TF-DWGNet: a directed weighted graph neural network with tensor fusion for multi-omics cancer subtype classification.

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Learn more: PMC Disclaimer | PMC Copyright Notice NAR Genom Bioinform . 2026 May 30;8(2):lqag054. doi: 10.1093/nargab/lqag054 TF-DWGNet: a directed weighted graph neural network with tensor fusion for multi-omics cancer subtype classification Tiantian Yang 1, 2, ✉ , Zhiqian Chen 3 Author information Article notes Copyright and License information 1 Department of Mathematics and Statistical Science, University of Idaho, 875 Perimeter Drive, Moscow, ID 83844, United States 2 Department of Bioinformatics and Computational Biology, University of Idaho, 875 Perimeter Drive, Moscow, ID 83844, United States 3 Department of Computer Science and Engineering, Mississippi State University, 665 George Perry Street, Starkville, MS 39762, United States ✉ To whom correspondence should be addressed. Email: tyang@uidaho.edu Roles Tiantian Yang : Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Software, Supervision, Validation, Visualization, Writing - original draft, Writing - review & editing Zhiqian Chen : Investigation, Methodology, Validation, Writing - review & editing Received 2025 Dec 16; Revised 2026 Apr 18; Accepted 2026 May 4; Collection date 2026 Jun. © The Author(s) 2026. Published by Oxford University Press. 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 reprints@oup.com for reprints and translation rights for reprints. All other permissions can be obtained through our RightsLink service via the Permissions link on the article page on our site—for further information please contact journals.permissions@oup.com PMC Copyright notice PMCID: PMC13221656  PMID: 42221877 Previous version available: This article is based on a previously available preprint posted on ArXiv on February 11, 2026: " TF-DWGNet: A Directed Weighted Graph Neural Network with Tensor Fusion for Multi-Omics Cancer Subtype Classification ". Abstract Integration and analysis of multi-omics data provide valuable insights for improving cancer subtype classification. However, such data are inherently heterogeneous, high-dimensional, and exhibit complex intra- and inter-modality dependencies. Graph neural networks provide a principled framework for modeling these structures, but existing approaches often rely on prior knowledge or predefined similarity networks that produce either undirected or unweighted graphs, failing to capture task-specific directionality and interaction strengths. Interpretability at both the modality and feature levels also remains limited. To address these challenges, we propose “TF-DWGNet,” a novel G raph Neural Net work framework that combines tree-based D irected W eighted graph construction with T ensor F usion for multiclass cancer subtype classification. TF-DWGNet introduces two key methodological innovations: (i) a supervised tree-based strategy that constructs directed weighted graphs tailored to each omics modality, and (ii) a tensor fusion mechanism that captures unimodal, bimodal, and trimodal interactions using low-rank decomposition for computational efficiency. Experiments on three real-world cancer datasets demonstrate that TF-DWGNet consistently outperforms state-of-the-art baselines across multiple metrics and statistical tests. In addition, the model provides biologically meaningful insights through modality level contribution scores and ranked feature importance. These results highlight that TF-DWGNet is an effective and interpretable solution for multi-omics integration in cancer research. Introduction Cancer remains a major global health challenge, marked by uncontrolled cell growth, genetic instability, and complex molecular alterations that vary across cancer types and among individual patients. This biological heterogeneity underscores the importance of accurate cancer subtype classification, which is central to precision medicine. Subtype classification enables personalized treatment strategies by leveraging individual molecular profiles, yet reliable prediction remains difficult due to the complexity and multilayered nature of tumor biology. Advances in high-throughput technologies now enable simultaneous profiling of multiple omics layers, such as DNA methylation, mRNA (messenger RNA) expression, and microRNA (miRNA) expression, providing a comprehensive molecular view of tumors [ 1 , 2 ]. These modalities offer complementary biological signals that, when jointly analyzed, can reveal disease mechanisms not captured by single-omics analysis. However, multi-omics data integration remains challenging due to high dimensionality, modality heterogeneity, and complex cross-modal interactions [ 3 ]. Addressing these issues requires computational frameworks that are capable of modeling hierarchical and directional biological processes while preserving model interpretability. Classical learning approaches such as random forest (RF) [ 4 ], eXtreme Gradient Boosting (XGBoost) [ 5 ], and deep feedforward networks (DFNs) [ 6 , 7 ] have been widely applied to omics data, but their Euclidean assumptions make it difficult to capture relational structures, handle severe feature dimensionality, or provide task-relevant interpretability [ 8 , 9 ]. Geometric deep learning, particularly graph neural networks (GNNs), provides a principled solution by modeling non-Euclidean structures such as biological networks [ 10 , 11 ]. GNNs learn node representations by aggregating information from graph neighborhoods [ 12 , 13 ], and architectures such as graph convolutional networks (GCNs) [ 14 ], graph attention networks (GATs) [ 15 ], graph isomorphism networks [ 16 ], and GraphSAGE [ 17 ] have demonstrated strong performance in biomedical tasks [ 18 , 19 ]. Meanwhile, multimodal fusion strategies, such as encoder–decoder designs, attention-based methods, and tensor-based models, aim to address heterogeneity across modalities [ 3 , 20 ], although many do not fully capture higher-order interactions. Several GNN-based frameworks have been proposed for multi-omics integration in cancer subtype classification and survival prediction. For example, methods such as MOGONET [ 21 ] employ GCNs to process multiple omics modalities and incorporate a cross-omics discovery tensor and a view correlation discovery network for the final prediction. Architectures such as MODILM [ 22 ] extend these ideas using graph attention mechanisms to strengthen intra-omics modeling. Other frameworks, including SUPREME [ 23 ] and MOGAT [ 24 ], highlight patient similarity networks, GCNs or GATs, and basic concatenation to integrate multiple omics types. Additional approaches, such as DeepMoIC [ 25 ] and MoGCN [ 26 ], leverage autoencoder-based feature extraction and similarity network fusion before applying GNN modules. These approaches differ in terms of how they construct graphs, fuse multimodal information, and ensure interpretability. Despite these advances, existing frameworks exhibit several key limitations: (i) suboptimal graph construction: most rely on predefined or correlation-based similarity networks that are undirected, unweighted, and insensitive to task-specific directionality and heterogeneous interaction strengths, both of which are essential for modeling biological regulatory flow; (ii) incomplete multi-omics integration: many approaches use simple concatenation or pairwise modeling and therefore fail to capture higher-order (unimodal, bimodal, trimodal) interactions that may reveal synergistic mechanisms among omics layers; and (iii) limited interpretability: interpretability often depends on post hoc analyses, such as feature ablation, attention inspection, or gradient-based methods, which may not reflect the model’s true decision-making process and can be computationally intensive. To address these challenges, we propose TF-DWGNet, a novel GNN-based framework for interpretable multi-omics integration and cancer subtype classification. The key novelties of TF-DWGNet lie in the integration of supervised directed weighted graph construction and low-rank tensor fusion, which together enable task-adaptive feature interaction modeling and efficient higher-order cross-omics learning. The primary contributions of this work are as follows: Supervised directed weighted graph construction: we leverage XGBoost’s tree-splitting structure to construct biologically meaningful, task-specific, and variable-sized (i.e. data-adaptively determined) directed weighted graphs that encode both directional influence and interaction strength while performing data-driven feature reduction. Tensor fusion for unimodal, bimodal, and trimodal interactions: we implement a low-rank tensor fusion module that efficiently captures higher-order cross-model interactions across all omics types, offering richer representational capacity than concatenation or pairwise fusion. Built-in interpretability at both feature and modality levels: TF-DWGNet inherently produces feature-importance rankings and modality contribution scores without requiring post hoc analyses. End-to-end supervised learning and modality flexibility: the framework operates end-to-end, from raw omics inputs and labels to final subtype predictions, and can naturally accommodate additional omics modalities through higher-order tensor fusion and low-rank canonical polyadic (CP) decomposition without requiring architectural modifications. To the best of our knowledge, TF-DWGNet is the first framework to jointly integrate supervised directed weighted graph construction, comprehensive low-rank tensor fusion, modality-flexible GNN encoding, and built-in interpretability for multi-omics cancer subtype classification. Methodology Problem definition Let denote the input feature matrices for three omics modalities, where with samples and modality-specific features. Let denote the corresponding multiclass subtype labels, where for classes. For each modality, we train an XGBoost classifier using as the target. From the same trained model, we simultaneously (i) select informative features, resulting in reduced datasets with , and (ii) construct a directed weighted feature graph based on tree-splitting dependencies. Here, denotes the set of selected feature nodes and denotes directed edges inferred from the root-to-leaf ordering of features. The corresponding adjacency matrix is , where encodes the presence (or strength) of an edge from node to node . We augment each graph with self-loops, giving , where is the identity matrix. The degree matrix is defined as a diagonal matrix with entries . The normalized adjacency matrix is . Given the dataset , the objective is to learn modality-specific graphs and a predictive function mapping to . Formally, we aim to learn minimizing the classification loss between the true labels and predicted labels . Supplementary Table S1 summarizes the key notations and symbols used throughout the manuscript. Framework overview Figure 1 illustrates the proposed “TF-DWGNet framework” ( T ensor F usion on D irected W eighted G raph Net work), which integrates four coordinated modules for effective multimodal representation learning: (i) an “XGBoost module” independently processes each omics modality, performs feature selection, and constructs task-specific directed weighted graphs that capture feature importance and directional relationships; (ii) a “GNN module” learns latent unimodal embeddings from each modality by jointly encoding the reduced feature matrix and its corresponding directed weighted graph; (iii) a “tensor fusion module” models all unimodal, bimodal, and trimodal interactions across modalities, followed by low-rank CP decomposition for computational efficiency; and (iv) a “Deep Residual Network module” takes the compact-fused embedding as input and performs multiclass classification using a residual architecture. Algorithm 1 summarizes the workflow. Overall, TF-DWGNet forms a fully supervised, end-to-end pipeline that takes raw multi-omics data and subtype labels as inputs, constructs supervised graphs, learns modality-specific embeddings, performs tensor fusion, and outputs subtype predictions under a unified training objective. Figure 1. Open in a new tab Overview of the TF-DWGNet framework, consisting of four key modules: (i) an XGBoost module for feature selection and supervised construction of directed weighted graphs within each omics modality; (ii) a GNN module that learns unimodal embeddings by jointly encoding graph topology and reduced feature matrices; (iii) a tensor fusion module that models unimodal, bimodal, and trimodal interactions, followed by low-rank CP decomposition; and (iv) a deep residual network for classification. All modules are trained only on the training split; validation and test sets use the learned graphs and parameters without information leakage. TF-DWGNet forms a fully supervised, end-to-end pipeline from raw multi-omics inputs to subtype predictions. Directed weighted graph construction We leverage XGBoost’s sequential boosting mechanism to capture task-specific feature dependencies and directional interaction patterns in multi-omics data. Unlike ensemble methods such as RF, which construct trees independently, XGBoost builds trees through successive gradient-based updates to the objective function, with each iteration refining the structure not captured by previous trees. This iterative refinement encourages the model to repeatedly focus on discriminative feature splits, leading to task-adaptive and stable feature interaction patterns when aggregated across the boosted ensemble. For each omics modality, an ensemble of boosted trees is trained. Each tree is viewed as a directed graph , where a directed edge indicates that feature appears before along the same root-to-leaf path. These edges do not represent causal biological relationships. Instead, they encode task-specific conditional dependencies among features: feature partitions patients into subgroups for which feature provides additional discriminative power. Aggregating the feature co-occurrence across all trees yields a single directed weighted graph as follows: where contains all selected features across trees and represents directional dependencies, with edge weights equal to their frequencies across trees. Graph sparsity arises naturally from the supervised construction process. Because XGBoost repeatedly emphasizes discriminative feature splits across the boosted ensemble, the resulting graphs concentrate edges among informative features rather than forming dense connections. Increasing the number of trees beyond the default setting (100) produces only moderate changes in graph sparsity, with diminishing effects as the number increases, as additional trees largely reinforce previously identified feature interactions. Denoting as the number of uniquely selected features, we obtain . This procedure is applied independently to each omics modality, resulting in three directed weighted graphs: , , and , with corresponding reduced feature matrices , , and , where and the reduced dimensions may differ across modalities. We refer to these XGBoost-selected subsets as reduced features. These graphs encode both feature importance and directional structure, providing informative, task-conditioned priors for downstream learning. Graph neural network Each modality is then processed using a modified graph-embedded deep feedforward networks (GEDFN) [ 27 ], which incorporates adjacency structure by restricting connections between the input and the first hidden layer. For the -th modality, let be the feature-reduced input and be the directed weighted adjacency matrix with self-loops. The first hidden layer is computed as follows: where has the same feature dimension as the reduced input , ; and denote trainable weights and biases between the input layer and the first hidden layer , respectively; denotes element-wise (Hadamard) multiplication; and is the degree matrix. Normalization with prevents the highly connected nodes from dominating the representation. By embedding directed weighted feature graphs directly into the first layer, this design captures topological dependencies while preserving architectural sparsity, which is an important consideration for modeling high-dimensional multi-omics data. Tensor fusion and decomposition Given the unimodal latent representations obtained from GNNs for samples, we employ a tensor fusion network(TFN) [ 28 ] to model both intra- and inter-modal interactions across the three omics modalities. To enable the unimodal, bimodal, and trimodal interaction terms, each latent representation is augmented with a constant feature: where , and the appended 1s facilitate the interaction terms involving fewer than three modalities. For each sample, the TFN constructs a third-order tensor using the outer product: This tensor encodes a complete set of intra- and inter-modality interactions, including all lower- and higher-order combinations: Unimodal terms ( , , and ): elements involving only one modality Bimodal terms ( , , and ): pairwise multiplicative interactions Trimodal term ( ): three-way interactions across all modalities These correspond to the seven structured regions illustrated in the tensor fusion module shown in Fig. 1 . Although expressive, this tensor scales as , which quickly becomes computationally intractable as the latent dimensions increase. To address this challenge, we apply a CP decomposition (also called CANDECOMP/PARAFAC) [ 29 – 31 ], which approximates the third-order tensor by the sum of rank-one tensors: where is a positive integer, , and for modalities . This formulation generalizes matrix singular value decomposition to higher-order tensors [ 32 ] and reduces the computational complexity to . Definition 1 (Rank-One Tensors). An -way tensor is rank-one if it can be written as the outer product of vectors: Definition 2 (Tensor Rank). The rank of a tensor is the smallest number of rank-one tensors that sum to . The maximum rank is defined as the largest attainable rank. Proposition 1. For , the maximum tensor rank satisfies [ 32 , 33 ] Determining the exact tensor rank is NP-hard in general [ 34 ], so is selected by fitting various rank- CP models and evaluating the performance empirically, consistent with prior tensor literature [ 30 , 31 ]. Low-rank fusion with modality-specific projections To support parameter sharing across samples and maintain computational efficiency, we introduce modality-specific projection matrices, as follows: where . This yields the compact CP-based fusion representation: where denotes the CP operator. Each term projects the augmented latent vector into rank- space. The element-wise product across modalities yields a fused tensor representation of for each sample. Novelty Compared to conventional fusion approaches that rely on simple concatenation, full tensor expansion, or low-capacity bilinear interactions, our tensor fusion module models high-order cross-omics interactions through low-rank CP decomposition retains modality-specific projections that capture shared and unique biological structure avoids the computational burden of full tensor expansion Extension to additional omics modalities The TF-DWGNet framework naturally generalizes to an arbitrary number of modalities. For omics datasets with unimodal embeddings , we construct augmented embeddings with . The tensor fusion module forms a -th order interaction tensor The same low-rank CP decomposition can then be applied: In practice, the modality-specific projection matrices map each augmented embedding into a shared rank- space before fusion, enabling efficient parameter sharing across modalities. Because the CP decomposition scales with the chosen rank and grows linearly with the number of modalities, the framework avoids the exponential growth associated with full tensor expansion. It remains computationally tractable as additional omics layers are incorporated. Modeling intra- and inter-omics dependencies In TF-DWGNet, dependencies are modeled through two complementary components. Within each modality, directed weighted graphs constructed via XGBoost capture structured, task-relevant relationships among features. Across modalities, interactions are modeled through the tensor fusion module, which operates on modality-specific embeddings and represents unimodal, bimodal, and trimodal interactions via outer products followed by low-rank CP decomposition. This design allows modality-specific graphs to encode intra-omics dependencies, while the tensor fusion module provides a unified mechanism to model both within-modality effects and higher-order cross-omics interactions without requiring a predefined cross-omics graph. Residual classification network The low-rank fused embedding produced by CP decomposition is then passed to a residual multilayer perceptron [ 35 ] for final subtype prediction. Each residual block is implemented as either an “identity block” or a “projection block.” In an identity block, the input and output dimensions match, allowing the original input to be added directly to the transformed output before applying the activation function. This yields a skip connection of the form: where denotes the block input, is the learnable weight matrix, and is the activation function (bias terms are omitted for simplicity). When the input and output dimensions differ, a projection block linearly transforms the input to match the hidden width before addition. In our architecture, the hidden width is fixed across all layers; therefore, only the first block requires projection and all subsequent blocks are identity blocks. This design stabilizes the optimization and enables deeper and more expressive feature transformations. Feature-level and modality-level interpretability To provide biologically meaningful interpretability at both the input feature (biomarker) and modality (omics) levels, enabling identification of influential biomarkers and quantifying the relative contribution of different omics sources, we develop feature- and modality-level importance measures derived directly from the learned model parameters. (1) Feature-level importance: for each omics modality, feature importance is derived from the connection weights between the input layer and the first hidden layer of the corresponding GNN. This definition builds upon connection weight-based importance measures [ 27 , 36 , 37 ], and we extend them to directed weighted feature graphs by explicitly decomposing each feature’s contribution into source and target roles under graph constraints. Since the feature graphs are directed, a feature can participate either as a source or a target node. For example, in edge , feature contributes as a source, while in , it contributes as a target. For feature in modality , its importance score is defined as follows: where indexes the selected features. For completeness, we assign to features not selected by XGBoost. Here, is the associated weight matrix and is the directed weighted adjacency matrix (with self-loops) for omics modality . (2) Modality-level importance: to quantify the relative importance of each omics source and understand which molecular layers drive subtype discrimination, we aggregate the feature-level scores within each modality. Specifically, the model dynamically estimates the relative importance of omics by where and . This yields a normalized measure of how strongly each omics modality contributes to the final classification, enabling transparent and biologically interpretable insights without requiring additional post hoc analysis or ablations. Technical details XGBoost is implemented using the xgboost Python library, and all GNN and DFN components are trained in TensorFlow . Models are optimized using mini-batch training with the Adam optimizer, with ReLU as the activation function throughout and softmax in the output layer for multiclass classification. Training minimizes the categorical cross-entropy loss combined with dynamic regularization applied to all trainable weights from the three GNN branches and the DFN with residual blocks: where is the true label; is the predicted probability; and is the regularization strength. We tune the number of estimators for XGBoost and optimize the architectural and training hyperparameters, including depth, hidden width, learning rate, batch size, and maximum epochs, for both the GNN and DFN components. Dropout, early stopping, and batch normalization are applied to mitigate overfitting in high-dimensional settings. Model evaluation is performed over 20 independent stratified training–validation–test splits (60%:20%:20%), with each run using a fixed random seed and different seeds across runs. These strategies act complementarily: dropout and regularization constrain model complexity during training, early stopping prevents excessive fitting to the training data, and repeated stratified splits provide a more robust assessment of generalization performance across different data partitions in high-dimensional multi-omics settings. In addition, we tune the tensor fusion rank and the number of residual blocks in the DFN. Unless otherwise specified, all components of TF-DWGNet fit exclusively on the training split. For each random split, XGBoost is trained on only the training samples, and directed weighted graphs are constructed from the tree-splitting paths on this data. The validation and test samples are then passed through the trained trees to obtain their reduced-feature representations, while the graph structure remains fixed within each split. This prevents any leakage of test information into the graph construction stage. All experiments are run on the Falcon supercomputer [ 38 ], with each job requesting four CPUs and 20 GB of memory. Real data experiments Datasets overview We evaluate TF-DWGNet on three widely used multi-omics cancer subtype datasets: BRCA, UCEC, and KIPAN. The processed BRCA dataset is obtained from Wang et al. [ 21 ], while the original UCEC and KIPAN datasets are downloaded from The Cancer Genome Atlas (TCGA) via the Broad GDAC Firehose and preprocessed following the procedure in Lu et al . [ 39 ]. BReast CArcinoma (BRCA) is used for PAM50 subtype classification with five molecular subtypes. Uterine corpus endometrial carcinoma (UCEC) contains three clinically relevant subgroups. KIPAN, the TCGA PAN-KIdney cohort, includes three kidney cancer subtypes. Table 1 summarizes the sample sizes, number of classes, class distributions, and feature dimensions before and after preprocessing. All datasets contain matched patient-level measurements across three omics modalities: DNA methylation, mRNA expression, and miRNA expression. Only samples with complete measurements in all modalities are retained for analysis. The datasets display varying degrees of class imbalance, including severely underrepresented subtypes such as HER2-enriched in BRCA and MSEAC in UCEC, offering a realistic and challenging evaluation setting. Although all datasets originate from TCGA, they represent biologically distinct cancer types with different imbalance structures and molecular characteristics. Figure 2 shows histograms for DNA methylation, mRNA, and miRNA features, illustrating substantial heterogeneity across datasets and modalities. These datasets also reflect the typical small- , large- regime of multi-omics studies, where a few hundred patients are measured on tens of thousands of features, making robust graph learning, fusion, and regularization essential. Table 1. Summary of BRCA, UCEC, and KIPAN datasets, including total sample size ( ), number of classes ( ), samples per class ( ), , original feature counts, and preprocessed feature dimensions for each omics modality Dataset Samples per class Original features Preprocessed features (meth:mRNA:miRNA) (meth:mRNA:miRNA) BRCA 875 5 Normal-like (115), basal-like (131), HER2-enriched (46), luminal A (436), luminal B (147) 20,531:20,106:503 1000:1000:503 UCEC 430 3 EEA (311), SEA (98), MSEAC (21) 20,118:20,531:554 2000:2000:554 KIPAN 707 3 KICH (66), KIPC (343), KIRP (298) 20,531:20,111:472 2000:2000:472 Open in a new tab BRCA = breast carcinoma; UCEC = uterine corpus endometrial carcinoma; and KIPAN = TCGA Pan-kidney cohort. Figure 2. Open in a new tab Histograms of DNA methylation, mRNA, and miRNA feature distributions in the preprocessed BRCA, UCEC, and KIPAN datasets. All features are scaled to the range [0, 1]. The heterogeneous distributions across modalities and datasets highlight the distinct biological characteristics captured by each omics type. Baselines, ablation studies, and experimental setting We compare TF-DWGNet with several conventional and deep learning baselines, including RF, XGBoost, DFN, GCN, and GEDFN, as well as “three ablation variants” designed to isolate the effects of graph construction, fusion strategy, and directionality: (i) replaces XGBoost with RF for graph construction, examining the effect of the tree-based estimator; (ii) replaces tensor fusion with direct concatenation, assessing the contribution of tensor modeling and CP decomposition; and (iii) symmetrizes the adjacency matrix to remove directional information, evaluating the importance of task-conditioned directionality. All baselines are trained on column-wise concatenated multi-omics features, unless the model inherently incorporates graph structure. For fairness, both GCN and GEDFN use XGBoost-derived graphs, consistent with TF-DWGNet. All models are evaluated using the same 20 stratified training, validation, and test splits (60%/20%/20%), where each run uses a fixed random seed, with different seeds across runs to ensure robustness. For tree-based baselines, XGBoost and RF use 100 estimators. DFN, GCN, and GEDFN each use two hidden layers of width 64. TF-DWGNet employs one graph-embedded hidden layer of width fixed to the reduced feature dimension in each GNN branch, and three residual blocks in the DFN module, each with three hidden layers of width 64. The default TF-DWGNet model uses 100 trees in the XGBoost-based graph-construction module. The tensor fusion module uses a CP rank of . All models are trained using a learning rate of 0.0001, a batch size of 64, and a maximum of 1000 training epochs. Regularization strategies include dropout (0.5), L2 regularization ( ), and early stopping (patience = 10 epochs; minimum validation-loss improvement = 0.001). Table 2 summarizes the hyperparameter search space and the selected optimal values. Table 2. Hyperparameter tuning ranges for TF-DWGNet on the BRCA, UCEC, and KIPAN datasets Hyperparameter Tuning range XGBoost parameters Number of trees ( ) 50, 100 , 200, 500, 1000 GNN architecture Depth (hidden layers) 1 , 2, 3 Width (neurons per layer) fixed to (feature-aligned) Activation function ReLU , leakyReLU Training parameters Learning rate 0.001, 0.0005, 0.0001 Batch size 8, 16, 32, 64 Training epochs 500, 1000 , 1500 Regularization and early stopping Dropout rate 0.2, 0.3, 0.5 L2 regularization lambda 0.01 Early stopping patience 5, 10 Early stopping min delta 0.001 , 0.005 Low-rank decomposition Rank ( ) 4, 8, 16, 24, 30, 32, 40, 48 , 50, 64 Deep residual network architecture Number of blocks 1, 2, 3 , 5 Depth per block 1, 2, 3 Width (neurons per layer) 32, 64 , 128 Open in a new tab The optimal choices selected for the final model are shown in bold. Evaluation metrics We evaluate the model performance using accuracy, F1-weighted, and F1-macro scores. F1-macro computes the F1-score for each class independently and averages them equally, providing a balanced view of performance across all classes. In contrast, the F1-weighted score computes a weighted average of per-class F1-scores, with weights proportional to class frequencies, making it more sensitive to class imbalance. For a classification problem with classes, the generalized F1-scores are computed as where The class weights follow for F1-macro and for F1-weighted, with the number of samples in class . We report the mean, standard deviation, and 95% confidence intervals (CIs) over 20 independent training-validation-test splits. To assess statistical significance, we also perform pairwise Welch’s -tests between TF-DWGNet and each comparator model, which account for unequal variances across runs. Results and discussion Constructed graph properties Table 3 summarizes the reduced feature dimensions for each omics modality ( ), which correspond to the number of nodes in the constructed directed weighted graphs. The results are reported using the default setting of XGBoost trees. The resulting feature sets range from roughly 150 to 900 nodes, reflecting adaptive compression across datasets. The table also reports edge counts , which includes self-loops. The learned graphs remain relatively sparse, with edge-to-node ratios ranging between two and nine, while being sufficiently connected to capture modality-specific feature dependencies. These properties enable effective message passing in downstream GNNs while maintaining computational efficiency. Table 3. Reduced features (nodes), edge counts ( ), and edge-to-node ratio ( ) for graphs constructed by TF-DWGNet on BRCA, UCEC, and KIPAN using the default setting of trees in XGBoost Dataset Nodes Edges Edge/node ( : : ) ( : : ) ( : : ) BRCA 878:825:494 4300:3658:4176 4.90:4.43:8.45 UCEC 381:336:272 882:763:794 2.31:2.27:2.92 KIPAN 311:244:155 749:552:437 2.41:2.26:2.82 Open in a new tab For each omics type , denotes the number of reduced features, the number of edges, and the edge-to-node ratio. Across the three datasets, the edge-to-node ratios remain within a similar range despite differences in feature dimensionality and class distributions, indicating that the supervised graph construction yields consistent task-driven feature interaction structures. In practice, a well-constructed graph should remain relatively sparse while preserving connectivity among informative features: overly dense graphs may introduce noisy interactions, whereas excessively sparse graphs may fail to capture meaningful dependencies among biomarkers. The evaluated datasets also reflect typical high-dimensional multi-omics settings with hundreds of samples and tens of thousands of candidate biomarkers before feature selection, indicating that TF-DWGNet can operate effectively in small- , large- regimes commonly encountered in multi-omics studies. As shown in Supplementary Tables S2 and S3 , we conducted a sensitivity analysis to assess the effect of graph construction parameters on sparsity and predictive performance by varying the number of XGBoost trees ( ). Specifically, Supplementary Table S2 summarizes the resulting graph properties, showing that increasing generally leads to denser graphs, particularly for BRCA, while graph structures for UCEC and KIPAN remain largely stable. Supplementary Table S3 reports the corresponding predictive performance, demonstrating that Accuracy, F1-weighted, and F1-macro remain consistently stable across all datasets despite variations in graph density. These results provide clear empirical evidence that TF-DWGNet is robust to moderate variations in graph sparsity. Furthermore, these findings support the use of as the default setting, providing a good balance among graph sparsity, interpretability, and predictive performance. Classification performance Table 4 presents the classification performance of TF-DWGNet, its three ablation variants, and all baselines on BRCA, UCEC, and KIPAN. We report accuracy, F1-weighted, and F1-macro scores, with the best results shown in bold. Overall, TF-DWGNet consistently achieves the strongest performance across all datasets and metrics, demonstrating robustness and predictive strength in heterogeneous multi-omics settings. On BRCA, TF-DWGNet achieves substantial gains, including more than an 8% increase in accuracy relative to the weakest baseline and even larger improvements in F1-macro and F1-weighted (~12% and 10%, respectively). These gains highlight the model’s ability to handle subtype imbalance and capture both intra- and inter-modal structure. All ablation variants underperform the full model, though to varying degrees: shows the largest degradation, reflecting the importance of XGBoost-based graph construction; performs significantly worse on accuracy and F1-weighted, underscoring the benefit of modeling directionality; and trails by a smaller margin, but TF-DWGNet still attains the highest means, suggesting additional benefit from tensor-based feature integration. On UCEC and KIPAN, TF-DWGNet again achieves the best performance across all metrics, with consistent but smaller margins of 1%–3%. The ablation variants remain competitive but uniformly below the full model, confirming that each architectural component contributes to predictive performance. To assess statistical significance, we perform pairwise Welch’s -tests based on 20 independent runs per model, comparing TF-DWGNet with each baseline and ablation. Superscripts in Table 4 indicate the corresponding significance levels. For BRCA, nearly all improvements are highly significant ( P -value .001), and the confidence intervals for TF-DWGNet lie distinctly above those of competing models. Two ablation variants ( and ) also differ significantly from TF-DWGNet, highlighting the added value of directional information and XGBoost-based graph construction. On UCEC, several differences with classical baselines remain significant, while none of the ablation variants differ significantly from the full model. For KIPAN, a few differences are significant, consistent with all models achieving strong performance on this simpler task. Table 4. Classification performance comparison on BRCA, UCEC, and KIPAN Dataset Model Accuracy F1-weighted F1-macro BRCA RF 0.747 ± 0.026 [0.735, 0.759] 0.736 ± 0.029 [0.722, 0.750] 0.671 ± 0.036 [0.654, 0.688] XGBoost 0.742 ± 0.021 [0.732, 0.752] 0.731 ± 0.024 [0.720, 0.742] 0.664 ± 0.041 [0.645, 0.683] DFN 0.751 ± 0.021 [0.741, 0.761] 0.750 ± 0.021 [0.740, 0.760] 0.705 ± 0.033 [0.689, 0.721] GCN 0.752 ± 0.028 [0.739, 0.765] 0.751 ± 0.029 [0.737, 0.765] 0.708 ± 0.043 [0.688, 0.728] GEDFN 0.790 ± 0.025 [0.778, 0.801] 0.792 ± 0.024 [0.781, 0.804] 0.748 ± 0.032 [0.733, 0.763] 0.789 ± 0.033 [0.774, 0.804] 0.791 ± 0.032 [0.776, 0.806] 0.748 ± 0.038 [0.730, 0.766] 0.806 ± 0.031 [0.791, 0.821] 0.814 ± 0.030 [0.800, 0.828] 0.782 ± 0.040 [0.763, 0.801] 0.800 ± 0.030 [0.786, 0.814] 0.804 ± 0.029 [0.790, 0.818] 0.765 ± 0.039 [0.747, 0.783] TF-DWGNet 0.821 ± 0.023 [0.810, 0.832] 0.826 ± 0.022 [0.816, 0.836] 0.785 ± 0.033 [0.769, 0.801] UCEC RF 0.852 ± 0.022 [0.842, 0.862] 0.824 ± 0.025 [0.812, 0.836] 0.544 ± 0.024 [0.533, 0.555] XGBoost 0.860 ± 0.026 [0.848, 0.872] 0.839 ± 0.026 [0.827, 0.851] 0.559 ± 0.022 [0.549, 0.569] DFN 0.842 ± 0.034 [0.826, 0.858] 0.837 ± 0.031 [0.822, 0.852] 0.566 ± 0.026 [0.554, 0.578] GCN 0.837 ± 0.038 [0.819, 0.855] 0.833 ± 0.035 [0.817, 0.849] 0.563 ± 0.033 [0.548, 0.578] GEDFN 0.859 ± 0.024 [0.848, 0.871] 0.848 ± 0.022 [0.838, 0.858] 0.571 ± 0.018 [0.562, 0.579] 0.859 ± 0.031 [0.844, 0.874] 0.846 ± 0.027 [0.833, 0.859] 0.574 ± 0.020 [0.565, 0.583] 0.864 ± 0.031 [0.849, 0.879] 0.853 ± 0.030 [0.839, 0.867] 0.587 ± 0.045 [0.566, 0.608] 0.866 ± 0.031 [0.851, 0.881] 0.853 ± 0.027 [0.840, 0.866] 0.578 ± 0.023 [0.567, 0.589] TF-DWGNet 0.869 ± 0.038 [0.851, 0.887] 0.858 ± 0.034 [0.842, 0.874] 0.598 ± 0.056 [0.572, 0.624] KIPAN RF 0.953 ± 0.023 [0.942, 0.964] 0.953 ± 0.023 [0.942, 0.964] 0.943 ± 0.033 [0.928, 0.958] XGBoost 0.952 ± 0.023 [0.941, 0.963] 0.952 ± 0.023 [0.941, 0.963] 0.942 ± 0.034 [0.926, 0.958] DFN 0.957 ± 0.017 [0.949, 0.965] 0.957 ± 0.017 [0.949, 0.965] 0.945 ± 0.024 [0.934, 0.956] GCN 0.955 ± 0.023 [0.944, 0.966] 0.955 ± 0.023 [0.944, 0.966] 0.945 ± 0.030 [0.931, 0.959] GEDFN 0.957 ± 0.017 [0.949, 0.965] 0.957 ± 0.017 [0.949, 0.965] 0.946 ± 0.025 [0.934, 0.957] 0.957 ± 0.020 [0.948, 0.966] 0.957 ± 0.020 [0.948, 0.966] 0.945 ± 0.024 [0.934, 0.956] 0.960 ± 0.016 [0.953, 0.967] 0.960 ± 0.016 [0.953, 0.967] 0.947 ± 0.024 [0.936, 0.958] 0.958 ± 0.020 [0.949, 0.967] 0.959 ± 0.019 [0.950, 0.968] 0.947 ± 0.026 [0.935, 0.959] TF-DWGNet 0.964 ± 0.017 [0.956, 0.972] 0.964 ± 0.017 [0.956, 0.972] 0.956 ± 0.024 [0.945, 0.967] Open in a new tab Metrics are reported as mean ± standard deviation (95% confidence interval) over 20 reproducible seeds. Best results are bolded. Superscripts , , and indicate that TF-DWGNet is statistically significantly better than the corresponding model ( P -values .05, .01, and .001, respectively) based on Welch’s -test. Note that KIPAN is included as a simpler, proof-of-concept dataset due to its relatively easy classification structure for all models. Figure 3 visualizes the distribution of TF-DWGNet and the baseline performance across 20 BRCA splits. TF-DWGNet achieves the highest median for all metrics and exhibits tight distributions centered near the median, indicating high stability. Most of TF-DWGNet’s individual run scores exceed the maximums achieved by other baselines, demonstrating that it outperforms them not only on average but also across nearly every test split. Figure 3. Open in a new tab Classification performance comparison of TF-DWGNet and baseline models across 20 independent seeds on the BRCA dataset using violin plots. Each violin displays score distributions, individual values (dots), and medians (white diamonds). Results are shown for accuracy ( A ), F1-weighted ( B ), and F1-macro ( C ). Feature interpretability Each input feature (biomarker) receives a ranking score after training. While the absolute magnitude of the score is not directly interpretable, the relative ranking provides meaningful insight into each feature’s contribution to subtype discrimination. High-ranking biomarkers may indicate strong statistical associations or potential mechanistic links to specific cancer types. To complement these rankings, we examine the global distribution of feature importance scores across datasets and modalities (see Supplementary Figs S1 – S3 ). The distributions are consistently right-skewed, indicating that the predictive signal is concentrated on a small subset of informative biomarkers. This pattern suggests that the model does not rely on diffuse or noisy contributions, but instead identifies a stable set of highly informative biomarkers that drive subtype discrimination. Consistent with this pattern, several top-ranked features in BRCA align with established cancer biology. For example, FAM134B , an ER-phagy receptor associated with hypoxic breast cancer survival, has been shown to suppress tumor proliferation when silenced [ 40 ]. Likewise, FOXC1 , commonly overexpressed in basal-like breast cancer (e.g., BRCA1-mutant tumors), promotes proliferation and increases sensitivity to PARP inhibitors [ 41 ]. Such concordance supports TF-DWGNet’s ability to highlight biologically meaningful, subtype-relevant markers. Although our interpretability validation focuses on agreement with known biomarkers, a more systematic analysis, such as pathway enrichment or comparisons with curated biological knowledge, remains outside the scope of this study. Accordingly, the identified biomarkers should be interpreted as data-driven candidates rather than confirmed mechanistic drivers. The observed concordance with known biomarkers provides supportive evidence of biological relevance, while comprehensive biological validation is beyond the methodological focus of this work. TF-DWGNet is designed to provide statistically grounded, biologically plausible signals that can guide downstream investigations. In practice, groups of highly ranked biomarkers may reveal coordinated pathway activity or shared regulatory mechanisms that warrant further follow-up. Modality interpretability Figure 4 shows the relative contribution of each omics modality to TF-DWGNet predictions, derived from the aggregated feature-importance scores. The model adaptively reweights each modality based on its predictive utility. In BRCA, DNA methylation, mRNA expression, and miRNA expression contribute at comparable levels, suggesting complementary and non-redundant information across modalities. In contrast, for UCEC and KIPAN, DNA methylation emerges as the dominant source of predictive signal, while miRNA plays a comparatively smaller role. These patterns demonstrate TF-DWGNet’s ability to integrate heterogeneous omic sources in a data-driven manner, allowing the model to emphasize biologically relevant modalities without manual weighting or hand-engineered assumptions. While our analysis emphasizes statistical interpretability, validating modality-level contributions against known molecular mechanisms or pathway-level evidence remains an important direction for future work. The supervised graph construction mechanism is designed to capture task-relevant dependencies, providing a principled basis for the learned modality contributions and directional interactions. Figure 4. Open in a new tab Relative importance of different omics types (DNA methylation, mRNA, and miRNA) estimated by TF-DWGNet across BRCA, UCEC, and KIPAN. Each bar reflects the normalized contribution of an omics graph, automatically learned by the model. Conclusion We propose TF-DWGNet, a novel GNN framework for multi-omics cancer subtype classification that integrates directed weighted graph learning with tensor-based feature fusion in an end-to-end supervised architecture. By jointly modeling modality-specific graph structures and higher-order cross-omics interactions, TF-DWGNet provides a unified and expressive approach to capture the complexities inherent in heterogeneous omics data. The framework is also modality-flexible, enabling the incorporation of additional omics types through higher-order tensor fusion and low-rank CP decomposition and supporting future extensions to broader multi-omics settings. Comprehensive experiments across three real-world cancer datasets demonstrate that TF-DWGNet consistently outperforms or matches strong classical and deep learning baselines across all major evaluation metrics. These results highlight the model’s robustness, predictive accuracy, and generalization ability across cancers with distinct molecular architectures and varying degrees of class imbalance. Beyond improved predictive performance, TF-DWGNet provides interpretable outputs, including modality-level contribution scores and biomarker-level importance rankings. These insights help reveal potential molecular drivers and dataset-specific patterns, offering interpretability essential for downstream biological investigations and clinical hypothesis generation. Overall, TF-DWGNet offers a powerful and interpretable solution for multi-omics integration, highlighting how task-adaptive graph learning and structured tensor fusion can jointly improve predictive modeling and biological interpretability. Supplementary Material lqag054_Supplemental_File lqag054_supplemental_file.pdf (421.2KB, pdf) Acknowledgements Author contributions : Tiantian Yang (Data curation [lead] Formal analysis [lead] Funding acquisition [lead] Investigation [lead] Methodology [lead] Software [lead] Supervision [lead] Validation [lead] Visualization [lead] Writing – original draft [lead] Writing – review & editing [lead]) and Zhiqian Chen (Investigation [supporting] Methodology [supporting] Validation [supporting] Writing – review & editing [supporting]) Contributor Information Tiantian Yang, Department of Mathematics and Statistical Science, University of Idaho, 875 Perimeter Drive, Moscow, ID 83844, United States; Department of Bioinformatics and Computational Biology, University of Idaho, 875 Perimeter Drive, Moscow, ID 83844, United States. Zhiqian Chen, Department of Computer Science and Engineering, Mississippi State University, 665 George Perry Street, Starkville, MS 39762, United States. Supplementary data Supplementary data is available at NAR Genomics & Bioinformatics online. Conflict of interest None declared. Funding This work was partially supported by the National Institute of General Medical Sciences of the National Institutes of Health (NIH/NIGMS) under Award No. P20GM104420. Data availability The original datasets used in this study are available from the TCGA Broad GDAC Firehose portal ( https://gdac.broadinstitute.org/ ). The preprocessing procedure follows Lu et al . [ 39 ]. The harmonized datasets used in our experiments (BRCA, UCEC, and KIPAN), along with the implementation and code of TF-DWGNet, are publicly available at Zenodo: https://doi.org/10.5281/zenodo.20044016 . The GitHub repository ( https://github.com/tiantiy/TF-DWGNet ) is also maintained for ongoing updates. References 1. Subramanian  I, Verma  S, Kumar  S  et al.  Multi-omics data integration, interpretation, and its application. 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Oncotarget. 2016;7:75729. 10.18632/oncotarget.12370 [ DOI ] [ PMC free article ] [ PubMed ] [ Google Scholar ] Associated Data This section collects any data citations, data availability statements, or supplementary materials included in this article. Supplementary Materials lqag054_Supplemental_File lqag054_supplemental_file.pdf (421.2KB, pdf) Data Availability Statement The original datasets used in this study are available from the TCGA Broad GDAC Firehose portal ( https://gdac.broadinstitute.org/ ). The preprocessing procedure follows Lu et al . [ 39 ]. The harmonized datasets used in our experiments (BRCA, UCEC, and KIPAN), along with the implementation and code of TF-DWGNet, are publicly available at Zenodo: https://doi.org/10.5281/zenodo.20044016 . The GitHub repository ( https://github.com/tiantiy/TF-DWGNet ) is also maintained for ongoing updates. 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