Figure 10b.
When λ is less than 0.5, the model tends to overestimate the impact of negative samples, thereby over-penalizing irrelevant relationships. When λ is greater than 0.5, the model has a greater risk of overfitting the positive samples. When λ is equal to 0.5, the model has the highest prediction accuracy. This balanced configuration allows the model to fairly learn the characteristics of positive and negative samples during training, thereby optimizing the performance of the link prediction task.

5. Conclusions

This paper aimed to address the challenge of link prediction in multiplex graphs, as the relationships between various graph structures are intricate. To effectively capture multiple semantic relationships, we proposed a novel laminated heterogeneous graph convolutional network (LHGCN). We first designed a laminate generation module that built laminates based on different node types. Subsequently, an adaptive convolution module was used to dynamically modify the receptive field of the node so that the convolution operator can adapt to the local changes of each node. Finally, a laminate fusion module was designed to ensure the effective integration of information across various laminates. Extensive experiments on various benchmarks demonstrated the superiority of our method.
Although this paper has made significant progress, there are still some limitations that need to be addressed. First, it remains unclear how outliers in the data are considered, and further studies should explore the distributions that present “new” trends which cannot be traced back to a structured benchmark. In addition, the efficiency of the model in processing extremely large graphs needs to be further optimized. Future research could also extend the present study using the concept of m-bipartite Ramsey numbers [52], which may provide a novel theoretical framework for analyzing and optimizing complex graph structures.

Author Contributions

Conceptualization, K.L.; methodology, M.K.; software, M.K.; validation, W.D., K.L. and X.L.; formal analysis, K.L.; investigation, X.L.; resources, K.L.; data curation, M.K.; writing—original draft preparation, W.D.; writing—review and editing, K.L.; visualization, X.L.; supervision, K.L.; project administration, K.L.; funding acquisition, K.L. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Fundamental Research Funds for the Central Universities No. 2024ZKPYZN01, and in part by the Key Research and Development Program of Shanxi Province under Grant 202102090301006.

Data Availability Statement

The data presented in this study are available at https://github.com/xuehansheng/DualHGCN and https://github.com/7thsword/MFPR-Datasets, reference number [46,47,48]. These data were derived from the following resources available in the public domain: http://jmcauley.ucsd.edu/data/amazon, https://www.alibaba.com, https://drugtargetcommons.fimm.fi, http://book.douban.com, and http://www.dianping.com.

Acknowledgments

We would like to express our heartfelt gratitude to the China University of Mining and Technology, Beijing, for providing the facilities and equipment essential for the completion of this research.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

ACMAdaptive Convolution Module
AUPRCArea Under the Precision-Recall Curve
AUROCArea Under the Receiver Operating Characteristic Curve
GATGraph Attention Networks
GCNGraph Convolutional Network
GNNsGraph Neural Networks
GRUGated Recurrent Units
LFMLaminate Fusion Module
LGMLaminate Generation Module
LHGCNLaminated Heterogeneous Graph Convolutional Network
LSTMLong Short-Term Memory

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Figure 1. Alibaba dataset structure pattern (left) and feature distribution (right).
Figure 1. Alibaba dataset structure pattern (left) and feature distribution (right).
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Figure 2. The overall architecture of LHGCN.
Figure 2. The overall architecture of LHGCN.
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Figure 3. Laminate generation module.
Figure 3. Laminate generation module.
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Figure 4. Adaptive convolution module (gate outputs z and r and controls 2 types of laminates enjoying different receptive fields); the detailed implementation of attention and gate is shown in Figure 5.
Figure 4. Adaptive convolution module (gate outputs z and r and controls 2 types of laminates enjoying different receptive fields); the detailed implementation of attention and gate is shown in Figure 5.
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Figure 5. Information flow in convolutional layers, exemplified by laminate U.
Figure 5. Information flow in convolutional layers, exemplified by laminate U.
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Figure 6. Laminate fusion module (the AC Layer is the adaptive convolution layer).
Figure 6. Laminate fusion module (the AC Layer is the adaptive convolution layer).
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Figure 7. Effects of the module design: full indicates the full model, -laminate excludes the LGM, -gate excludes the ACM, and base omits both modules mentioned above.
Figure 7. Effects of the module design: full indicates the full model, -laminate excludes the LGM, -gate excludes the ACM, and base omits both modules mentioned above.
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Figure 8. Comparison of experimental results between GRU and LSTM.
Figure 8. Comparison of experimental results between GRU and LSTM.
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Figure 9. Multiplex effectiveness analysis.
Figure 9. Multiplex effectiveness analysis.
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Figure 10. Parameter sensitivity analysis.
Figure 10. Parameter sensitivity analysis.
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Table 1. Statistics of three e-commerce datasets and two generalization validation datasets.
Table 1. Statistics of three e-commerce datasets and two generalization validation datasets.
Datasetse-Commerce DatasetsGeneralization
AmazonAlibaba-sAlibabaDTIDouban
Nodes953015,21822,649483734,890
Eedges60,65827,03645,73416,458307,088
Edges types23356
Features4N/A18N/AN/A
Table 2. Experimental results (%) on datasets for the link predication task in terms of node features initialized using adjacency matrices.
Table 2. Experimental results (%) on datasets for the link predication task in terms of node features initialized using adjacency matrices.
MethodsAmazonAlibaba-sAlibabaDTIDouban
AUROCAUPRCAUROCAUPRCAUROCAUPRCAUROCAUPRCAUROCAUPRC
Louvain87.0584.7933.9941.4132.0542.4953.8693.3149.6283.28
Node2vec50.3055.4450.4351.4250.1051.5250.8857.4550.2760.61
GCN64.9377.4563.0879.5956.8777.6656.9576.0085.4492.66
GAT66.7070.1653.2854.2955.3854.4976.3380.6467.1387.29
GraphSAGE69.9969.3964.9165.7666.4960.3679.3482.3678.1592.40
FiLM88.8092.4190.9887.1085.9981.4692.7493.5982.6290.33
HGT93.7595.5990.6883.4783.5971.7593.1692.2371.0790.09
HAN71.0781.5667.0852.8467.2955.8481.2584.9463.1986.41
PSHGCN87.1092.2290.8482.7185.3982.7190.2692.7576.3892.78
DualHGCN86.6988.6987.5789.0285.5487.5193.8595.0084.8685.17
BPHGNN93.7993.6793.6489.4190.5386.5093.9193.5389.2090.17
ours95.8295.9093.3494.1691.3693.0694.8794.9387.0396.15
The best results are highlighted in bold, and the second-best are underlined.
Table 3. Experimental results (%) on the Amazon and Alibaba datasets for the link predication task in terms of node features initialized using attributes.
Table 3. Experimental results (%) on the Amazon and Alibaba datasets for the link predication task in terms of node features initialized using attributes.
MethodsAmazonAlibaba
AUROCAUPRCAUROCAUPRC
Louvain87.0584.7932.0542.49
Node2vec50.3055.4450.1051.52
GCN56.2658.1971.3869.16
GAT62.4467.1159.1259.73
GraphSAGE85.7078.8684.4181.40
FiLM63.8173.6985.0883.36
HGT77.8984.9786.7286.00
HAN69.4572.2250.5152.09
PSHGCN57.1774.2577.2575.68
DualHGCN86.5588.5586.7688.59
BPHGNN81.8582.2085.0887.55
ours90.4691.0289.7592.33
The best results are highlighted in bold, and the second-best are underlined.
Table 4. Experimental results of different attention strategies.
Table 4. Experimental results of different attention strategies.
MethodsAmazonAlibaba-sAlibabaDTIDouban
AUROCAUPRCAUROCAUPRCAUROCAUPRCAUROCAUPRCAUROCAUPRC
static-one96.1396.0193.2394.1990.9892.5793.8693.6585.0095.42
static-multi95.6095.4692.9593.8891.1092.7994.5294.6986.3895.95
dynamic95.8295.9093.3494.1691.3693.0694.8794.9387.0396.15
The best results are highlighted in bold.
Table 5. Experimental results of different aggregation methods across edges.
Table 5. Experimental results of different aggregation methods across edges.
MethodsAmazonAlibaba-sAlibabaDTIDouban
AUROCAUPRCAUROCAUPRCAUROCAUPRCAUROCAUPRCAUROCAUPRC
proportion95.6895.7192.0993.3390.2092.4693.6393.0886.5395.99
semantic95.8195.7692.7293.7690.2391.8493.9394.0086.8296.09
average95.8295.9093.3494.1691.3693.0694.8794.9387.0396.15
The best results are highlighted in bold.
Table 6. Experimental results of active and non-active models.
Table 6. Experimental results of active and non-active models.
MethodsAmazonAlibaba-sAlibabaDTIDouban
AUROCAUPRCAUROCAUPRCAUROCAUPRCAUROCAUPRCAUROCAUPRC
active87.9589.2984.6185.9284.4982.6988.9591.6583.3895.13
non-active95.8295.9093.3494.1691.3693.0694.8794.9387.0396.15
The best results are highlighted in bold.
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Liu, K.; Kang, M.; Li, X.; Dai, W. LHGCN: A Laminated Heterogeneous Graph Convolutional Network for Modeling User–Item Interaction in E-Commerce. Symmetry 2024, 16, 1695. https://doi.org/10.3390/sym16121695

AMA Style

Liu K, Kang M, Li X, Dai W. LHGCN: A Laminated Heterogeneous Graph Convolutional Network for Modeling User–Item Interaction in E-Commerce. Symmetry. 2024; 16(12):1695. https://doi.org/10.3390/sym16121695

Chicago/Turabian Style

Liu, Kang, Mengtao Kang, Xinyu Li, and Wenqing Dai. 2024. "LHGCN: A Laminated Heterogeneous Graph Convolutional Network for Modeling User–Item Interaction in E-Commerce" Symmetry 16, no. 12: 1695. https://doi.org/10.3390/sym16121695

APA Style

Liu, K., Kang, M., Li, X., & Dai, W. (2024). LHGCN: A Laminated Heterogeneous Graph Convolutional Network for Modeling User–Item Interaction in E-Commerce. Symmetry, 16(12), 1695. https://doi.org/10.3390/sym16121695

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