{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T03:02:13Z","timestamp":1769828533183,"version":"3.49.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T00:00:00Z","timestamp":1765756800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 62172299"],"award-info":[{"award-number":["No. 62172299"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 62172299"],"award-info":[{"award-number":["No. 62172299"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003399","name":"Shanghai Science and Technology Committee","doi-asserted-by":"crossref","award":["No. 22511105500"],"award-info":[{"award-number":["No. 22511105500"]}],"id":[{"id":"10.13039\/501100003399","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s10994-025-06944-5","type":"journal-article","created":{"date-parts":[[2025,12,15]],"date-time":"2025-12-15T22:29:06Z","timestamp":1765837746000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Enhancing Low-Degree Graph Neural Networks via Joint Training and Improved Message Passing"],"prefix":"10.1007","volume":"115","author":[{"given":"Zedong","family":"Sun","sequence":"first","affiliation":[]},{"given":"Jian","family":"Li","sequence":"additional","affiliation":[]},{"given":"Guanjun","family":"Liu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,12,15]]},"reference":[{"key":"6944_CR1","unstructured":"Ali, A., Wolf, L., Cevikalp, H. (2024). Degree-based stratification of nodes in graph neural networks. In: Proceedings of the Asian Conference on Machine Learning (ACML), pp. 15\u201327."},{"key":"6944_CR2","unstructured":"Balcilar, M., H\u00e9roux, P., Gauzere, B., Vasseur, P., Adam, S., Honeine, P. (2021). Breaking the limits of message passing graph neural networks. In: Proceedings of the International Conference on Machine Learning (ICML), pp. 599\u2013608."},{"key":"6944_CR3","first-page":"19314","volume":"33","author":"Y Chen","year":"2020","unstructured":"Chen, Y., Wu, L., & Zaki, M. (2020). Iterative deep graph learning for graph neural networks: Better and robust node embeddings. Advances in Neural Information Processing Systems, 33, 19314\u201319326.","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"6","key":"6944_CR4","doi-asserted-by":"publisher","first-page":"2001","DOI":"10.3934\/dcds.2024155","volume":"45","author":"M Fazly","year":"2025","unstructured":"Fazly, M., Wei, J., & Yang, W. (2025). Classification of finite morse index solutions of higher-order gelfand-liouville equation. Discrete and Continuous Dynamical Systems, 45(6), 2001\u20132044.","journal-title":"Discrete and Continuous Dynamical Systems"},{"key":"6944_CR5","unstructured":"Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., Dahl, G. E. (2017). Neural message passing for quantum chemistry. In: Proceedings of the International Conference on Machine Learning (ICML), pp. 1263\u20131272."},{"key":"6944_CR6","unstructured":"Hamilton, W., Ying, Z., Leskovec, J. (2017). Inductive representation learning on large graphs. Advances in Neural Information Processing Systems 30."},{"key":"6944_CR7","doi-asserted-by":"publisher","DOI":"10.1016\/j.egyai.2022.100201","volume":"10","author":"X Han","year":"2022","unstructured":"Han, X., Jia, M., Chang, Y., Li, Y., & Wu, S. (2022). Directed message passing neural network (d-mpnn) with graph edge attention (gea) for property prediction of biofuel-relevant species. Energy and AI, 10, Article 100201.","journal-title":"Energy and AI"},{"key":"6944_CR8","first-page":"22118","volume":"33","author":"W Hu","year":"2020","unstructured":"Hu, W., Fey, M., Zitnik, M., Dong, Y., Ren, H., Liu, B., Catasta, M., & Leskovec, J. (2020). Open graph benchmark: Datasets for machine learning on graphs. Advances in Neural Information Processing Systems, 33, 22118\u201322133.","journal-title":"Advances in Neural Information Processing Systems"},{"key":"6944_CR9","doi-asserted-by":"crossref","unstructured":"Jin, W., Derr, T., Wang, Y., Ma, Y., Liu, Z., Tang, J. (2021). Node similarity preserving graph convolutional networks. In: Proceedings of the 14th ACM International Conference on Web Search and Data Mining (WSDM), pp. 148\u2013156.","DOI":"10.1145\/3437963.3441735"},{"key":"6944_CR10","doi-asserted-by":"crossref","unstructured":"Jin, W., Ma, Y., Liu, X., Tang, X., Wang, S., Tang, J. (2020). Graph structure learning for robust graph neural networks. In: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining (KDD), pp. 66\u201374.","DOI":"10.1145\/3394486.3403049"},{"key":"6944_CR11","unstructured":"Ju, M., Zhao, T., Yu, W., Shah, N., Ye, Y. (2024). Graphpatcher: mitigating degree bias for graph neural networks via test-time augmentation. Advances in Neural Information Processing Systems, 36"},{"key":"6944_CR12","unstructured":"Kipf, T. N., Welling, M. (2017). Semi-supervised classification with graph convolutional networks. In: Proceedings of the International Conference on Learning Representations (ICLR) ."},{"key":"6944_CR13","doi-asserted-by":"crossref","unstructured":"Li, Q., Han, Z., Wu, X.-M. (2018). Deeper insights into graph convolutional networks for semi-supervised learning. In: Proceedings of the AAAI Conference on Artificial Intelligence (AAAI), vol. 32.","DOI":"10.1609\/aaai.v32i1.11604"},{"key":"6944_CR14","doi-asserted-by":"crossref","unstructured":"Li, G., Muller, M., Thabet, A., Ghanem, B. (2019). Deepgcns: Can gcns go as deep as cnns? In: Proceedings of the IEEE\/CVF International Conference on Computer Vision (ICCV), pp. 9267\u20139276.","DOI":"10.1109\/ICCV.2019.00936"},{"key":"6944_CR15","doi-asserted-by":"crossref","unstructured":"Liu, Z., Nguyen, T.-K., Fang, Y. (2021). Tail-gnn: Tail-node graph neural networks. In: Proceedings of the 27th ACM SIGKDD Conference On Knowledge Discovery & Data Mining (KDD), pp. 1109\u20131119.","DOI":"10.1145\/3447548.3467276"},{"issue":"9","key":"6944_CR16","doi-asserted-by":"publisher","first-page":"2624","DOI":"10.1109\/JPROC.2012.2197809","volume":"100","author":"W Liu","year":"2012","unstructured":"Liu, W., Wang, J., & Chang, S.-F. (2012). Robust and scalable graph-based semisupervised learning. Proceedings of the IEEE, 100(9), 2624\u20132638.","journal-title":"Proceedings of the IEEE"},{"issue":"4","key":"6944_CR17","doi-asserted-by":"publisher","first-page":"824","DOI":"10.1109\/TPAMI.2018.2889473","volume":"42","author":"YA Malkov","year":"2018","unstructured":"Malkov, Y. A., & Yashunin, D. A. (2018). Efficient and robust approximate nearest neighbor search using hierarchical navigable small world graphs. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(4), 824\u2013836.","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"issue":"2","key":"6944_CR18","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1137\/S003614450342480","volume":"45","author":"ME Newman","year":"2003","unstructured":"Newman, M. E. (2003). The structure and function of complex networks. SIAM Review, 45(2), 167\u2013256.","journal-title":"SIAM Review"},{"key":"6944_CR19","doi-asserted-by":"crossref","unstructured":"Peng, H., Gurevin, D., Huang, S., Geng, T., Jiang, W., Khan, O., Ding, C. (2022). Towards sparsification of graph neural networks. In: Proceedings of the 40th IEEE International Conference on Computer Design (ICCD), pp. 272\u2013279.","DOI":"10.1109\/ICCD56317.2022.00048"},{"key":"6944_CR20","unstructured":"Rangesh, A., Maheshwari, P., Gebre, M., Mhatre, S., Ramezani, V., Trivedi, M. M. (2021). Trackmpnn: A message passing graph neural architecture for multi-object tracking. arXiv:2101.04206."},{"key":"6944_CR21","doi-asserted-by":"crossref","unstructured":"Schlichtkrull, M., Kipf, T.N., Bloem, P., Van Den\u00a0Berg, R., Titov, I., Welling, M. (2018). Modeling relational data with graph convolutional networks. In: Proceedings of the 15th International Conference on the Semantic Web (ESWC), pp. 593\u2013607.","DOI":"10.1007\/978-3-319-93417-4_38"},{"key":"6944_CR22","doi-asserted-by":"crossref","unstructured":"Sun, J., Xie, Y., Zhang, H., Faloutsos, C. (2007). Less is more: Compact matrix decomposition for large sparse graphs. In: Proceedings of the 2007 SIAM International Conference on Data Mining, pp. 366\u2013377.","DOI":"10.1137\/1.9781611972771.33"},{"key":"6944_CR23","unstructured":"Swanson, K. (2019). Message passing neural networks for molecular property prediction. PhD thesis, Massachusetts Institute of Technology. https:\/\/dspace.mit.edu\/handle\/1721.1\/122721."},{"key":"6944_CR24","doi-asserted-by":"crossref","unstructured":"Tang, X., Yao, H., Sun, Y., Wang, Y., Tang, J., Aggarwal, C., Mitra, P., Wang, S. (2020). Investigating and mitigating degree-related biases in graph convoltuional networks. In: Proceedings of the 29th ACM International Conference on Information and Knowledge Management (CIKM), pp. 1435\u20131444.","DOI":"10.1145\/3340531.3411872"},{"key":"6944_CR25","first-page":"4","volume":"1050","author":"P Velickovic","year":"2018","unstructured":"Velickovic, P., Cucurull, G., Casanova, A., Romero, A., Lio, P., & Bengio, Y. (2018). Graph attention networks. Stat, 1050, 4.","journal-title":"Stat"},{"key":"6944_CR26","doi-asserted-by":"crossref","unstructured":"Wang, R., Mou, S., Wang, X., Xiao, W., Ju, Q., Shi, C., Xie, X. (2021). Graph structure estimation neural networks. In: Proceedings of the Web Conference 2021 (WWW), pp. 342\u2013353.","DOI":"10.1145\/3442381.3449952"},{"issue":"2","key":"6944_CR27","doi-asserted-by":"publisher","first-page":"758","DOI":"10.1109\/TBDATA.2022.3172060","volume":"9","author":"C Wang","year":"2022","unstructured":"Wang, C., Zhu, H., Hu, R., Li, R., & Jiang, C. (2022). Longarms: Fraud prediction in online lending services using sparse knowledge graph. IEEE Transactions on Big Data, 9(2), 758\u2013772.","journal-title":"IEEE Transactions on Big Data"},{"key":"6944_CR28","doi-asserted-by":"crossref","unstructured":"Ying, R., He, R., Chen, K., Eksombatchai, P., Hamilton, W.L., Leskovec, J. (2018). Graph convolutional neural networks for web-scale recommender systems. In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), pp. 974\u2013983.","DOI":"10.1145\/3219819.3219890"},{"key":"6944_CR29","unstructured":"Zhang, M., Chen, Y. (2018). Link prediction based on graph neural networks. Advances in Neural Information Processing Systems, 31."},{"key":"6944_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.ces.2022.117624","volume":"254","author":"J Zhang","year":"2022","unstructured":"Zhang, J., Wang, Q., & Shen, W. (2022). Message-passing neural network based multi-task deep-learning framework for cosmo-sac based $$\\sigma $$-profile and vcosmo prediction. Chemical Engineering Science, 254, Article 117624.","journal-title":"Chemical Engineering Science"},{"key":"6944_CR31","doi-asserted-by":"crossref","unstructured":"Zhao, Z., Li, J., Liu, G. (2025). Automated graph contrastive learning based on node-level and edge-level learnable augmentation. IEEE Transactions on Computational Social Systems.","DOI":"10.1109\/TCSS.2025.3551250"},{"key":"6944_CR32","unstructured":"Zheng, W., Huang, E. W., Rao, N., Katariya, S., Wang, Z., Subbian, K. (2021). Cold brew: Distilling graph node representations with incomplete or missing neighborhoods. In: Proceedings of the International Conference on Learning Representations (ICLR)."},{"key":"6944_CR33","doi-asserted-by":"crossref","unstructured":"Zhong, H., Wu, J., Chen, C., Huang, J., Deng, M., Nie, L., Lin, Z., Hua, X.-S. (2021). Graph contrastive clustering. In: Proceedings of the IEEE\/CVF International Conference on Computer Vision (ICCV), pp. 9224\u20139233.","DOI":"10.1109\/ICCV48922.2021.00909"},{"issue":"3","key":"6944_CR34","doi-asserted-by":"publisher","first-page":"824","DOI":"10.1109\/TNET.2014.2306416","volume":"23","author":"X Zhou","year":"2014","unstructured":"Zhou, X., Zhang, Z., Wang, G., Yu, X., Zhao, B. Y., & Zheng, H. (2014). Practical conflict graphs in the wild. IEEE\/ACM Transactions on Networking, 23(3), 824\u2013835.","journal-title":"IEEE\/ACM Transactions on Networking"}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06944-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10994-025-06944-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06944-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T14:05:52Z","timestamp":1769781952000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10994-025-06944-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,15]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["6944"],"URL":"https:\/\/doi.org\/10.1007\/s10994-025-06944-5","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"value":"0885-6125","type":"print"},{"value":"1573-0565","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,15]]},"assertion":[{"value":"20 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 October 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 November 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 December 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"5"}}