
==== Front
Brief Bioinform
Brief Bioinform
bib
Briefings in Bioinformatics
1467-5463
1477-4054
Oxford University Press

10.1093/bib/bbae443
bbae443
Problem Solving Protocol
AcademicSubjects/SCI01060
Global-local aware Heterogeneous Graph Contrastive Learning for multifaceted association prediction in miRNA–gene–disease networks
https://orcid.org/0000-0003-2828-5231
Si Yuxuan Department of Ophthalmology, Sir Run Run Shaw Hospital, Zhejiang University School of Medicine, East Qingchun Road, 310016 Zhejiang, China
College of Computer Science and Technology, Zhejiang University, 38 Zheda Road, 310027 Zhejiang, China

Huang Zihan College of Computer Science and Technology, Zhejiang University, 38 Zheda Road, 310027 Zhejiang, China

Fang Zhengqing Department of Ophthalmology, Sir Run Run Shaw Hospital, Zhejiang University School of Medicine, East Qingchun Road, 310016 Zhejiang, China
College of Computer Science and Technology, Zhejiang University, 38 Zheda Road, 310027 Zhejiang, China

Yuan Zhouhang Department of Ophthalmology, Sir Run Run Shaw Hospital, Zhejiang University School of Medicine, East Qingchun Road, 310016 Zhejiang, China
College of Computer Science and Technology, Zhejiang University, 38 Zheda Road, 310027 Zhejiang, China

Huang Zhengxing College of Computer Science and Technology, Zhejiang University, 38 Zheda Road, 310027 Zhejiang, China

Li Yingming College of Information Science and Electronic Engineering, Zhejiang University, 38 Zheda Road, 310027 Zhejiang, China

Wei Ying College of Computer Science and Technology, Zhejiang University, 38 Zheda Road, 310027 Zhejiang, China

Wu Fei College of Computer Science and Technology, Zhejiang University, 38 Zheda Road, 310027 Zhejiang, China

Yao Yu-Feng Department of Ophthalmology, Sir Run Run Shaw Hospital, Zhejiang University School of Medicine, East Qingchun Road, 310016 Zhejiang, China
Department of Ophthalmology, The Fourth Affiliated Hospital of Soochow University, 215000 Suzhou, China

Corresponding author. Department of Ophthalmology, The Fourth Affiliated Hospital of Soochow University, 215000 Suzhou, China; Department of Ophthalmology, Sir Run Run Shaw Hospital, Zhejiang University School of Medicine, East Qingchun Road, 310016 Zhejiang, China. E-mail: yaoyf@zju.edu.cn
9 2024
10 9 2024
10 9 2024
25 5 bbae44303 3 2024
11 8 2024
30 8 2024
© The Author(s) 2024. Published by Oxford University Press.
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (https://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com

Abstract

Unraveling the intricate network of associations among microRNAs (miRNAs), genes, and diseases is pivotal for deciphering molecular mechanisms, refining disease diagnosis, and crafting targeted therapies. Computational strategies, leveraging link prediction within biological graphs, present a cost-efficient alternative to high-cost empirical assays. However, while plenty of methods excel at predicting specific associations, such as miRNA–disease associations (MDAs), miRNA–target interactions (MTIs), and disease–gene associations (DGAs), a holistic approach harnessing diverse data sources for multifaceted association prediction remains largely unexplored. The limited availability of high-quality data, as vitro experiments to comprehensively confirm associations are often expensive and time-consuming, results in a sparse and noisy heterogeneous graph, hindering an accurate prediction of these complex associations. To address this challenge, we propose a novel framework called Global-local aware Heterogeneous Graph Contrastive Learning (GlaHGCL). GlaHGCL combines global and local contrastive learning to improve node embeddings in the heterogeneous graph. In particular, global contrastive learning enhances the robustness of node embeddings against noise by aligning global representations of the original graph and its augmented counterpart. Local contrastive learning enforces representation consistency between functionally similar or connected nodes across diverse data sources, effectively leveraging data heterogeneity and mitigating the issue of data scarcity. The refined node representations are applied to downstream tasks, such as MDA, MTI, and DGA prediction. Experiments show GlaHGCL outperforming state-of-the-art methods, and case studies further demonstrate its ability to accurately uncover new associations among miRNAs, genes, and diseases. We have made the datasets and source code publicly available at https://github.com/Sue-syx/GlaHGCL.

graph contrastive learning
heterogeneous data sources
miRNA–disease association
miRNA–gene interaction
disease–gene association
National Key Research and Development Program of China 10.13039/501100012166 2023YFE0204200 National Natural Science Foundation of China 10.13039/501100001809 U20A20387
==== Body
pmcIntroduction

MicroRNAs (miRNAs) are small (approximately 22 nucleotides) non-coding regulatory RNAs that play a crucial role in post-transcriptional gene expression regulation [1]. They are involved in various cellular processes such as growth, development, and the onset of human diseases, including cardiovascular diseases, autoimmune diseases, neurodegenerative diseases, and cancer [1–3]. For instance, a study by Ma et al. [4] demonstrated that miR-10b promotes the invasion and migration of breast tumors by targeting the tumor suppressor HOXD10 and increasing the expression of the metastasis-related protein RhoC. Researchers have developed comprehensive resources to chart the landscape of miRNA regulatory networks. For example, a recent study provided a detailed map of miRNA-gene interactions across multiple cancers [5], while the mirTrans database [6] documents the transcriptional regulation of miRNAs in diverse human cell lines. Additionally, studies have systematically elucidated the topological features and modular structure of gene regulatory networks in major human tissues, underscoring the critical roles of miRNAs and other regulatory factors [7]. These resources have deepened our understanding of the regulatory functions of miRNAs in physiological and pathological processes. Despite the increasing recognition of the importance of miRNAs in diseases, traditional experimental approaches have limitations in discovering new associations due to long cycles, high costs, and low success rates. Therefore, there is a need for efficient computational methods that can leverage available biological data to accelerate the prediction and validation of associations among miRNAs, genes, and diseases.

In recent years, computational methods have gained prominence in predicting miRNA–target interactions (MTIs), miRNA–disease associations (MDAs), and disease–gene associations (DGAs). These methods employ techniques like matrix completion [8], machine learning [9], and deep learning [10], to integrate omics data and uncover novel associations. Deep learning, particularly through the use of graph neural networks (GNNs) [11], has emerged as a powerful tool for handling unstructured data in bioinformatics, with notable advancements in graph representation learning tasks like node classification and link prediction. However, prior methods have typically treated the prediction of MDAs, MTIs, and DGAs as independent tasks, overlooking the inherent correlations among these diverse associations. Recent studies [12,13] have highlighted the importance of simultaneously considering these interrelated associations. For instance, researchers calculated the similarity between miRNAs via shared target genes [14], the similarity between diseases by measuring functional similarities of disease-related genes [15], and the relevance of miRNAs to diseases through the correlation in associated genes [16–18]. The work of [12] further incorporated multi-task learning to simultaneously infer MDA and DGAs.

Considering the scarcity of annotated associations in existing databases, exemplified by the HMDD v2.0 database which comprises merely 578 miRNA nodes and 383 disease nodes, alongside the well-documented seesaw conundrum inherent in multitasking scenarios, recent research [19] has adopted self-supervised learning to derive generic node features for tasks involving lncRNA, miRNA, and diseases. This approach contrasts the global representation of the original graph against local graph patches from a perturbed version. However, two critical challenges persist. Firstly, current methods corrupt the feature matrix by row-wise shuffling, resulting in a corrupted graph that diverges markedly from the original and functions as ‘easy negatives’, which have been deemed limited in the literature [20]. Secondly, the consistency of node representations when integrating diverse data sources is fundamentally dictated by their interconnections within the graph structure. Yet, the issue of sparsity critically hampers the scope of these interlinkages, resulting in a significant portion of node representations being trained with no effective guidance.

To address these challenges, we introduce an innovative framework dubbed Global-local aware Heterogeneous Graph Contrastive Learning (GlaHGCL), designed to infer multifaceted associations among miRNAs, genes, and diseases. Concretely, first, we augment the original three-layer heterogeneous graph constructed, consisting of miRNA, gene, and disease nodes, via imposing a nuanced perturbation on the adjacency matrix. Employing a Multi-Hop Attention Diffusion (MhAD) encoder [21] enables the capture of both long-range and even global structure information across the graph. Further maximization of the agreement between representations of the original graph and those of the augmented one promotes robustness of learned representations, wherein the augmented graph serves as effective ‘hard positives’. Furthermore, we methodically introduce local contrastive learning to bridge nodes from diverse data sources, directly addressing the issue of sparsity. This approach extends beyond merely constructing nodes that are directly connected across different layers as positive pairs (e.g. connecting a gene node directly to an anchoring miRNA node). It also embraces nodes that, although not directly connected, exhibit significant similarity, thereby broadening the scope for consistency maximization and effectively mitigating the limitations posed by data sparsity.

The overall workflow of GlaHGCL is presented in Fig. 1. In summary, our proposed method contributes to the field in the following ways:

A novel contrastive learning framework, termed GlaHGCL, is meticulously crafted to excel in the domain of feature representation learning for miRNAs, genes, and diseases. The cornerstone ambition of this framework is to cultivate an encoder capable of not only reconstructing the comprehensive graph structure from its augmented counterpart but also optimizing the mutual information across node representation features, all within the confines of a triple-wise association constraint. Distinctively, this encoder is engineered to assimilate high-order or global structural insights via a MhAD mechanism, rendering the GlaHGCL framework adaptable to a vast array of sparse heterogeneous graphs derived from multiple data sources.

Subsequent to the initial phase, the node representation features undergo a meticulous refinement process in a supervised setting, aimed at enhancing their applicability to downstream tasks. These tasks encompass predictions of MTIs, MDAs, and DGAs, thereby covering a comprehensive spectrum of biological inquiry.

Our rigorous empirical analysis, conducted across two distinct datasets, serves to validate the efficacy of our proposed framework. The findings unequivocally demonstrate GlaHGCL’s superiority over contemporary state-of-the-art methodologies, achieving significant improvements by margins of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $2.62$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $3.73\%$\end{document}, respectively.

Figure 1 The overall framework of GlaHGCL. GlaHGCL consists of four main modules: (a) a three-layer heterogeneous graph with miRNA, gene, and disease nodes, along with its augmented version, are constructed. (b) A MhAD encoder takes the constructed graphs as input and generates the nodes’ representation. (c) A global contrastive learning task aims to maximize of the agreement between the representations of the original graph and those of the augmented one. (d) A local contrastive learning task optimizes the mutual information across node representation features within the confines of a triple-wise association constraint. Finally, GlaHGCL is fine-tuned under supervision for three downstream tasks: MDA, MTI, and DGA prediction.

Materials and methods

Datasets

In this study, for Dataset1, we collected MDAs from HMDD v2.0 [22], MTIs from RAIN [23], and DGAs from DISEASES [24]. For Dataset2, we utilized MDAs from HMDD v3.2 [25], miRNA-gene associations from mirTarBase v8.0 [26], and disease–gene relationships from DisGeNET v7.0 [27]. For the MTIs, we included only those involving protein-coding genes (PCGs) associated with at least one Reactome pathway [28], to examine biologically relevant interactions. The statistics for each dataset is shown in Table 1.

Table 1 The statistics for each dataset.

	MiRNAs	Genes	Diseases	MDAs	MTIs	DGAs	
Dataset1	442	9236	309	4592	2629	27 115	
Dataset2	742	10 116	545	11 972	4676	84 038	

The heterogeneous graph representing the relationships among miRNA, gene, and disease entities comprises distinct layers corresponding to miRNAs, genes, and diseases, respectively. Two types of edges exist, including those linking nodes across different layers and those connecting nodes within the same layer. For the former, we mark the known associations as positive edges, denoted as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $1$\end{document}. The remaining edges indicate no verified associations, denoted as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $0$\end{document}. The latter represents similarities between the nodes in the same layer. In this study, we calculate similarities for miRNA–miRNA, gene–gene, and disease–disease from multiple perspectives including miRNA sequence similarity, miRNA functional similarity, gene functional interaction probability, disease semantic similarity, and Gaussian interaction profile kernel similarity for miRNAs and diseases. Detailed description of the construction of miRNA–gene–disease heterogeneous graph can be found in Supplementary Materials. At last, the three-layer heterogeneous graph can be denoted as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal{G}=(A, X)$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $A$\end{document} is the adjacency matrix and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $X$\end{document} is the attribute matrix obtained by normalizing each row of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $A$\end{document}.

MhAD-based encoder

We follow the MhAD encoder [21] to facilitate comprehensive graph representations that encompass long-range and global relational contexts. It computes attention scores across all edges, utilizing a diffusion process to assign attention values to nodes connected indirectly. We illustrate the encoding process using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal{G}=(A, X)$\end{document} as an example.

Edge attention computation. Within each layer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $l$\end{document} of the MhAD encoder, a vector message is computed for each node pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $(v_{i}, v_{j})$\end{document}. The attention score \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $q$\end{document} for the edge between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v_{i}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v_{j}$\end{document} is computed as following:

(1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& {q}_{i, j}^{(l)}=\operatorname{LeakyReLU}\ \left(\boldsymbol{u}^{(l)} \tanh \left(\boldsymbol{W}^{(l)} {Z}_ {i}^{(l)}\|\boldsymbol{W}^{(l)} {Z}_{j}^{(l)} \right)\right),\end{align*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol{u}^{(l)} \in \mathbb{R}^{1 \times 2 d^{(l)}}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol{W}^{(l)} \in \mathbb{R}^{d^{(l)} \times d^{(l)}}$\end{document} are the trainable weights for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $l$\end{document}th layer. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${Z}_{i}^{(l)} \in \mathbb{R}^{d^{(l)}}$\end{document} represents the embedding of node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $i$\end{document} at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $l$\end{document}th layer, and particularly \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${Z}_{i}^{(0)}={X}_{i}$\end{document} is the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $i$\end{document}th row of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $X$\end{document}. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $a \| b$\end{document} denotes concatenation of embedding vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $a$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $b$\end{document}. Applying Equation (1) on each edge of the graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\mathcal{G}}$\end{document} gives the attention score matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal{Q}^{(l)}$\end{document} whose entries equal to

(2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \mathcal{Q}_{i,j}^{(l)}= \begin{cases}s_{i, j}^{(l)}, & \text{ if} \left(v_{i}, v_{j}\right) \text{ in}\ {\mathcal{G}}, \\ -\infty, & \text{ otherwise. }\end{cases}\end{align*}\end{document}

We obtain the final attention matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol{\tilde{Q}}^{(l)}$\end{document} by performing row-wised softmax over the score matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal{Q}^{(l)}:\boldsymbol{\tilde{Q}}^{(l)} =\operatorname{softmax}(\mathcal{Q}^{(l)})$\end{document}. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol{\tilde{Q}}^{(l)}_{ij}$\end{document} denotes the attention value at layer \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $l$\end{document} when aggregating messages from node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $j$\end{document} to node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $i$\end{document}.

Multi-hop Attention Diffusion. In the second stage, the scope of attention for each node is broadened to encompass nodes in the graph that lack direct connections. This is implemented by (1) starting from the 1-hop attention matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\tilde{\boldsymbol{Q}}^{(l)}$\end{document}, (2) employing an iterative power-raising method to simulate graph diffusion where the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $k$\end{document}th power \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $(\tilde{\boldsymbol{Q}}^{(l)})^{k}$\end{document} takes two nodes that are indirectly connected via \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $k$\end{document} hops into consideration. The diffused attention matrix

(3) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \tilde{\mathcal{Q}}^{(l)} =\sum_{k=0}^{K} \theta_{k} (\tilde{\boldsymbol{Q}}^{(l)})^{k}\ \text{ where}\ \sum_{k=0}^{K} \theta_{k}=1 \text{ and}\ \theta_{k}>0.\end{align*}\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $K$\end{document} is the maximum number of attention hops, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\theta _{k}$\end{document} is the attention decay factor satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\theta _{k}>\theta _{k+1}$\end{document}. Specifically, we employ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\theta _{k} = \alpha (1-\alpha )^{k}$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha \in (0,1]$\end{document}. As a result, an inductive bias is incorporated, assigning less weight to distant nodes during message aggregation. Note that nodes with varying path lengths to the target node are independently weighted.

Encoder. Equipped with the above multi-hop attention, we proceed to finally compute the node embeddings \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Z^{l+1}$\end{document} at the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $(l+1)$\end{document}th layer throughout the following procedures, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Z^{(0)} = X$\end{document} and the node embedding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Z^{l}$\end{document} passed through Layer Normalization [29] first.

(4) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \begin{aligned} & {Z}_{\text{MH}}^{(l+1)} = \left(\|_{i=1}^{H} \tilde{\mathcal{Q}}^{(l)}_{i}\right) \boldsymbol{W}^{(l)}_{0}, \\ & Z^{(l+1)}=\boldsymbol{W}_{2}^{(l)} \operatorname{ReLU}\left(\boldsymbol{W}_{1}^{(l)} \mathcal{Z}_{\mathrm{LN}}\right)+{Z}_{\text{MH}}^{(l+1)} + Z^{(l)}, \end{aligned}\end{align*}\end{document}

where ∥ denotes concatenation, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal{Q}^{(l)}_{i}$\end{document} represents the diffusion attention matrix via the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $i$\end{document}th (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $(1 \leq i \leq H)$\end{document}) head following the multi-head mechanism. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $H$\end{document} is the total number of heads. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol{W}^{(l)}_{0}$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol{W}^{(l)}_{1}$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol{W}^{(l)}_{2}$\end{document} represent the parameter matrices.

Provided with a total number of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $L$\end{document} layers, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Z^{L}$\end{document} after the last layer of the encoder presents the final node embeddings, i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $$Z^{L}=\left [\begin{array}{l}Z^{m} \\ Z^{g} \\ Z^{d}\end{array}\right ]$$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Z^{m}, Z^{g}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Z^{d}$\end{document} indicate embeddings of miRNAs, genes, and diseases, respectively.

SSL task1: global contrastive learning

The construction procedure for the augmented graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hat{\mathcal{G}}$\end{document} and its corresponding adjacency matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hat{A}$\end{document} is as follows:

(5) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \hat{A}=A \odot(1-L)+(1-A) \odot L.\end{align*}\end{document}

The matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $L$\end{document} represents perturbation locations, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $L_{ij}=1$\end{document} indicating a planned perturbation between nodes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v_{i}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v_{j}$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $L_{ij}=0$\end{document} indicating no perturbation. The augmented graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hat{\mathcal{G}}$\end{document} is expressed as

(6) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \hat{\mathcal{G}} = (\hat{A},X).\end{align*}\end{document}

Reconstruction of an edge between nodes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $i$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $j$\end{document} reduces to predicting the association probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hat{A}^{r}_{i,j}$\end{document} from node embeddings \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hat{Z}$\end{document} via Equation (4), i.e.

(7) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \hat{A}_{ij}^{r}=\operatorname{Sigmoid}\left({\hat{Z}[i]^{\top}\mathbf{V} \hat{Z}[j]}\right),\end{align*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathbf{V}$\end{document} denotes a trainable parameter matrix. To measure the sum of reconstruction errors for all edges involved in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal{G}$\end{document}, a binary cross-entropy loss function is employed to maximize the similarity between the reconstructed association matrix and the original one. The global contrastive learning loss is thus formulated as

(8) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \mathcal{L}_{\text{gcl}}=-\sum_{i,j\in{\mathcal{G}}}\left(A_{ij} \log \hat{A}_{ij}^{r}+\left(1-A_{ij}\right) \log \left(1-\hat{A}_{ij}^{r}\right)\right).\end{align*}\end{document}

SSL task2: local contrastive learning

Local contrastive learning task is based on the assumption that when a miRNA is associated with a disease, the miRNA’s target genes are likely to share similar or identical functions with the disease-related genes. Concretely, for each miRNA–disease pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $(v^{m}_{i}, v^{d}_{j})$\end{document} in the graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal{G}$\end{document}, we first employ the inner product to calculate the probability that a gene node is associated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v^{m}_{i}$\end{document}. The gene nodes with the top-N highest probability scores are then identified and included in the positive sample set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P^{m^{+}}_{g}$\end{document}, thereby providing an enriched relational context for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v^{m}_{i}$\end{document}. For the remaining pool of gene nodes, we exclude those already in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P^{m^{+}}_{g}$\end{document} and randomly select \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $N$\end{document} nodes to form the hard negative sample set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P^{m^{-}}_{g}$\end{document}. Applying a parallel strategy for disease nodes, we similarly enhance the context for a disease node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v^{d}_{j}$\end{document} by identifying and selecting gene nodes that are most likely associated, creating positive (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P^{d^{+}}_{g}$\end{document}) and negative (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P^{d^{-}}_{g}$\end{document}) sets, respectively.

In each training batch, we construct sets of positive and negative samples for miRNA–disease pairs. Formally, we adopt the InfoNCE [30] loss function, following previous studies [31,32]. For each miRNA–disease pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $(v^{m}_{i}, v^{d}_{j})$\end{document}, we designate the gene node connected to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v^{m}_{i}$\end{document} within the adjacency matrix as our anchor sample, whose embedding is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Z[g]$\end{document}. According to the assumption we mentioned above, the obtained positive set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P^{d^{+}}_{g}$\end{document} with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v^{d}_{j}$\end{document} should be close to the anchor while the negative set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P^{d^{-}}_{g}$\end{document} should be pushed apart. The local contrastive loss from the miRNA-gene perspective is defined as follows:

(9) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \mathcal{L}_{\text{lcl1}}=-\log\frac{{\sum}_{v_{i}\in P^{d^{+}}_{g}} \exp\left(s\left( Z[g], Z[i]\right)/\tau\right)}{{\sum}_{v_{j}\in \{P^{d^{+}}_{g} \cup P^{d^{-}}_{g} \}} \exp\left(s\left( Z[g],Z[j]\right)/\tau \right)},\end{align*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\tau $\end{document} serves as a critical hyper-parameter that modulates the discrimination degree, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $s(\cdot ,\cdot )$\end{document} denotes the cosine similarity function. In a similar vein, we have the local contrastive loss from the disease–gene perspective, i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mathcal{L}_{\text{lcl2}}$\end{document}. The overall local contrastive loss is written as

(10) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \mathcal{L}_{\text{lcl}}=\sum_{(v^{m}_{i},v^{d}_{j}) \in{\mathcal{G}}} \left[\rho \cdot \mathcal{L}_{\text{lcl1}}+(1-\rho) \cdot \mathcal{L}_{\text{lcl2}}\right]\end{align*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\rho $\end{document} is a hyper-parameter that balances the two terms.

Supervised fine-tuning

The above contrastive learning objectives yield feature representations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Z^{m}$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Z^{g}$\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $Z^{d}$\end{document} for miRNAs, genes, and diseases, respectively. These representations are utilized for supervised fine-tuning in downstream tasks, such as predicting MDAs, MTIs, and DGAs. To capture the interactions between pairs of nodes, we construct pairwise representations for miRNA–disease, miRNA-gene, and gene-disease pairs by element-wise multiplication of their node features. This approach allows us to predict the presence or absence of association between them. For instance, for a pair consisting of a miRNA node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v^{m}_{i}$\end{document} and a disease node \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v^{d}_{j}$\end{document}, their joint feature representation is obtained by element-wise multiplication of their features, represented as

(11) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \mathbf{z}^{md}={Z}^{m}[i]\odot{Z}^{d}[j],\end{align*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${Z}^{m}[i]$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} ${Z}^{d}[j]$\end{document} correspond to the node features for nodes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v^{m}_{i}$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $v^{d}_{j}$\end{document}, respectively. We compute the probability of an association between the miRNA and disease pair as

(12) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \hat{y}^{md}=\operatorname{Sigmoid}\left((\boldsymbol{W}^{md})^{T} \mathbf{z}^{m d}\right)\end{align*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\boldsymbol{W}^{md}$\end{document} is a learnable weight matrix. Thus, we fine-tune with the following binary cross-entropy loss function:

(13) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{align*}& \mathcal{L}_{\text{MDA}}=-\sum_{m,d}\left(y^{md} \log \hat{y}^{md}+\left(1-y^{md}\right) \log \left(1-\hat{y}^{md}\right)\right)\end{align*}\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $y_{m d}$\end{document} denote the groundtruth association label known for the corresponding pair. The MTI and DGA prediction tasks share the similar fine-tuning process with different learnable weight matrices.

Experiments

Experiments settings

For each dataset, the known MTIs, MDAs, and DGAs are used as positive samples, while an equal number of negative samples are randomly selected to ensure a balanced dataset. Each dataset is then randomly split into a 90% cross-validation (CV) set and a 10% held-out independent test set. We perform 5-fold cross-validation on the CV set. The samples are randomly divided into five subsets. Each subset serves as the validation set in turn, while the remaining ones are used for training. We train and validate on the CV set, and independently test on the corresponding held-out test set. In addition, we train on Dataset1 and independently test on a subset of Dataset2 (excluding any entities present in Dataset1), ensuring unbiased evaluation on unseen data.

The evaluation metrics include Area Under the Curve (AUC), Area Under the Precision-Recall Curve (AUPR), accuracy, F1-score, recall, and precision. AUC and AUPRC provide a comprehensive evaluation of model quality by plotting the ROC and PR curves, respectively, where higher values indicate better overall performance. The F1-score measures the harmonic mean of precision and recall.

Parameter analysis

In this section, we investigate the impact of the key hyperparameters on the performance of the GlaHGCL model. We leverage 5-fold CV and analyze the evaluation metrics to identify the optimal hyperparameter settings. The primary hyperparameters examined include (1) the depth of the GNN encoder, (2) the hop number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $K$\end{document}, the teleport probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha $\end{document}, and the head number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $H$\end{document} in the encoder module.

Model depth

To examine model depth, we vary the number of layers in GCN, Diffusion-GCN, GAT, and GlaHGCL. The results (shown in Fig. 2(a)) reveal that GCN, Diffusion-GCN, and GAT suffer from performance degradation due to over-smoothing [33]. In contrast, GlaHGCL consistently achieves the best results, even with 24 layers, demonstrating its robustness and expressive power. Notably, a 12-layer GAT fails to outperform a 3-layer GlaHGCL with a hop number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $K$\end{document} of 4.

Figure 2 Parameter analysis of GlaHGCL. (a) Effect of GNN encoder depth on performance. (b) Effect of hop number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $K$\end{document} on performance. (c) Effect of teleport probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha $\end{document}. (d) Effect of head number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $H$\end{document}.

Effect of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $K$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha $\end{document}, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $H$\end{document}

We examine the effect of the hop number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $K$\end{document}, teleport probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha $\end{document}, and head number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $H$\end{document} in the MhAD encoder (Fig. 2(b–d)). Incorporating multi-hop neighbor information (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $K>1$\end{document}) significantly increases performance, as GlaHGCL considers previously overlooked nodes to exploit more global structure, unlike prior aggregation methods. However, we observe diminishing returns for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $K \geq 4$\end{document}. Larger \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha> 0.25$\end{document} values lead to a low-pass effect, while excessively small \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha $\end{document} narrows the focus to large-scale graph structure, both resulting in lower performance. Setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $H$\end{document} to 8 yields the highest scores, as a small number of heads hampers self-attention, while a large number leads to over-smoothing.

Comparison with other baseline methods

Analysis of MDA prediction

To demonstrate the effectiveness of our method in prediction MDAs, we conducted a comparative analysis with seven state-of-the-art methods, including NIMCGCN [34], GAEMDA [35], HGANMDA [36], MINIMDA [37], AMHMDA [38], MTLMDA [12], and GCLMTP [19]. These comparisons are made under consistent experimental conditions and datasets to ensure a fair and objective assessment. NIMCGCN uses GCNs for feature extraction and a neural inductive completion model for association prediction. GAEMDA applies random walks, an MLP, and bilinear decoding for link prediction. HGANMDA constructs a heterogeneous miRNA–disease–lncRNA graph, propagates node information through meta-paths, and uses attention mechanisms and bilinear decoding. MINIMDA leverages high-order neighborhood information for node embeddings and an MLP for association prediction. AMHMDA introduces hypernodes and attention-aware multi-view similarity networks. MTLMDA integrates miRNA–disease and gene-disease sub-networks using multi-task learning. GCLMTP utilizes contrastive learning to learn node embeddings and trains multiple classifiers for MDA prediction.

Table 2 and Fig. 3(a–b) summarize the performance of GlaHGCL and the baseline models. GlaHGCL outperforms other methods, achieving the highest average scores on the evaluation metrics. The superior performance of GlaHGCL demonstrates the benefits of its efficient feature extractor and well-designed self-supervised tasks for learning general representations, followed by supervised fine-tuning for MDA prediction. Compared to the GCN-based NIMCGCN, GlaHGCL’s improved accuracy proves the enhanced aggregation effect of multi-hop attention layers. GlaHGCL also outperforms metapath-based models like GAEMDA and HGAMDA, highlighting the importance of multi-hop diffusion in detecting implicit interactions among indirectly connected nodes. Furthermore, GlaHGCL’s global contrastive learning task enhances the robustness of feature representation by capturing intricate structural interconnections, in contrast to MINIMDA’s reliance on high-order neighborhood information. Lastly, GlaHGCL’s superiority over MTLMDA emphasizes the effectiveness of local consistency constraints with miRNA–gene–disease associations in integrating multiple data sources for improved generalization.

Table 2 Prediction performance (mean\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}standard deviation) for MDA prediction on dataset1 under 5-fold CV.

Method	AUC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	AUPR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	Accuracy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	Precision\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	Recall\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	F1-score\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	
NIMCGCN [34]	0.9172\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0004	0.9186\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0013	0.8438\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0016	0.8162\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0012	0.8886\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0011	0.8504\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0007	
GAEMDA [35]	0.9307\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0044	0.9267\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0065	0.8522\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0095	0.8229\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0198	0.8988\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0117	0.8588\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0071	
HGANMDA [36]	0.9346\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0041	0.9310\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0056	0.8570\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0060	0.8737\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0113	0.8351\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0125	0.8536\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0059	
MINIMDA [37]	0.9444\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0089	0.9296\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0150	0.8797\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0121	0.8545\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0180	0.9160\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0144	0.8839\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0123	
AMHMDA [38]	0.9438\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0062	0.9414\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0043	0.8752\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0087	0.8834\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0083	0.8699\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0209	0.8766\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0109	
MTLMDA [12]	0.9417\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0057	0.9357\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0050	0.8688\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0065	0.8763\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0046	0.8774\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0104	0.8693\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0054	
GCLMTP [19]	0.9366\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0006	0.9393\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0074	0.8665\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0004	0.8820\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0142	0.8462\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0125	0.8637\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0006	
Ours	0.9498\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0039	0.9508\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0044	0.8812\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0084	0.9012\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0100	0.8948\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0086	0.9134\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0091	
Note: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document} means higher is better; The bold areas indicate the highest value for each column metric.

Figure 3 Comparison of ROC and PR curves for MDA, MTI, and DGA prediction on dataset2 under 5-fold CV.

Analysis of MTI prediction

As mentioned previously, the association of a miRNA with multiple diseases can provide valuable insights into its target genes. In this section, we validate the effectiveness of our model in predicting miRNA–target gene interactions, using the same experimental parameters as before.

We compare our model with DeepWalk [39], LINE [40], GraRep [41], SDNE [42], and SG-LSTM [43]. DeepWalk uses random walks to capture local neighborhood structure, while LINE models first-order and second-order proximities. SDNE resorts to semi-supervised learning to model both global and local structure. GraRep is a matrix factorization-based technique that incorporates global structural information. SG-LSTM crafts geometric and sequential embeddings for miRNA–target gene prediction.

Table 3 and Fig. 3(c–d) summarize GlaHGCL’s performance in predicting miRNA–target gene interactions. For dataset1, GlaHGCL achieves an average AUC of 0.9142 and an AUPR of 0.9122; for dataset2, the values are 0.9164 AUC and 0.9055 AUPR. GlaHGCL also demonstrates high accuracy, precision, recall, and F1-score. The superior performance of GlaHGCL can be attributed to its fusion of miRNA and disease-associated gene information, as well as the application of graph contrastive learning. Furthermore, graph neural network-based approaches, including SG-LSTM and GlaHGCL, outperform traditional machine learning methods, highlighting the benefits of integrating node attributes and network topology through graph neural networks.

Table 3 Prediction performance mean\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}standard deviation) for MTI prediction on dataset1 under 5-fold CV.

Method	AUC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	AUPR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	Accuracy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	Precision\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	Recall\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	F1-score\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	
DeepWalk [39]	0.8219\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0020	0.8560\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0020	0.7638\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0030	0.8071\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0029	0.6934\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0037	0.7459\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0029	
LINE [40]	0.8425\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0077	0.8556\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0076	0.7822\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0119	0.7764\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0144	0.7928\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0370	0.7845\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0165	
GraRep [41]	0.8435\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0010	0.8695\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0015	0.7947\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0008	0.8071\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0013	0.7744\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0017	0.7904\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0010	
SDNE [42]	0.8699\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0015	0.8767\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0009	0.8052\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0017	0.8386\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0011	0.7560\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0014	0.7952\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0023	
SG-LSTM [43]	0.8880\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0026	0.8953\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0035	0.8149\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0037	0.8742\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0042	0.7357\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0029	0.7990\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0065	
Ours	0.9142\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0037	0.9122\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0062	0.8333\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0044	0.8542\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0123	0.8039\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0134	0.8283\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0036	
Note: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document} means higher is better. The bold areas indicate the highest value for each column metric.

Analysis of DGA prediction

To comprehensively compare GlaHGCL, we select several baselines for the gene-disease association prediction task: DeepWalk [39], which we introduced earlier for MTI prediction; GCN [44], a semi-supervised Graph Convolutional Network trained on the adjacency matrix; HOGCN [45], which aggregates neighborhood information at varying distances for disease–gene prediction; HNEEM [46], which concatenates disease and gene features from heterogeneous networks and uses a random forest for association prediction; HerGePred [47], which utilizes random walks on heterogeneous networks to capture connectivity patterns between diseases and genes; dgn2vec [48], a random walk-based network embedding algorithm for disease–gene heterogeneous networks. We conduct experiments with the same dataset and parameters as specified for each baseline.

Table 4 and Fig. 3(e–f) present a summary of GlaHGCL’s performance in predicting DGAs. The 5-fold cross-validation average and AUPR of GlaHGCL are reported as 0.9322 and 0.9227, respectively, for dataset1. For dataset2, the corresponding values are 0.9517 for AUC and 0.9450 for AUPR. Experimental results show that GlaHGCL can improve the ability of predicting disease genes, especially for AUC, AUPR, Accuracy, and F1-score, compared to the baseline methods.

Table 4 Prediction performance (mean\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}standard deviation) for DGA prediction on dataset1 under 5-fold CV.

Method	AUC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	AUPR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	Accuracy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	Precision\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	Recall\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	F1-score\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	
DeepWalk [39]	0.8534\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0021	0.8622\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0023	0.7836\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0029	0.7669\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0041	0.8149\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0038	0.7902\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0035	
GCN [44]	0.9158\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0012	0.8889\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0014	0.8393\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0014	0.8347\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0016	0.8462\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0018	0.8404\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0013	
HOGCN [45]	0.9179\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0045	0.9204\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0040	0.8402\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0091	0.8837\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0190	0.7836\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0125	0.8306\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0068	
HNEEM [46]	0.8746\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0037	0.8851\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0038	0.8131\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0051	0.8195\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0142	0.8029\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0072	0.8112\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0005	
HerGePred [47]	0.8848\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0149	0.8892\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0089	0.8122\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0119	0.8445\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0175	0.7652\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0143	0.8029\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0121	
dgn2vec [48]	0.9259\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0040	0.9201\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0049	0.8499\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0070	0.8151\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0094	0.9052\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0117	0.8578\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0117	
Ours	0.9322\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0049	0.9227\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0062	0.8667\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0087	0.8611\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0085	0.8745\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0200	0.8677\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\pm $\end{document}0.0098	
Note: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document} means higher is better. The bold areas indicate the highest value for each column metric.

Independent test analysis

To further evaluate the robustness and generalization ability of the proposed GlaHGCL framework, we conduct a comparative analysis of all methods on independent test sets. For the independent test sets derived from Dataset1 (held-out dataset1) and Dataset2 (held-out dataset2), we train the models on the corresponding cross-validation (CV) sets and then evaluate their performance on the held-out datasets for the three downstream prediction tasks. The model parameters are selected through 5-fold cross-validation on the CV sets, ensuring the training process is independent of the held-out datasets. This approach allows us to assess the prediction performance of the models on unseen data. The results, including AUC, AUPR, and F1-score, are presented in Table 5. Supplementary Figs 1 and 2 provide visualizations of the ROC and PR curves for held-out dataset1 and held-out dataset2, respectively. The results demonstrate that GlaHGCL achieves the best prediction performance, indicating its superior generalization ability.

Table 5 The independent test results of GlaHGCL and other baseline methods.

	Methods	Held-out dataset 1	Held-out dataset 2	Indep. dataset 3	
		AUC	AUPR	F1	AUC	AUPR	F1	AUC	AUPR	F1	
MDA	NIMCGCN	0.9053	0.8964	0.8401	0.9173	0.9246	0.8536	0.7943	0.7713	0.6910	
	GAEMDA	0.8792	0.8799	0.8137	0.9107	0.8984	0.8117	0.8096	0.7784	0.6611	
	HGANMDA	0.9178	0.9228	0.8426	0.9088	0.9215	0.8459	0.7903	0.7901	0.7007	
	MINIMDA	0.9127	0.9162	0.8582	0.9232	0.9225	0.8469	0.7905	0.7610	0.6905	
	AMHMDA	0.9221	0.9205	0.8739	0.9468	0.9431	0.8751	0.8147	0.7748	0.7189	
	MTLMDA	0.9185	0.9196	0.8702	0.9269	0.9231	0.8601	0.8212	0.7912	0.7622	
	GCLMTP	0.9305	0.9298	0.8753	0.9349	0.9368	0.8762	0.8224	0.8020	0.7601	
	Ours	0.9440	0.9400	0.8800	0.9525	0.9460	0.8895	0.8403	0.8175	0.7652	
MTI	DeepWalk	0.7814	0.7634	0.6927	0.8034	0.7960	0.7133	0.6317	0.6647	0.5619	
	LINE	0.8289	0.8148	0.7178	0.8324	0.8273	0.7434	0.6970	0.6699	0.6012	
	GraRep	0.8322	0.7898	0.7694	0.8255	0.7860	0.7647	0.6848	0.6567	0.6236	
	SDNE	0.8314	0.8335	0.7250	0.8400	0.8469	0.7243	0.7196	0.7067	0.6507	
	SG-LSTM	0.8665	0.8629	0.7838	0.8654	0.8678	0.7964	0.7349	0.7084	0.6488	
	Ours	0.8820	0.8788	0.8275	0.8903	0.8841	0.8364	0.7694	0.7419	0.6800	
DGA	DeepWalk	0.8579	0.8492	0.7244	0.8486	0.8539	0.7539	0.6594	0.6842	0.5721	
	GCN	0.8960	0.8952	0.8171	0.9070	0.9048	0.8397	0.7567	0.7346	0.6691	
	HOGCN	0.9107	0.9031	0.8290	0.9107	0.8984	0.8359	0.7993	0.7748	0.7039	
	HNEEM	0.8579	0.8492	0.7650	0.8772	0.8666	0.8017	0.7138	0.6842	0.6532	
	HerGePred	0.8694	0.8780	0.8016	0.8814	0.8812	0.8271	0.6990	0.6511	0.5990	
	dgn2vec	0.9074	0.9050	0.8155	0.9106	0.8986	0.8241	0.7713	0.7346	0.6865	
	Ours	0.9127	0.9162	0.8273	0.9342	0.9291	0.8550	0.8290	0.7966	0.7111	

The Venn diagram in Supplementary Fig. 4 shows 3624 overlapping MDAs between Dataset1 and Dataset2, and no overlapping MTIs or DGAs. We create an independent dataset3 by excluding from Dataset2 any entities present in Dataset1. This independent dataset3 contains all associations present in Dataset2 but not in Dataset1. We then use Dataset1 for training and this independent dataset3 for testing, ensuring an unbiased evaluation on previously unseen data. As demonstrated in Table 5 and Supplementary Fig. 3, GlaHGCL achieves outstanding and stable performance, further emphasizing the importance of integrating multi-source data for robust and accurate prediction.

Ablation experiments

In this section, we conduct ablation experiments to analyze the individual contributions of the self-supervised training framework, the MhAD encoder, the global contrastive learning task, and the local contrastive learning task. We divide the experiments into four groups: (1) training models from scratch (TFS) on the downstream tasks of MDA, MTI, and DGA prediction, using the MhAD encoder to extract node embeddings; (2) validating the effectiveness of the MhAD encoder; (3) examining the necessity of the global contrastive learning objective; and (4) assessing the importance of the local contrastive learning task.

The results in Table 6 reveal several key observations. Adopting the unified self-supervised framework and fine-tuning outperforms training from scratch, indicating improved generalization. The MhAD enhances the capture of valuable contextual information and structural features beyond one-hop connections. Global contrastive learning improves robustness by aligning original and augmented graphs, while local contrastive learning enforces triple-wise association consistency, enhancing representation learning with local semantic information. Additionally, Fig. 4(a–b) shows that a perturbation rate of 25% yields optimal results for GlaHGCL. These insights collectively contribute to the observed improvements in overall performance.

Table 6 The ablation experimental results of GlaHGCL under 5-fold CV.

Framework	MhAD	Task 1	Task 2	MDA	MTI	DGA	
				AUC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	AUPR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	AUC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	AUPR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	AUC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	AUPR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}	
TFS	✓	\	\	0.8860	0.8820	0.8494	0.8471	0.8716	0.8747	
	✗	\	\	0.8736	0.8754	0.8200	0.8225	0.8560	0.8653	
	✗	✓	✓	0.9346	0.9322	0.9004	0.9053	0.9230	0.9208	
SSL	✓	✗	✓	0.9016	0.9081	0.8796	0.8592	0.9176	0.9128	
	✓	✓	✗	0.9320	0.9373	0.8876	0.8987	0.9155	0.9195	
	✓	✓	✓	0.9498	0.9508	0.9142	0.9122	0.9322	0.9227	
Note: ‘TFS’ means the model is trained from scratch, and ‘SSL’ means the model is first trained under self-supervised learning framework and then fine-tuned for downstream tasks. The up arrow (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\uparrow $\end{document}) indicates that higher values indicate better performance. The gray area represents the model that achieved the highest performance.

Figure 4 Performance versus augmentation strength on dataset1 for MDA prediction. The two figures illustrate the impact of edge perturbation with different ratios on the AUC and the AUPR.

Case studies

We conduct case studies to demonstrate the reliability of the GlaHGCL model. The primary focus of these case studies is to identify potential disease-associated miRNAs, disease-related genes, and miRNA-gene interactions. To this end, we use all the known MDAs, MTIs, and DGAs from Dataset1 (Dataset2) (i.e. associations validated by databases) as the training set. We then employ the remaining, unknown miRNA–disease, miRNA–target, and disease–gene pairs from the datasets for prediction. The predicted candidates are sorted in descending order by their association scores, and further analyses are conducted on the findings.

The first case study focuses on identifying potential disease-associated miRNAs. We train the model on the known MDAs in Dataset1. From the remaining unknown miRNA–disease pairs, we predict candidate miRNAs for breast cancer and pancreatic cancer, and rank the top 18 based on their association scores. Breast cancer, a prevalent malignant tumor, has well-established miRNA involvement, such as the downregulation of miR-125b and miR-145 in breast tumor lesions [49]. Pancreatic cancer, known for its low survival rates, has demonstrated associations between pancreatic neoplasms and the expression of miR-21, miR-31, and miR-143 [50]. We validate the findings against four external MDA databases: HMDD3.2, miRCancer [51], dbDEMC3.0 [52], and miR2Diabetes [53]. The results show that all of the top 18 predicted miRNAs are confirmed by at least one of the four databases (see Tables 7 and 8).

Table 7 The top 18 miRNAs associated with breast neoplasms predicted by GlaHGCL.

MiRNA	Evidence	MiRNA	Evidence	MiRNA	Evidence	
hsa-mir-125b	HMDD3.2, dbDEMC3.0, miRCancer	hsa-mir-205	HMDD3.2, dbDEMC3.0	hsa-mir-222	dbDEMC3.0	
hsa-mir-29a	HMDD3.2, dbDEMC3.0, miRCancer	hsa-mir-29b	HMDD3.2, dbDEMC3.0	hsa-mir-34c	dbDEMC3.0, miRCancer	
hsa-mir-21	HMDD3.2, dbDEMC3.0, miRCancer	hsa-mir-223	HMDD3.2, dbDEMC3.0, miRCancer	hsa-mir-24	miRCancer	
hsa-mir-146b	HMDD3.2, dbDEMC3.0, miRCancer	hsa-mir-375	HMDD3.2, dbDEMC3.0	hsa-mir-192	HMDD3.2, dbDEMC3.0	
hsa-mir-155	HMDD3.2, dbDEMC3.0, miRCancer	hsa-mir-34a	HMDD3.2	hsa-mir-150	HMDD3.2, dbDEMC3.0	
hsa-mir-92a	HMDD3.2, dbDEMC3.0	hsa-mir-27a	HMDD3.2	hsa-mir-191	HMDD3.2, dbDEMC3.0	

Table 8 The top 18 miRNAs associated with pancreatic neoplasms predicted by GlaHGCL.

MiRNA	Evidence	MiRNA	Evidence	MiRNA	Evidence	
hsa-mir-155	HMDD3.2, dbDEMC3.0, miRCancer	hsa-mir-92a	HMDD3.2, dbDEMC3.0	hsa-mir-27a	HMDD3.2, dbDEMC3.0	
hsa-mir-145	HMDD3.2, dbDEMC3.0, miRCancer	hsa-mir-34c	HMDD3.2, dbDEMC3.0	hsa-mir-122	miRCancer	
hsa-mir-21	dbDEMC3.0, miRCancer	hsa-mir-31	dbDEMC3.0	hsa-mir-27b	miRCancer	
hsa-mir-222	HMDD3.2	hsa-mir-205	HMDD3.2	hsa-mir-26a	HMDD3.2	
hsa-mir-223	HMDD3.2, dbDEMC3.0, miRCancer	hsa-mir-9	dbDEMC3.0	hsa-mir-196a	HMDD3.2, dbDEMC3.0	
hsa-mir-210	dbDEMC3.0	hsa-mir-133a	dbDEMC3.0	hsa-mir-146b	HMDD3.2, dbDEMC3.0	

The second case study evaluates the model’s ability to identify potential miRNA–target gene interactions. To predict potential miRNA targets or miRNA candidates, we first train the model using all the known MTIs in Dataset2. We then predict on the remaining unknown miRNA-gene pairs, sort the results by descending prediction scores, and manually validate the top 10 candidates using the PubMed literature database. Hsa-miR-122-5p and hsa-miR-335-5p, known oncogenic miRNAs [54,55], are of particular interest. Table 8 presents the top 10 predicted target genes. Literature validation confirms 3 out of the top 10 hsa-miR-122-5p-targeted genes and 5 out of the top 10 hsa-miR-335-5p-targeted genes. For example, hsa-miR-122-5p has been shown to modulate SOCS1 to regulate Type I interferon expression in HCV-infected patients [56]. Experimental evidence supports the activation of the TGF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $\beta $\end{document} signaling pathway by miR-335 through targeting SMAD7 [57]. CDKN1A and BRCA1, crucial genes involved in cellular processes and associated with cancer risk, are also selected for investigation [58,59]. More than half of the top 10 predicted associations (as shown in Table 9) with CDKN1A or BRCA1 have supporting evidence from published papers. Notably, the known MTIs used to train our model were sourced from the comprehensive miRTarBase database, updated in 2022. We then predicted on miRNA-gene pairs from Dataset2 that were not present in miRTarBase to identify potential MTIs. Thus, the number of newly published literature confirming these predicted interactions may be limited.

Table 9 The top 10 candidate taget genes/miRNAs predicted by GlaHGCL for hsa-miR-122-5p, hsa-miR-335-5p, CDKN1A, and BRCA1.

hsa-miR-122-5p	hsa-miR-335-5p	CDKN1A	BRCA1	
Taget gene	Evidence	Taget gene	Evidence	MiRNA	Evidence	MiRNA	Evidence	
SOCS1	PMID:23348614	SMAD7	PMID:31248450	hsa-miR-214	PMID:23100276	hsa-miR-222	PMID:23570906	
HHIP	–	PDIA6	–	hsa-miR-300	PMID:38070141	hsa-miR-214	PMID:25483088	
CHRDL1	–	RASA1	PMID:33767913	hsa-miR-21	–	hsa-miR-150	–	
CTDNEP1	PMID:28287811	VPS33A	–	hsa-miR-423-5p	PMID:32264887	hsa-miR-181a-5p	PMID:23622248	
FOXO3	–	H3F3B	PMID:27347075	hsa-miR-375	–	hsa-miR-520a-3p	–	
IGF1R	–	DUSP4	–	hsa-miR-193a-5p	PMID:33352502	hsa-mir-375	PMID:37291133	
CCNG1	PMID: 30428381	PTEN	PMID:29307835	hsa-miR-92a-3p	PMID:26482648	hsa-miR-191	PMID:20167074	
LDLR	–	BACH1	–	hsa-miR-222	–	hsa-miR-302a-3p	–	
IGF1	–	STAT5A	PMID:32791489	hsa-miR-299-3p	PMID:28600498	hsa-miR-373-3p	–	
XIAP	–	IL6R	–	hsa-miR-608	–	hsa-mir-372	–	

The third case study delves into identifying potential gene targets associated with specific diseases, focusing on Alzheimer’s disease (AD) and Parkinson’s disease (PD). These prevalent neurodegenerative disorders are known to be linked to gene abnormalities or mutations. Table 10 lists the top 10 genes predicted to be associated with AD and PD, supported by relevant literature. For example, the PSEN1 mutations cause Alzheimer’s disease by affecting the metabolism and aggregation of amyloid-beta protein, leading to increased production of beta-amyloid precursor protein and its deposition in the brain [60]. In the context of PD, the LRRK2 mutations cause Parkinson’s disease by impairing cellular processes, such as autophagy and mitochondrial function, leading to the degeneration of dopaminergic neurons in the brain [61].

Table 10 The top 10 related genes of AD and PD predicted by GlaHGCL.

Alzheimer’s disease (AD)	Parkinson’s disease (PD)	
Gene	Evidence	Gene	Evidence	Gene	Evidence	Gene	Evidence	
PSEN1	PMID:33440141	GAB2	PMID:19204163	LRRK2	PMID:33526455	TOMM40	PMID:33737565	
CYP26B1	PMID:31884477	EPB41L5	–	CD33	PMID:33582190	ABCA7	PMID:27066581	
DCP1A	PMID:12605101	SMARCAD1	–	PICALM	PMID:32914893	IQCB1	–	
TAGAP	PMID:16375659	BIN1	PMID:35526014	SORL1	PMID:35364126	TDRD6	–	
CD33	PMID:31301936	ABCA7	PMID:24878767	DLG2	PMID:32652860	PARP12	PMID: 33387580	

The fourth case study aims to evaluate the performance of the GlaHGCL model in predicting novel disease-associated miRNAs for breast cancer and pancreatic cancer. We investigate the interactions between the predicted miRNAs and canonical driver genes (CDGs) to validate the model’s potential in capturing the multifaceted associations between miRNA–gene–disease. The CDGs consist of 591 well-characterized genes demonstrated to play key roles in cancer initiation and progression [62]. The results (Fig. 5a–b) show that in both cancers, the newly predicted disease-associated miRNAs that interact with more CDGs also exhibit higher predicted disease association scores. Specifically, in the breast cancer analysis, the Spearman correlation coefficient between the number of miRNA-CDG interactions and the predicted disease association scores reaches 0.722 (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} $P$\end{document}-value<0.001). Similar trends are observed in pancreatic cancer analysis. In contrast, we find no significant correlation between the predicted disease-associated miRNAs’ interactions with general genes (other than the CDGs) and the disease association scores generated by the model (Supplementary Fig. 5). This suggests that the GlaHGCL model may leverage more complex heterogeneous relationships between miRNAs, genes, and diseases, rather than solely relying on interaction frequencies. This finding provides strong support for the reliability of the GlaHGCL model in predicting MDAs. The positive correlation between miRNA predicted disease association and their interactions with cancer driver genes implies that these newly predicted miRNAs may be involved in key cancer pathways. This lays the foundation for further in-depth exploration of the roles of miRNAs in cancer molecular mechanisms.

Figure 5 Kernel density estimate plot of the distribution of predicted miRNAs. The x-axis represents the rank of a miRNA according to its predicted score, and the y-axis represents the rank of a miRNA according to its interaction number with CDGs. MiRNAs that interact more with CDGs tend to have higher predicted scores (left-bottom) and vice versa (right-top). Distribution plot above the x-axis represents the predicted score of miRNA. Distribution plot on the right of the y-axis represents the number of interactions between the miRNA and CDGs. (a) Results for breast cancer and (b) for pancreatic cancer.

Overall, our study contributes to the expanding knowledge of miRNA-mediated regulation and its impact on disease. However, further research is necessary to validate these findings and unravel the underlying molecular mechanisms driving miRNA–gene–disease interactions.

Conclusion

We propose GlaHGCL, a framework for feature representation learning in a heterogeneous graph of miRNAs, genes, and diseases. Our objective is to derive generic node features for multiple downstream prediction tasks. To enhance the graph, we introduce a perturbation to the adjacency matrix and utilize a MhAD encoder. Through global contrastive learning, we align representations of the original and augmented graphs, promoting robust encodings. To address data sparsity, we incorporate contrasts between local node representations from diverse data sources, leveraging connectivity and contextual nuances for richer representations. In a supervised setting, we refine the node representations for improved performance in predicting MDA, MTI, and DGAs. In conclusion, GlaHGCL is a comprehensive framework that mitigate noise and sparsity using global and local contrastive learning strategies. Experiments show its efficacy, outperforming existing methods. Case studies further validate its potential for discovering new associations.

Despite its success, GlaHGCL still has certain limitations. First, MTIs and MDAs can be context-dependent. The associations predicted by current general and global approach may not be functionally relevant across all tissue or disease contexts. Constructing network graphs based on well-chosen tissue-specific data in future work could help mitigate such context-dependent biases and reduce the number of false positive predictions. Additionally, the local contrastive learning component relies on the assumption that miRNA target genes share functional similarity with disease-associated genes. However, due to the complexity of miRNA regulation, this assumption may not hold true in all cases. While GlaHGCL’s integrated framework helps alleviate this limitation to some degree, integrating additional features, such as miRNA/gene sequence information and protein structural data, could further enhance the model’s ability to identify relevant relationships. Notably, exploring the incorporation of protein 3D structural information is a particularly promising direction, as structural similarity and protein-protein interaction interfaces can provide valuable structural insights to enhance the model’s ability to make more accurate and comprehensive predictions of miRNA–gene–disease associations.

Key Points

We introduce GlaHGCL, which amalgamates three types of nodes and six types of edges into a heterogeneous graph to learn a comprehensive and well-generalizable feature representations. These representations can be adeptly fine-tuned to extend their applicability across a wide spectrum of biomedical applications.

By aligning global representations from both the original and an augmented graph, captured via a multi-hop diffusion encoder, our global contrastive learning strategy significantly mitigates noise within heterogeneous graphs. This method ensures the cultivation of robust representations, enhancing the model’s resilience and reliability in capturing graph features.

GlaHGCL employs local constrastive learning across diverse data sources to address the challenge of sparsity in inter-source connections, meantime fostering consistent, and coherent feature representations across different data sources.

The prowess of GlaHGCL is unequivocally demonstrated through its outstanding performance, surpassing existing state-of-the-art methods in prediction of MDA, MTI, and DGA. Further validation through case studies on disease-related miRNAs underscores its exceptional ability to unearth new potential candidate associations, showcasing its practical relevance and impact in the field.

Supplementary Material

supplementary_materials_bbae443

Funding

This work was supported by the National Key Research and Development Program of China (2023YFE0204200) and the National Natural Science Foundation of China (U20A20387).

Conflict of interest: None declared.

Data availability

The source code and databases are available at https://github.com/Sue-syx/GlaHGCL.
==== Refs
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