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Article

Dynamic Link Prediction in Jujube Sales Market: Innovative Application of Heterogeneous Graph Neural Networks

by
Yichang Wu
†,
Liang Heng
*,†,
Fei Tan
,
Jingwen Yang
and
Li Guo
College of Information Science and Technology, Shihezi University, Shihezi 832003, China
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Appl. Sci. 2024, 14(20), 9333; https://doi.org/10.3390/app14209333
Submission received: 9 September 2024 / Revised: 7 October 2024 / Accepted: 11 October 2024 / Published: 13 October 2024
Browse Figures
Figure 1
<p>An example of a heterogeneous graph for Xinjiang jujube sales. (<b>a</b>) Three types of nodes: producers, distributors, and retailers. (<b>b</b>) A heterogeneous graph representing Xinjiang jujube sales with three node types and two types of connections.</p> "> Figure 2
<p>Illustration of the node labeling algorithm for a graph. In the depicted graph, the source node is denoted as 0, and the target node as 1. The algorithm, showcased in each iteration, involves Step 1, which calculates a distinctive string for each node by recording the indices of the nodes and their respective neighbors. Following this, Step 2 orchestrates the re-labeling of nodes in adherence to the devised node labeling algorithm.</p> "> Figure 3
<p>The HMAGNN framework for link prediction. The process involves node labeling, two Multi-Layer Perceptrons (MLPs), and a multi-head mechanism for node feature generation. The structural vector <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="bold">R</mi> </mrow> <mrow> <mi>N</mi> <mo>×</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> is transformed into the structural matrix <math display="inline"><semantics> <mrow> <msup> <mrow> <mi>X</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msup> <mo>∈</mo> <msup> <mrow> <mi mathvariant="bold">R</mi> </mrow> <mrow> <mi>N</mi> <mo>×</mo> <mi>N</mi> </mrow> </msup> </mrow> </semantics></math>, and similarity scores are computed and adaptively combined using the parameter α. (<b>a</b>) HMAGNN first transforms the heterogeneous graph into the same feature space through a transformation matrix, and then labels the nodes in the graph; (<b>b</b>) HMAGNN learns structural features from the adjacency matrix and considers a multi-head mechanism to generate structural feature vectors; (<b>c</b>) Diagonalizes the structural feature vectors to construct a diagonal matrix; (<b>d</b>) Computes the loss based on the two node representations Z and h obtained from HMAGNN and GNN, respectively.</p> "> Figure 4
<p>Comparison of model results on five datasets. (<b>a</b>) OGB-PPA dataset; (<b>b</b>) OGB-DDI dataset; (<b>c</b>) OGB-Collab dataset; (<b>d</b>) OGB-Citation2 dataset; (<b>e</b>) Jujube dataset.</p> "> Figure 5
<p>Impact of node label size on link prediction. (<b>a</b>) OGB-PPA dataset; (<b>b</b>) OGB-DDI dataset; (<b>c</b>) OGB-Collab dataset; (<b>d</b>) OGB-Citation2 dataset.</p> "> Figure 6
<p>Influence of multi-head attention mechanism on link prediction. (<b>a</b>) OGB-PPA dataset; (<b>b</b>) OGB-DDI dataset; (<b>c</b>) OGB-Collab dataset; (<b>d</b>) OGB-Citation2 dataset.</p> "> Figure 6 Cont.
<p>Influence of multi-head attention mechanism on link prediction. (<b>a</b>) OGB-PPA dataset; (<b>b</b>) OGB-DDI dataset; (<b>c</b>) OGB-Collab dataset; (<b>d</b>) OGB-Citation2 dataset.</p> ">
Review Reports Versions Notes

Abstract

:
Link prediction is crucial in forecasting potential distribution channels within the dynamic and heterogeneous Xinjiang jujube sales market. This study utilizes knowledge graphs to represent entities and constructs a complex network model for market analysis. Graph neural networks (GNNs) have shown excellent performance in handling graph-structured data, but they do not necessarily significantly outperform in link prediction tasks due to an overreliance on node features and a neglect of structural information. Additionally, the Xinjiang jujube dataset exhibits unique complexity, including multiple types, attributes, and relationships, distinguishing it from typical GNN datasets such as DBLP and protein-protein interaction datasets. To address these challenges, we introduce the Heterogeneous Multi-Head Attention Graph Neural Network model (HMAGNN). Our methodology involves mapping isomeric nodes to common feature space and labeling nodes using an enhanced Weisfeiler–Lehman (WL) algorithm. We then leverage HMAGNN to learn both structural and attribute features individually. Throughout our experimentation, we identify the critical influence of local subgraph structure and size on link prediction outcomes. In response, we introduce virtual nodes during the subgraph extraction process and conduct validation experiments to underscore the significance of these factors. Compared to alternative models, HMAGNN excels in capturing structural features through our labeling approach and dynamically adapts to identify the most pertinent link information using a multi-head attention mechanism. Extensive experiments on benchmark datasets consistently demonstrate that HMAGNN outperforms existing models, establishing it as a state-of-the-art solution for link prediction in the context of jujube sales market analysis.

1. Introduction

The Xinjiang dried fruit industry, with jujube as a standout product known for its exceptional quality, is a cornerstone of economic development in Xinjiang, China. These jujubes, valued not only in domestic markets but also internationally, are acclaimed for their rich nutritional value and superior taste, significantly contributing to China’s agricultural exports. This sector plays a pivotal role in fostering local economic growth and enhancing farmers’ livelihoods. However, as the Xinjiang jujube industry expands and attracts more participants, it faces several challenges. Chief among these is a reliance on a single sales channel and the presence of information asymmetry in the supply and distribution markets. This limited market access impedes farmers’ ability to quickly adapt to changing market dynamics, leading to increased costs due to intermediaries in the sales process. Furthermore, the issue of information asymmetry compounds these challenges within the supply and distribution chain. Jujube farmers often do not have access to timely and accurate market information, which hampers their ability to understand demand fluctuations and predict price trends effectively. Consequently, their decision-making is compromised, affecting their profitability and livelihoods.
To address these challenges, we propose a machine learning model specifically designed for the Xinjiang jujube market, incorporating a novel link prediction algorithm. Link prediction aims to leverage existing market data to anticipate and unveil valuable potential distribution chains. The data within the jujube sales market often manifest as a graph structure characterized by multiple attributes and objects, with multidimensional and dynamic links existing among market entities. Conventional link prediction methods are highly reliant on pre-defined feature vectors and algorithmic models, potentially struggling to discern underlying patterns and relationships within complex jujube sales markets. Consequently, we advocate for integrating Graph Neural Networks (GNNs) technology to address the limitations of traditional link prediction methods within the jujube sales market [1].
Graph-structured data are prevalent across various domains, including social networks [2,3], biology [4,5,6], and recommendation systems [7,8]. GNNs possess the adaptive capability to learn and represent intricate graph structure information, exhibiting exceptional performance in handling graph-structured data. A myriad of GNN variants has been proposed, achieving state-of-the-art performance across tasks such as link prediction [9,10,11], graph classification [12,13,14], node classification [15,16,17], and recommendation systems [18,19]. For instance, Xia et al. [20] introduced an unexpected interest recommendation system based on GNNs to offer users a novel and diverse experience. Yu et al. [21] proposed the PiGCN, which splits a given cascade into a sequence of sub-cascade graphs and then uses a GCN to learn the local structure of each sub-cascade for predicting information propagation in complex networks.
Although GNNs can adaptively learn the most relevant features in the data for link prediction through an end-to-end learning process, surpassing traditional heuristic methods [22,23,24] and embedding feature methods [25,26], these methods often rely excessively on node features while neglecting structural information. SEAL [27] incorporates structural information into link prediction by considering relative distances between target nodes and their neighborhoods. While promising, methods such as SEAL incur computational costs for subgraph extraction, and others like GraiL [28] necessitate prior knowledge or heuristic rules for subgraph construction and inference. Thus, we propose the Heterogeneous Multi-Head Attention Graph Neural Network (HMAGNN) to comprehensively capture both structural and attribute information through node labels and a multi-head attention mechanism.
In summary, our contributions in this study are substantial and can be outlined as follows:
  • We propose a groundbreaking GNN-based model called HMAGNN. This model is designed to seamlessly integrate node attribution and graph structural information. By embedding the unique structural features and utilizing a multi-head mechanism to assign diverse weights to sales characteristics, it achieves dynamic link prediction. HMAGNN represents a novel advancement in leveraging both node features and structural information for enhanced link prediction within the jujube sales market.
  • Taking into account the crucial structural information in links, we introduce an enhanced node labeling method, building upon the classical Weisfeiler–Lehman (WL) algorithm. By refining the node labeling process, we significantly improve the efficiency of capturing the influence of predicted nodes on their surrounding neighbor nodes. This advancement is crucial in accurately characterizing the complex relationships within the Xinjiang jujube sales market.
  • Our study includes extensive experiments conducted on diverse and complex datasets, showcasing the robustness and versatility of our proposed approach. Furthermore, we undertake practical validation specifically on the Xinjiang jujube dataset. This empirical validation not only solidifies the theoretical foundation of our approach but also demonstrates its effectiveness in addressing the unique challenges posed by the Xinjiang jujube sales market.

2. Materials and Methods

2.1. Related Works

In constructing our study, we harnessed a comprehensive dataset tailored to the Xinjiang jujube sales market. This dataset, meticulously compiled and organized, comprises 2118 nodes and 43,417 edges. The primary sources of our jujube market data stem from the production statistics provided by the Xinjiang Production and Construction Corps, supplemented by data from offline supermarket sales surveys and e-commerce sales. The Xinjiang jujube sales dataset encapsulates a rich array of attributes, contributing to the depth and granularity of our analysis. These attributes encompass various facets, including the types, origins, grades, and prices of jujubes sold by producers. Additionally, the dataset incorporates crucial details such as the geographical locations of distributors and retailers, purchase prices, and the total sales volumes of the jujubes. The intricate interplay of these attributes forms the foundation of our research, enabling a nuanced exploration of the Xinjiang jujube sales market. Moreover, within this dataset, the edges indicate sales relationship links, establishing connections among the three distinct types of sales objects. This interconnected network of sales relationships is critical in unraveling the dynamics and complexities inherent in the Xinjiang jujube sales market. By leveraging this meticulously curated dataset, our study aims to contribute novel insights and advancements to the burgeoning field of dynamic link prediction within the agricultural sector.
Graph neural networks represent a powerful class of machine learning algorithms that process graph-structured data [29,30]. Initially proposed by Scarelil et al. [31], GNNs are designed to capture and understand the complex structural nuances inherent in graphs by encoding embeddings for both nodes and edges. This methodology enables the creation of compact vector representations for individual nodes, thereby facilitating more efficient analysis and comprehension of the underlying graph topology. The cornerstone of a GNN is the message passing module, a critical component facilitating information exchange within the graph. In this module, each node actively aggregates information from its neighboring nodes. The aggregated information is then concatenated with the node’s feature vector, comprehensively representing the node’s local context. This amalgamated information is subsequently transmitted to the next layer of the network. Complementary to the message passing module, the feature update module is crucial in refining and enhancing the accuracy of node features. The aggregated information, now enriched with insights from neighboring nodes, is employed to update the node’s feature vector. This iterative process ensures that the node representations evolve to capture nuanced structural patterns within the graph, making the overall node features more accurate and informative.
GNNs, with their ability to seamlessly incorporate both node-level and structural information, have demonstrated remarkable success across various domains, including social networks, biology, and recommendation systems. Their adaptability and effectiveness make them a compelling choice for tasks such as link prediction, graph classification, and node classification. The dynamic interplay between the message passing and feature update modules empowers GNNs to learn and represent intricate relationships within graph-structured data, positioning them as a versatile tool in the realm of machine learning.
Heterogeneous Graph Neural Networks (HGNN) represent a specialized approach tailored to heterogeneous graph representations’ complexities. In the heterogeneous network of the Xinjiang jujube market, a plethora of diverse object relationships exist, with the attribute features of different objects playing distinct roles in the link prediction process. This diversity enables nodes and edges to provide a wide range of attributes and semantic information, enhancing the depth and complexity of the graph model. In contrast to traditional GNNs, heterogeneous graph neural networks take into account the relationships between various types of nodes and edges alongside node features and adjacency information. This comprehensive approach enables the optimal representation of the intricate and multifaceted information present within the network. Heterogeneous graph neural networks are applied in various agricultural areas, including crop yield prediction [32], pest and disease identification [33], and price forecasting [4], etc. Beyond agriculture, heterogeneous graphs have found extensive applications in other domains, highlighting their versatility and effectiveness in capturing intricate relationships. Notable applications include recommendation systems [34], knowledge graphs [35,36], bioinformatics analysis [37], and social network analysis [38,39].
Link prediction is a pivotal task in network analysis, involving the determination of whether a connection exists between two target nodes based on the structural and feature information of those nodes [40]. This process is crucial in forecasting potential relationships within a network, such as predicting links within the Xinjiang jujube sales market. Such predictions are invaluable in expanding sales channels and refining sales strategies. In the context of agricultural trade, links represent existing trade relationships, with new links emerging and others being terminated annually. It is worth noting that different links hold varying trade values. Consequently, the pursuit of identifying optimal sales channels that correspond to diverse trade demands emerges as an ongoing area of research.
Notable heuristic methods such as Katz [41], PageRank [42], and SimRank [43] have shown effectiveness in link prediction. However, these methods face limitations, especially in their reliance on manually set link similarity scores, which makes them less adaptable to dynamic and complex networks. Overcoming the manual setting constraints of heuristic methods, graph embedding-based methods [44] such as DeepWalk [25] and node2vec [26] leverage random walks to generate node embeddings using Skip-Gram [45]. While successful, these methods may face challenges in link prediction due to their focus primarily on local structures, potentially neglecting broader, global patterns. GNNs [46], viewed as extensions of DeepWalk and node2vec, bring a transformative approach to link prediction. They deploy multi-layer convolutional neural networks to learn node embeddings, capturing intricate structural information within the graph. Unlike graph embedding-based methods, GNNs excel in representing both local and global structures, addressing overfitting concerns. Moreover, GNNs perform link prediction without explicitly computing similarity matrices between node pairs, making them well-suited for large-scale graph applications.

2.2. Preliminaries

This section lays the groundwork for understanding the proposed model. Our model is designed to effectively capture both structural and attribute features of nodes within the dimensional space following a heterogeneous graph transformation. By leveraging this transformed representation, HMAGNN calculates scores for edge pairs (u, v) based on information derived from neighboring nodes, facilitating accurate link prediction in complex graph structures.
We define fundamental concepts pertinent to our model, providing the necessary background to comprehend its architecture and functionalities. Additionally, we offer a concise overview of previous computational methods utilized in link prediction, highlighting key approaches and their respective strengths and limitations.
Subsequently, we present the HMAGNN model, outlining its architecture, key components, and novel contributions. Through this comprehensive exposition, readers will gain insight into the motivations behind our proposed approach and its potential to address the challenges posed by dynamic and heterogeneous graph data, particularly in the context of the Xinjiang jujube sales market.

2.2.1. Heterogeneous Nodes Transformation

In the context of our study, a heterogeneous graph, denoted as G = (V, E), serves as a foundational representation, where V = { v 1 , v 2 , ⋯ , v k } denotes the set of node types and E = { e 1 , e 2 , ⋯ , e n } signifies the set of edge types. Within this heterogeneous graph, the union of nodes and edges (V + E > 2) reflects the rich diversity inherent in the graph structure.
Figure 1 provides a visual representation of a heterogeneous graph illustrating the sales process of Xinjiang jujubes. This graph encompasses various node types, including producers (P), distributors (D), and retail markets (R). The connections within the graph capture the intricate relationships between nodes, with producers supplying through distributors and directly to retailers.
This heterogeneous graph model serves as a pivotal framework for our analysis, allowing us to encapsulate the multifaceted nature of the Xinjiang jujube sales market and providing the foundation upon which our proposed HMAGNN operates.
Within a heterogeneous graph G, nodes of different types often possess varying feature dimensions. To standardize the processing of these diverse node features, a transformation matrix is applied to project them into a shared feature space. This transformation is mathematically expressed as:
h i ′ = W ϕ i ⋅ h i
Here, W ϕ i represents the mapping matrix, ϕ i denotes the mapping specific to type i , and h i ′ signifies the node feature following the projection mapping.
This transformation process ensures that node features across distinct types are brought into alignment, enabling consistent analysis and computation in subsequent stages. The mapping matrix W ϕ i encapsulates the learned relationships and dependencies required to effectively project features from type-specific spaces to a unified feature space.
By employing this transformation, the HMAGNN can seamlessly navigate the intricacies of the Xinjiang jujube sales market, where diverse node types play vital roles in the sales process. The uniform representation of features facilitates a more robust and comprehensive analysis, contributing to the efficacy of the proposed model in handling the complexity inherent in heterogeneous graph structures.
To deepen the understanding of the structural intricacies within a graph, HMAGNN initiates the extraction of a local subgraph tailored for the target link, serving as a pivotal component for subsequent training phases. This localized subgraph encapsulates pertinent neighbor information relative to the target link. Local subgraph extraction offers several advantages over global calculations that span the entire graph. First, it reduces computational complexity, making the modeling process more efficient. Second, by focusing on local environments, it enables a more nuanced understanding of node characteristics and attributes within specific contexts. This nuanced understanding, in turn, enhances the model’s ability to generalize to unseen data while concurrently reducing the risk of overfitting. Consequently, local subgraph extraction is pivotal in enhancing the model’s generalization performance. The size of this local subgraph is dynamically determined by a hyperparameter, denoted as K, which represents the number of nodes in the graph.
The process commences by including first-order neighbors pertaining to the target node pair (x, y) into the local subgraph. Subsequently, higher-order neighbors are sequentially incorporated. Should the total number of nodes within the local subgraph remain below the predefined K, virtual nodes are strategically introduced. Conversely, if the number of neighbor nodes surpasses the stipulated K, a preference strategy is applied: removing the highest-order neighbor nodes. This strategic decision is grounded in the rationale that nodes proximate to the target node pair inherently offer a superior representation of the graph’s topological structure, underscoring the significance of one-hop neighbors over their two-hop counterparts.

2.2.2. Node Labeling

Node labeling has emerged as a crucial enhancement strategy, significantly contributing to the refinement of graph topological structures in various experiments An effective node labeling methodology strives for two critical objectives: ensuring that similar nodes garner comparable rankings across diverse subgraphs and maintaining a structured labeling process. Specifically, the latter implies that if node i receives a lower label than node j during the node labeling procedure, this hierarchical relationship should persist in the final node labeling output.
While the classic WL algorithm adeptly addresses the former objective, ensuring similarity among nodes, it falls short in ensuring the orderly arrangement of labels. It fails to pinpoint the target node within the labeled local subgraph. In response, our devised node labeling algorithm introduces a meticulous approach. It assigns a label of 0 to the source node x in the link prediction target node pair (x, y) and designates the target node y with a label of 1. Subsequently, an initial string is generated by examining the connections of neighboring nodes, with the length of the string corresponding to the number of connected nodes. Then, nodes are relabeled in ascending order based on this initial string. This process is iterated multiple times until each node obtains a unique label, as illustrated in Figure 2.
This systematic labeling approach imparts distinct characteristics to the labeled local subgraph, notably assigning the label pair (0, 1) to the link prediction target node pair, thereby explicitly distinguishing them from ordinary nodes within the graph.

2.2.3. Model

We present the HMAGNN designed for link prediction, elucidating its effective utilization and learning of both structural and attribute information. The HMAGNN architecture comprises three integral components, illustrated in Figure 3.
In the node transformation and labeling phase, the heterogeneous graph undergoes a process to obtain the feature matrix X ∈ R N × F and the adjacency matrix A ∈ R N × N . We use the traditional feature-based GNN to compute the feature-based node representation H through the joint calculation of the adjacency matrix and the feature matrix.
H = G N N ( X , Ã ; W )
where X is the original feature matrix, Ã is the normalized adjacency matrix, and W is the parameter matrix. This operation utilizes a GNN to effectively capture and encode the structural information within the heterogeneous graph, providing a learned feature representation H that will be further utilized in subsequent stages of the model.
The multi-head attention mechanism is employed in the node feature generator to enhance the utilization of node information and generate refined feature vectors. This mechanism is introduced to improve predictions for sales links of Xinjiang jujube by considering various aspects of the data.
In this mechanism, different features are treated as distinct “heads”, each assigned its level of importance and correlations among different features. By employing the multi-head attention mechanism, attention can be directed simultaneously to these various features, resulting in a more comprehensive representation of the relationships between nodes. For instance, one head may prioritize sales volume, while another focuses on profit maximization. Each head learns a unique weight matrix to calculate attention for different features, enabling the model to capture multiple aspects of the data simultaneously.
The fused feature vector representation h i t for node i at layer t is computed as follows:
h i t = R e L U ∑ j ∈ N i   ∑ k = 1 K   α i , j , k t W k t h j t − 1
Here, ReLU (Rectified Linear Unit) is a commonly used activation function, h i t represents the feature representation of node i in the t -th layer, W k t denotes the weight matrix of the k -th head in the t -th layer, K is the number of heads, and α i , j , k t represents the k-th attention coefficient between node i and node i . The attention coefficient α i , j , k t is computed using a LeakyReLU activation function as follows:
α i , j , k t t = e x p ( L e a k y R e L U ( a k   T [ W h t h i t − 1 | | W h t h j t − 1 ] ) ) ∑ j ∈ N i   e x p L e a k y R e L U ( a k   T [ W h t h i t − 1 | | W h t h i t − 1 ] )
where LeakyReLU is an improvement on ReLU, which will give a very small non-zero output. When the input is negative, α k is the attention vector of the k -th head and ∥ represents the vector concatenation operation. This mechanism enables the model to focus on relevant features and relationships within the graph, enhancing its ability to extract meaningful representations for downstream tasks.
Generating edge and structural features is crucial in enhancing the model’s understanding of the graph’s topology. Both the edge feature generator and node feature generator are implemented as Multi-Layer Perceptrons (MLPs) to capture intricate patterns and relationships within the data.
The edge feature e i , j t for nodes i and j in layer t is computed using the following equation:
e i , j t = R e L U W e t h i t − 1 ∥ h j t − 1
Here, W e t represents the weight matrix associated with the edge feature generator, h i t − 1 and h j t − 1 denote the feature representations of nodes i and j in the previous layer ( t − 1 ), respectively, and ∥ represents the vector concatenation operation.
The structural feature vector x i for node i is then formed by combining its node feature representation h i t and the mean of edge feature features across its neighbors N i .
x i = h i t ∥ e i , j t j ∈ N i
Here, e i , j t represents the mean edge feature of neighboring nodes. This combination enables the model to leverage both node-centric and neighborhood-wide information when constructing the structural feature vector. The mean operation ensures a comprehensive representation that considers the collective influence of neighboring nodes on the target node.
Once the structural feature vector x is computed for each node, a new structural feature matrix X s ∈ R N × N is constructed to encapsulate the structural information of the heterogeneous graph. This matrix X s is defined as follows:
X s = X i , j = 0 , ( i ≠ j ) X i , j = x i , ( i = j )
In essence, each diagonal element X i , j of X s represents the structural feature vector x i of the corresponding node i , while all off-diagonal elements are set to zero. After obtaining the structural feature matrix X s , it is multiplied with the initial adjacency matrix A to compute the structural representation Z :
Z = A X s
This operation integrates the structural information derived from node features into the adjacency matrix, thereby enriching the representation of edges in the graph. The resulting structural representation Z captures the combined influence of node attributes and graph topology, facilitating more informed predictions in subsequent stages of the model.
To incorporate higher-order neighbor information and enhance the structural representation Z , we refine the computation outlined in Equation (8). The improved formulation is expressed as Z = f ( Φ , ∑ l = 1 L   { A l , X s l } ) . Here, f is a function, Φ is a parameter, A l denotes the l -th adjacency matrix, and X s l represents the structural feature matrix at the l -th hop. This adjustment aims to capture the influence of higher-order neighbors on the structural representation.
Specifically, we iteratively update Z by applying a MLP denoted as g :
Z = g Z 0 , Z 1 , … , Z l
In this equation, Z 0 represents the initial structural features, and Z l is computed as Z l = ∑ i = 0 l − 1   β i A i X s i Z 0 . Here, β is a hyperparameter controlling the weight ratio between nearby and distant neighbors. This formulation adapts the structural representation to incorporate information from different hops, enabling the model to capture nuanced relationships and dependencies within the graph.
We calculate the similarity scores for a given target link pair ( i , j ) based on the computed values h i , h j , z i , and z j . By adjusting the weights of structure and attributes using parameter α , the y ^ i j is determined as follows:
y ^ i j = α ⋅ σ z i T z j + 1 − α ⋅ σ s h a i , h a j
This equation combines the structural similarity z i T z j and attribute-based similarity s h a i , h a j with weights controlled by α .
The model is jointly trained using a composite loss function consisting of three standard binary cross-entropy terms:
L = ∑ i , j ∈ D   λ 1 B C E y ^ i j , y i j + λ 2 B C E σ z i T z j , y i j + λ 3 B C E σ s h a i , h a j , y i j
Here, BCE represents the binary cross-entropy loss, σ is the activation function, and λ 1 , λ 2 , λ 3 are utilized to adjust the weight of each loss term. This comprehensive loss function guides the model to simultaneously optimize structural, attribute, and combined similarities during training.

3. Results

In this comprehensive experimental analysis, we meticulously evaluate the performance of our proposed HMAGNN compared to a diverse set of well-established link prediction methods. The primary objective is to showcase the efficacy of HMAGNN by achieving and, in certain cases, surpassing state-of-the-art performance on various datasets. The ensuing section delves into a detailed exposition of the datasets utilized, elucidates the experimental configurations, and thoroughly analyzes the results.

3.1. Datasets

We utilize a diverse set of datasets to evaluate the effectiveness of our approach across various domains:
The PPA dataset we used consists of 576,289 nodes and 30,326,273 edges. Each node has 50 features. The data are derived from a protein-protein interaction graph containing proteins from 37 organisms. Each node is characterized by attributes such as protein type, composition count, and composition size. Edges signify confidence levels in specific protein interactions, including gene co-occurrence, fusion events, and co-expression.
The DDI dataset contains 4267 nodes and 1,334,889 edges, with an average degree of 500.5. This dataset represents a drug-drug interaction network, where nodes represent drugs and edges indicate interaction relationships. Its goal is to predict interactions between drugs. Positive samples indicate the presence of interaction and negative samples indicate absence. It is aimed at aiding pharmaceutical research.
The Collab dataset has 235,868 nodes and 1,285,465 edges. Each node has 128 features. It represents a collaboration network among authors. Nodes stand for authors and edges represent collaborations. Each node is characterized by 128-dimensional features obtained by averaging word embeddings from published papers. The task in this dataset involves predicting future author collaborations based on past interactions.
The Citation2 dataset has 2,927,963 nodes and 30,561,187 edges. Each node has 128 features. It encompasses multiple citation networks from diverse academic fields. The goal is to predict missing citation relationships based on existing citations. The dataset includes citation networks from the Computer Science Department at New York University (from 2003 to 2013) and the Max Planck Institute for Mathematics in Germany (from 1993 to 2003), comprising all papers from each institution.
The Jujube dataset contains 2118 nodes and 43,417 edges. Each node has 32 features. It is a proprietary dataset related to the sales links of Xinjiang jujube. It is sourced from statistical data on the jujube market from the Xinjiang Production and Construction Corps and local market research. In experimentation, the dataset will be divided into training and testing sets, with 247 nodes selected for validation and testing purposes.
The metrics reported in Table 1 include Hits@100, Hits@20, and MRR (Mean Reciprocal Rank) for a comprehensive evaluation of the link prediction performance.

3.2. Baselines

We employ a comprehensive set of baseline methods to benchmark the performance of HMAGNN across various dimensions. The baselines include traditional heuristic methods, embedding-based approaches, and GNN methods. For traditional heuristic methods, we employed three models. The Common Neighbors (CN) measures the number of shared neighbors between two nodes. The Adamic-Adar (AA) assigns weights to shared neighbors based on their degrees [47]. The Resource Allocation (RA) allocates a resource value between nodes according to shared neighbors. Additionally, the Heuristic Learning Method MF is a heuristic learning approach that leverages graph structural features, as mentioned in [48]. For Embedding-Based methods, there are two main approaches. Node2Vec is an embedding-based method that learns continuous representations of nodes in the graph. MLP utilizes a neural network with multiple layers to capture non-linear relationships in the graph. In addition, we compare our method to four GNN-Based Methods. The GCN (Graph Convolutional Network) is a standard GNN that operates on graph-structured data. The GraphSAGE (Graph Sample and Aggregation) Aggregates features from a node’s local neighborhood using sampling. SEAL (Structural Edge Attribute Learning) is a GNN that incorporates structural features of subgraphs for enhanced performance. The Neo is a GNN-based method that, similar to SEAL, learns both node representations and structural features for improved link prediction [49]. These baselines collectively cover a spectrum of methods, from classical heuristics to advanced embedding and GNN techniques. Comparing the performance of HMAGNN against these baselines allows us to assess its effectiveness in capturing complex relationships and structural features for link prediction.

3.3. Experimental Details

For heuristic methods such as Common neighbors, Adamic Adar, Resource allocation, and Matrix Factorization, we implemented them using PyTorch 1.8.1 according to the papers in the references. For Node2Vec, GCN, GraphSAGE, SEAL, and Neo, we implemented them using the methods provided in GitHub. We use GCN as a feature-based GNN model and jointly train it with HMAGNN. For HMAGNN, we set the batch size to 2048. The number of epochs is 40. The learning rate is set to 0.001 and the weight decay is 1 × 10−5. For all graph neural network-based models, we set the number of layers to 3 and the dimension to 256. All experiments are conducted on two RTX 3090s (24 GB).

3.4. Results on Link Prediction

Figure 4 and Table 2 present the performance comparison of HMAGNN with various baseline methods on multiple datasets, including those from the Open Graph Benchmark (OGB) and the Jujube dataset, in the context of link prediction. Methods leveraging closed subgraphs like SEAL, Neo, and HMAGNN consistently outperform traditional heuristic methods, emphasizing the efficacy of learned heuristics over manually designed ones. Across datasets, GNN-based methods generally outshine traditional heuristic methods, highlighting the ability of GNNs to automatically capture node and edge feature representations. Notably, HMAGNN exhibits superior performance among the GNN-based methods, showcasing its effectiveness in capturing intricate relationships and structural features. On the OGB-PPA dataset, feature-based GNNs demonstrate suboptimal performance, suggesting that these methods might not fully exploit structural information. In contrast, SEAL, Neo, and HMAGNN, which incorporate structural information, outperform GCN and GraphSAGE. Though SEAL achieves strong performance overall, it faces challenges on the OGB-DDI dataset, indicating potential issues in integrating input node and structural features for this specific dataset. HMAGNN performs exceptionally well on the Jujube dataset, surpassing other methods. However, there is a noticeable decrease in prediction accuracy across all models on the Jujube dataset compared to other OGB datasets. This decline may be attributed to the smaller size of the Jujube dataset, limiting the amount of available training data. HMAGNN demonstrates adaptability by combining effectively with GCN, showcasing improved performance. The collaborative utilization of both methods contributes to enhanced link prediction accuracy. The results highlight HMAGNN’s robust performance, particularly in scenarios involving closed subgraphs and datasets with complex structural information. Further investigations, possibly through statistical significance tests, could enrich understanding of the observed differences in model performances.

3.5. Ablation Studies and Experimental Sensitivity Analysis

In this section, we delve into ablation studies to meticulously assess the impact and effectiveness of various components within the HMAGNN framework. We executed experiments across diverse datasets, meticulously evaluating the individual contributions of GCN, HMAGNN, and their combined model. At the same time, we verify the impact of the trainable parameter α on the combined model. This parameter dynamically adjusts the weighting between the structural of HMAGNN and the attribute features of GCN.
As illustrated in Table 3, which summarizes the impact of combining HMAGNN with GCN on link prediction, the value of the trainable parameter α varies for each dataset. On the PPA dataset, the model performed optimally when α was set to 0.92. This specific configuration led to a notable enhancement in prediction accuracy, with HMAGNN (w/GCN) achieving 49.72%, outperforming both HMAGNN (w/o GCN) at 47.26% and the standalone GCN at 16.98%. This underscored the efficacy of HMAGNN in effectively utilizing structural features. On the other hand, when evaluating the DDI dataset, HMAGNN (w/o GCN) achieved an accuracy of 37.07%, which the standalone GCN surpassed at 44.60%. The DDI dataset, characterized by a higher number of node features than other datasets, favored the performance of GCN. However, the integration of GCN with HMAGNN significantly improved accuracy to 64.66%, reinforcing the capability of HMAGNN to effectively learn both structural and attribute features.
Additionally, we explored the influence of local subgraph size on the experiments, constrained by the node labels’ k-value. Figure 5 showcases the results obtained with different values of k (4, 8, 12, and 16) on four datasets. The overall performance was optimal when k was set to 8. On the PPA dataset, HMAGNN consistently outperformed other cases, demonstrating its superiority throughout the training. However, on the DDI and Collab datasets, the model’s prediction accuracy slightly improved as the subgraph size increased from k = 8 to 16 in the initial 25 training rounds, after which the accuracy started declining. This suggests that HMAGNN can extract sufficient structural information from local subgraphs without learning from the entire graph. Importantly, the findings indicate that excessively large or small local subgraphs are not conducive to extracting structural features.
To assess the impact of the multi-head attention mechanism (MHA) on HMAGNN, we conducted a comparative experiment between HMAGNN (with MHA) and HMAGNN (without MHA) on the OGB dataset. As depicted in Figure 6, HMAGNN (with MHA) consistently outperformed HMAGNN (without MHA), outperforming it in all datasets. Particularly noteworthy was the 45% improvement in accuracy on the PPA dataset when the multi-head attention mechanism was integrated. This emphasizes the beneficial role of the multi-head attention mechanism in enhancing the accuracy of link prediction within the HMAGNN framework.
Ablation studies provide valuable insights into HMAGNN’s robustness and sensitivity to key components. The optimal performance achieved under specific configurations and the nuanced relationships observed highlight the intricate interplay between structural and attribute features in link prediction tasks.

4. Conclusions

In this study, we introduced the Heterogeneous Graph Attention Neural Network as a powerful framework for enhancing link prediction accuracy in the complex sales network of Xinjiang jujubes. Our approach uniquely combines the exploitation of both structural and attribute information, addressing the intricacies of the heterogeneous graph representing the jujube sales network.
HMAGNN introduces an innovative strategy for node labeling within closed subgraphs, a pivotal step in the learning process. This labeling scheme facilitates the intuitive identification of target and source nodes, crucial for accurate link prediction. Moreover, our exploration revealed that the optimal subgraph size (k = 8) significantly influences the predictive accuracy of HMAGNN, underscoring the importance of this parameter in the network’s performance.
To harness the full potential of node attribute information, we seamlessly integrated HMAGNN with GCN. This hybrid model effectively integrates both structural and attribute information, resulting in a notable enhancement in performance.
Our experimental results showcase the superiority of HMAGNN over a spectrum of algorithms, including heuristic methods, embedding-based approaches, and GNN. The adaptability of HMAGNN to diverse datasets and its outperformance of existing methods underscore its efficacy in real-world applications.
Looking ahead, our future work will focus on expanding and standardizing the jujube dataset. We will actively seek more data sources related to Xinjiang jujube sales. By increasing the size of the dataset, we expect to improve the generalization ability of our model and capture more complex patterns in the sales network. This expansion will also enable us to further optimize the structure of HMAGNN and improve its performance in link prediction. In addition, we will continue to optimize hyperparameters, explore additional features, and improve the model architecture to continuously improve the accuracy of link prediction in the dynamic environment of Xinjiang jujube sales. Through these endeavors, we aim to contribute to advancing predictive modeling in agricultural product sales network.

Author Contributions

Conceptualization, Y.W. and L.H.; methodology, Y.W.; software, Y.W. and J.Y.; investigation, Y.W. and L.H.; data curation, Y.W. and F.T.; writing—original draft preparation, Y.W.; writing—review and editing, Y.W. and L.G. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Bingtuan Science and Technology Program under Grant 2021AB003 and Bingtuan Guidance Plan Project under Grant 2023ZD047.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. An example of a heterogeneous graph for Xinjiang jujube sales. (a) Three types of nodes: producers, distributors, and retailers. (b) A heterogeneous graph representing Xinjiang jujube sales with three node types and two types of connections.
Figure 1. An example of a heterogeneous graph for Xinjiang jujube sales. (a) Three types of nodes: producers, distributors, and retailers. (b) A heterogeneous graph representing Xinjiang jujube sales with three node types and two types of connections.
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Figure 2. Illustration of the node labeling algorithm for a graph. In the depicted graph, the source node is denoted as 0, and the target node as 1. The algorithm, showcased in each iteration, involves Step 1, which calculates a distinctive string for each node by recording the indices of the nodes and their respective neighbors. Following this, Step 2 orchestrates the re-labeling of nodes in adherence to the devised node labeling algorithm.
Figure 2. Illustration of the node labeling algorithm for a graph. In the depicted graph, the source node is denoted as 0, and the target node as 1. The algorithm, showcased in each iteration, involves Step 1, which calculates a distinctive string for each node by recording the indices of the nodes and their respective neighbors. Following this, Step 2 orchestrates the re-labeling of nodes in adherence to the devised node labeling algorithm.
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Figure 3. The HMAGNN framework for link prediction. The process involves node labeling, two Multi-Layer Perceptrons (MLPs), and a multi-head mechanism for node feature generation. The structural vector x ∈ R N × 1 is transformed into the structural matrix X s ∈ R N × N , and similarity scores are computed and adaptively combined using the parameter α. (a) HMAGNN first transforms the heterogeneous graph into the same feature space through a transformation matrix, and then labels the nodes in the graph; (b) HMAGNN learns structural features from the adjacency matrix and considers a multi-head mechanism to generate structural feature vectors; (c) Diagonalizes the structural feature vectors to construct a diagonal matrix; (d) Computes the loss based on the two node representations Z and h obtained from HMAGNN and GNN, respectively.
Figure 3. The HMAGNN framework for link prediction. The process involves node labeling, two Multi-Layer Perceptrons (MLPs), and a multi-head mechanism for node feature generation. The structural vector x ∈ R N × 1 is transformed into the structural matrix X s ∈ R N × N , and similarity scores are computed and adaptively combined using the parameter α. (a) HMAGNN first transforms the heterogeneous graph into the same feature space through a transformation matrix, and then labels the nodes in the graph; (b) HMAGNN learns structural features from the adjacency matrix and considers a multi-head mechanism to generate structural feature vectors; (c) Diagonalizes the structural feature vectors to construct a diagonal matrix; (d) Computes the loss based on the two node representations Z and h obtained from HMAGNN and GNN, respectively.
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Figure 4. Comparison of model results on five datasets. (a) OGB-PPA dataset; (b) OGB-DDI dataset; (c) OGB-Collab dataset; (d) OGB-Citation2 dataset; (e) Jujube dataset.
Figure 4. Comparison of model results on five datasets. (a) OGB-PPA dataset; (b) OGB-DDI dataset; (c) OGB-Collab dataset; (d) OGB-Citation2 dataset; (e) Jujube dataset.
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Figure 5. Impact of node label size on link prediction. (a) OGB-PPA dataset; (b) OGB-DDI dataset; (c) OGB-Collab dataset; (d) OGB-Citation2 dataset.
Figure 5. Impact of node label size on link prediction. (a) OGB-PPA dataset; (b) OGB-DDI dataset; (c) OGB-Collab dataset; (d) OGB-Citation2 dataset.
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Figure 6. Influence of multi-head attention mechanism on link prediction. (a) OGB-PPA dataset; (b) OGB-DDI dataset; (c) OGB-Collab dataset; (d) OGB-Citation2 dataset.
Figure 6. Influence of multi-head attention mechanism on link prediction. (a) OGB-PPA dataset; (b) OGB-DDI dataset; (c) OGB-Collab dataset; (d) OGB-Citation2 dataset.
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Table 1. Statistics and evaluation metrics of link prediction datasets.
Table 1. Statistics and evaluation metrics of link prediction datasets.
OGB-PPAOGB-DDIOGB-CollabOGB-Citation2Jujube
Nodes576,2894267235,8682,927,9632118
Edges30,326,2731,334,8891,285,46530,561,18743,417
Features5023212812832
Training403,4023413216,9982,869,4031624
Validation115,258427943529,280247
Test57,629427943529,280247
MetricHits@100Hits@20Hits@100MRRHits@20
Table 2. Link prediction performance of HMAGNN and baselines on different datasets. Bold indicates the best performance.
Table 2. Link prediction performance of HMAGNN and baselines on different datasets. Bold indicates the best performance.
PPADDICollabCitation2Jujube
CN27.65 ± 0.0017.73 ± 0.0050.06 ± 0.0076.20 ± 0.0018.97 ± 0.00
AA32.45 ± 0.0018.61 ± 0.0053.00 ± 0.0076.12 ± 0.0021.19 ± 0.00
RA49.33 ± 0.006.23 ± 0.0052.89 ± 0.0076.20 ± 0.0019.88 ± 0.00
MF32.29 ± 0.0033.70 ± 0.0348.96 ± 0.0051.89 ± 0.0437.46 ± 0.02
MLP0.47 ± 0.05——19.98 ± 0.9628.99 ± 0.1622.47 ± 0.11
Node2Vec17.24 ± 0.7621.95 ± 1.5841.36 ± 0.6953.47 ± 0.1222.13 ± 0.86
GCN16.98 ± 1.3344.60 ± 8.8747.01 ± 0.7984.79 ± 0.2443.17 ± 4.38
GraphSAGE13.93 ± 2.3848.01 ± 9.0248.60 ± 0.4682.62 ± 0.0153.61 ± 5.46
SEAL48.15 ± 4.1726.25 ± 6.0054.37 ± 0.0286.32 ± 0.5262.78 ± 0.43
Neo49.13 ± 0.6063.57 ± 3.5257.52 ± 0.3787.26 ± 0.8463.61 ± 0.77
HMAGNN49.72 ± 3.0664.66 ± 7.7457.63 ± 0.7286.54 ± 0.3465.28 ± 0.22
Table 3. Impact of combining HMAGNN with GCN on link prediction.
Table 3. Impact of combining HMAGNN with GCN on link prediction.
αHMAGNN (w/GCN)HMAGNN (w/o GCN)GCN
PPA0.92 ± 0.01249.72 ± 3.0647.26 ± 0.5616.98 ± 1.33
COLLAB0.59 ± 0.01557.63 ± 0.7255.87 ± 0.4147.01 ± 0.79
DDI0.57 ± 0.02464.66 ± 7.7437.07 ± 3.0544.60 ± 8.87
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Wu, Y.; Heng, L.; Tan, F.; Yang, J.; Guo, L. Dynamic Link Prediction in Jujube Sales Market: Innovative Application of Heterogeneous Graph Neural Networks. Appl. Sci. 2024, 14, 9333. https://doi.org/10.3390/app14209333

AMA Style

Wu Y, Heng L, Tan F, Yang J, Guo L. Dynamic Link Prediction in Jujube Sales Market: Innovative Application of Heterogeneous Graph Neural Networks. Applied Sciences. 2024; 14(20):9333. https://doi.org/10.3390/app14209333

Chicago/Turabian Style

Wu, Yichang, Liang Heng, Fei Tan, Jingwen Yang, and Li Guo. 2024. "Dynamic Link Prediction in Jujube Sales Market: Innovative Application of Heterogeneous Graph Neural Networks" Applied Sciences 14, no. 20: 9333. https://doi.org/10.3390/app14209333

APA Style

Wu, Y., Heng, L., Tan, F., Yang, J., & Guo, L. (2024). Dynamic Link Prediction in Jujube Sales Market: Innovative Application of Heterogeneous Graph Neural Networks. Applied Sciences, 14(20), 9333. https://doi.org/10.3390/app14209333

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