{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:39:40Z","timestamp":1759970380386,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,13]],"date-time":"2025-01-13T00:00:00Z","timestamp":1736726400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100013317","name":"Key Research and Development Program of Shanxi Province","doi-asserted-by":"publisher","award":["202102090301006","2024ZKPYZN01"],"award-info":[{"award-number":["202102090301006","2024ZKPYZN01"]}],"id":[{"id":"10.13039\/501100013317","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["202102090301006","2024ZKPYZN01"],"award-info":[{"award-number":["202102090301006","2024ZKPYZN01"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Graph Neural Network (GNN) is an effective model for processing graph-structured data. Most GNNs are designed to solve homophilic graphs, where all nodes belong to the same category. However, graph data in real-world applications are mostly heterophilic, and homophilic GNNs cannot handle them well. To address this, we propose a novel hybrid-learning framework based on Node Homophily and Spectral Heterophily (NHSH) for node classification in graph networks. NHSH is designed to achieve state-of-the-art or superior performance on both homophilic and heterophilic graphs. It includes three core modules: homophilic node extraction (HNE), heterophilic spectrum extraction (HSE) and node feature fusion (NFF). More specifically, HNE identifies symmetric neighborhoods of nodes with the same category, extracting local features that reflect these symmetrical structures. Then, HSE uses filters to analyze the high and low-frequency information of nodes in the graph and extract the global features of the nodes. Finally, NFF fuses the above two node features to obtain the final node features in graphs. Moreover, an elaborate loss function drives the network to preserve critical symmetries and structural patterns in the graph. Experiments on eight benchmark datasets validate that NHSH performs comparably or better than existing methods across diverse graph types.<\/jats:p>","DOI":"10.3390\/sym17010115","type":"journal-article","created":{"date-parts":[[2025,1,13]],"date-time":"2025-01-13T07:40:08Z","timestamp":1736754008000},"page":"115","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["NHSH: Graph Hybrid Learning with Node Homophily and Spectral Heterophily for Node Classification"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8413-123X","authenticated-orcid":false,"given":"Kang","family":"Liu","sequence":"first","affiliation":[{"name":"School of Artificial Intelligence, China University of Mining and Technology-Beijing, Beijing 100083, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-3357-3485","authenticated-orcid":false,"given":"Wenqing","family":"Dai","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, China University of Mining and Technology-Beijing, Beijing 100083, China"}]},{"given":"Xunyuan","family":"Liu","sequence":"additional","affiliation":[{"name":"Lab of Intelligent Social Computing, University of International Relations, Beijing 100091, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-1344-1657","authenticated-orcid":false,"given":"Mengtao","family":"Kang","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, China University of Mining and Technology-Beijing, Beijing 100083, China"}]},{"given":"Runshi","family":"Ji","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, China University of Mining and Technology-Beijing, Beijing 100083, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Pan, C.H., Qu, Y., Yao, Y., and Wang, M.J.S. (2024). HybridGNN: A Self-Supervised Graph Neural Network for Efficient Maximum Matching in Bipartite Graphs. Symmetry, 16.","DOI":"10.20944\/preprints202410.1354.v1"},{"key":"ref_2","unstructured":"Kipf, T.N., and Welling, M. (2016). Semi-supervised classification with graph convolutional networks. arXiv."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"12358","DOI":"10.1109\/TNNLS.2023.3257325","article-title":"Homophily-enhanced self-supervision for graph structure learning: Insights and directions","volume":"35","author":"Wu","year":"2023","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"476","DOI":"10.3390\/analytics3040027","article-title":"NPI-WGNN: A Weighted Graph Neural Network Leveraging Centrality Measures and High-Order Common Neighbor Similarity for Accurate ncRNA\u2013Protein Interaction Prediction","volume":"3","author":"Khoushehgir","year":"2024","journal-title":"Analytics"},{"key":"ref_5","unstructured":"Hamilton, W., Ying, Z., and Leskovec, J. (2017). Inductive representation learning on large graphs. Adv. Neural Inf. Process. Syst., 30."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Wang, Y., Xiang, S., and Pan, C. (2023, January 10\u201314). Improving the Homophily of Heterophilic Graphs for Semi-Supervised Node Classification. Proceedings of the 2023 IEEE International Conference on Multimedia and Expo (ICME), Brisbane, Australia.","DOI":"10.1109\/ICME55011.2023.00320"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"11634","DOI":"10.1109\/TNNLS.2024.3370918","article-title":"Permutation equivariant graph framelets for Heterophilous graph learning","volume":"35","author":"Li","year":"2024","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Chen, R. (2024). Preserving Global Information for Graph Clustering with Masked Autoencoders. Mathematics, 12.","DOI":"10.3390\/math12101574"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Huang, W., Guan, X., and Liu, D. (2023). Revisiting homophily ratio: A relation-aware Graph Neural Network for homophily and heterophily. Electronics, 12.","DOI":"10.3390\/electronics12041017"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"34421","DOI":"10.1109\/ACCESS.2023.3264596","article-title":"Multi-Duplicated Characterization Of Graph Structures Using Information Gain Ratio For Graph Neural Networks","volume":"11","author":"Oishi","year":"2023","journal-title":"IEEE Access"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Park, H.S., and Park, H.M. (2024, January 18\u201321). Enhancing Heterophilic Graph Neural Network Performance Through Label Propagation in K-Nearest Neighbor Graphs. Proceedings of the 2024 IEEE International Conference on Big Data and Smart Computing (BigComp), Bangkok, Thailand.","DOI":"10.1109\/BigComp60711.2024.00060"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Guan, X., Wang, D., Xiong, C., Li, S., and Chen, Y. (2022, January 21\u201325). PBGAN: Path Based Graph Attention Network for Heterophily. Proceedings of the 2022 26th International Conference on Pattern Recognition (ICPR), Montreal, QC, Canada.","DOI":"10.1109\/ICPR56361.2022.9956658"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Sun, J., Zhang, L., Zhao, S., and Yang, Y. (December, January 28). Improving your graph neural networks: A high-frequency booster. Proceedings of the 2022 IEEE International Conference on Data Mining Workshops (ICDMW), Orlando, FL, USA.","DOI":"10.1109\/ICDMW58026.2022.00102"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1109\/TKDE.2023.3303212","article-title":"Enhancing Locally Adaptive Smoothing of Graph Neural Networks Via Laplacian Node Disagreement","volume":"36","author":"Wang","year":"2023","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Gu, M., Yang, G., Zhou, S., Ma, N., Chen, J., Tan, Q., and Bu, J. (2023, January 21\u201325). Homophily-enhanced structure learning for graph clustering. Proceedings of the 32nd ACM International Conference on Information and Knowledge Management, Birmingham, UK.","DOI":"10.1145\/3583780.3614915"},{"key":"ref_16","unstructured":"Bruna, J., Zaremba, W., Szlam, A., and LeCun, Y. (2013). Spectral networks and locally connected networks on graphs. arXiv."},{"key":"ref_17","unstructured":"Wu, F., Souza, A., Zhang, T., Fifty, C., Yu, T., and Weinberger, K. (2019, January 9\u201315). Simplifying graph convolutional networks. Proceedings of the International Conference on Machine Learning, PMLR, Long Beach, CA, USA."},{"key":"ref_18","unstructured":"Lingam, V., Ragesh, R., Iyer, A., and Sellamanickam, S. (2021). Simple truncated svd based model for node classification on heterophilic graphs. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Xu, B., Shen, H., Cao, Q., Cen, K., and Cheng, X. (2020). Graph convolutional networks using heat kernel for semi-supervised learning. arXiv.","DOI":"10.24963\/ijcai.2019\/267"},{"key":"ref_20","unstructured":"Wang, X., and Zhang, M. (2022, January 17\u201323). How powerful are spectral graph neural networks. Proceedings of the International Conference on Machine Learning, PMLR, Baltimore, MD, USA."},{"key":"ref_21","first-page":"14239","article-title":"Bernnet: Learning arbitrary graph spectral filters via bernstein approximation","volume":"34","author":"He","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_22","unstructured":"Chien, E., Peng, J., Li, P., and Milenkovic, O. (2020). Adaptive universal generalized pagerank graph neural network. arXiv."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Awasthi, A.K., Garov, A.K., Sharma, M., and Sinha, M. (2023, January 27\u201329). GNN model based on node classification forecasting in social network. Proceedings of the 2023 International Conference on Artificial Intelligence and Smart Communication (AISC), Greater Noida, India.","DOI":"10.1109\/AISC56616.2023.10085118"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"3982","DOI":"10.1109\/TETCI.2024.3380481","article-title":"Node Classification in Weighted Complex Networks Using Neighborhood Feature Similarity","volume":"8","author":"Shetty","year":"2024","journal-title":"IEEE Trans. Emerg. Top. Comput. Intell."},{"key":"ref_25","unstructured":"Veli\u010dkovi\u0107, P., Cucurull, G., Casanova, A., Romero, A., Lio, P., and Bengio, Y. (2017). Graph attention networks. arXiv."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Pan, J., Lin, H., Dong, Y., Wang, Y., and Ji, Y. (2022). MAMF-GCN: Multi-scale adaptive multi-channel fusion deep graph convolutional network for predicting mental disorder. Comput. Biol. Med., 148.","DOI":"10.1016\/j.compbiomed.2022.105823"},{"key":"ref_27","unstructured":"Gasteiger, J., Bojchevski, A., and G\u00fcnnemann, S. (2018). Predict then propagate: Graph neural networks meet personalized pagerank. arXiv."},{"key":"ref_28","unstructured":"Pei, H., Wei, B., Chang, K.C.C., Lei, Y., and Yang, B. (2020). Geom-gcn: Geometric graph convolutional networks. arXiv."},{"key":"ref_29","unstructured":"Abu-El-Haija, S., Perozzi, B., Kapoor, A., Alipourfard, N., Lerman, K., Harutyunyan, H., and Galstyan, A. (2019, January 9\u201315). Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. Proceedings of the international Conference on Machine Learning, PMLR, Long Beach, CA, USA."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Roy, K.K., Roy, A., Rahman, A.M., Amin, M.A., and Ali, A.A. (2021, January 18\u201322). Node embedding using mutual information and self-supervision based bi-level aggregation. Proceedings of the 2021 International Joint Conference on Neural Networks (IJCNN), Shenzhen, China.","DOI":"10.1109\/IJCNN52387.2021.9533715"},{"key":"ref_31","first-page":"11168","article-title":"Graph neural networks with heterophily","volume":"35","author":"Zhu","year":"2021","journal-title":"Proc. Aaai Conf. Artif. Intell."},{"key":"ref_32","unstructured":"Ma, Y., Liu, X., Shah, N., and Tang, J. (2021). Is homophily a necessity for graph neural networks?. arXiv."},{"key":"ref_33","first-page":"3438","article-title":"Measuring and relieving the over-smoothing problem for graph neural networks from the topological view","volume":"34","author":"Chen","year":"2020","journal-title":"Proc. Aaai Conf. Artif. Intell."},{"key":"ref_34","first-page":"3950","article-title":"Beyond low-frequency information in graph convolutional networks","volume":"35","author":"Bo","year":"2021","journal-title":"Proc. Aaai Conf. Artif. Intell."},{"key":"ref_35","unstructured":"Brody, S., Alon, U., and Yahav, E. (2021). How attentive are graph attention networks?. arXiv."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Chen, Y., Luo, Y., Tang, J., Yang, L., Qiu, S., Wang, C., and Cao, X. (2023). LSGNN: Towards general graph neural network in node classification by local similarity. arXiv.","DOI":"10.24963\/ijcai.2023\/395"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Zheng, X., Zhang, M., Chen, C., Zhang, Q., Zhou, C., and Pan, S. (May, January 30). Auto-heg: Automated graph neural network on heterophilic graphs. Proceedings of the ACM Web Conference 2023, Austin, TX, USA.","DOI":"10.1145\/3543507.3583498"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Taud, H., and Mas, J.F. (2018). Multilayer perceptron (MLP). Geomatic Approaches for Modeling Land Change Scenarios, Springer.","DOI":"10.1007\/978-3-319-60801-3_27"},{"key":"ref_39","unstructured":"Chen, M., Wei, Z., Huang, Z., Ding, B., and Li, Y. (2020, January 13\u201318). Simple and deep graph convolutional networks. Proceedings of the International Conference on Machine Learning, PMLR, Virtual."},{"key":"ref_40","unstructured":"Zheng, Y., Zhang, H., Lee, V., Zheng, Y., Wang, X., and Pan, S. (2023, January 23\u201329). Finding the missing-half: Graph complementary learning for homophily-prone and heterophily-prone graphs. Proceedings of the International Conference on Machine Learning, PMLR, Honolulu, HI, USA."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/115\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:27:51Z","timestamp":1759919271000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/115"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,13]]},"references-count":40,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010115"],"URL":"https:\/\/doi.org\/10.3390\/sym17010115","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,1,13]]}}}