{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T20:56:52Z","timestamp":1760648212608,"version":"3.44.0"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T00:00:00Z","timestamp":1748390400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T00:00:00Z","timestamp":1748390400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100006447","name":"University of Zurich","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006447","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2025,7]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Signed graphs allow for encoding positive and negative relations between nodes and are used to model various online activities. Node representation learning for signed graphs is a well-studied task with important applications such as sign prediction. While the size of datasets is ever-increasing, recent methods often sacrifice scalability for accuracy. We propose a novel message-passing layer architecture called graph spring network (GSN) modeled after spring forces. We combine it with a graph neural ordinary differential equations (ODEs) formalism to optimize the system dynamics in embedding space to solve a downstream prediction task. Once the dynamics is learned, embedding generation for novel datasets is done by solving the ODEs in time using a numerical integration scheme. Our GSN layer leverages the fast-to-compute edge vector directions and learnable scalar functions that only depend on nodes\u2019 distances in latent space to compute the nodes\u2019 positions. Conversely, graph convolution and graph attention network layers rely on learnable vector functions that require the full positions of input nodes in latent space. We propose a specific implementation called spring-neural-network using a set of small neural networks mimicking attracting and repulsing spring forces that we train for link sign prediction. Experiments show that our method achieves accuracy close to the state-of-the-art methods with node generation time speedups factor of up to 28,000 on large graphs.<\/jats:p>","DOI":"10.1007\/s10994-025-06794-1","type":"journal-article","created":{"date-parts":[[2025,5,28]],"date-time":"2025-05-28T17:22:24Z","timestamp":1748452944000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Graph spring neural ODEs for link sign prediction"],"prefix":"10.1007","volume":"114","author":[{"given":"Andrin","family":"Rehmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexandre","family":"Bovet","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,28]]},"reference":[{"key":"6794_CR1","doi-asserted-by":"crossref","unstructured":"Ansel, J., Yang, E., He, H., Gimelshein, N., Jain, A., Voznesensky, M., & et al. (2024). Pytorch 2: faster machine learning through dynamic python bytecode transformation and graph compilation. In Proceedings of the 29th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 2 (pp. 929\u2013947).","DOI":"10.1145\/3620665.3640366"},{"key":"6794_CR2","unstructured":"Blakely, D., Lanchantin, J., & Qi, Y. (2021). Time and space complexity of graph convolutional networks. [Online]. https:\/\/qdata.github.io\/deep2Read\/\/talks-mb2019\/Derrick_201906_ GCN_complexityAnalysis-writeup.pdf. Accessed 13 Dec 2024."},{"key":"6794_CR3","unstructured":"Bradbury, J., Frostig, R., Hawkins, P., Johnson, M. J., Leary, C., Maclaurin, D., & Zhang, Q. (2018). JAX: composable transformations of Python+NumPy programs. http:\/\/github.com\/google\/jax"},{"key":"6794_CR4","unstructured":"Bronstein, M. M., Bruna, J., Cohen, T., & Veli\u010dkovi\u0107, P. (2021). Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. (Preprint available at https:\/\/arxiv.org\/abs\/2104.13478)"},{"issue":"5","key":"6794_CR5","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1037\/h0046049","volume":"63","author":"D Cartwright","year":"1956","unstructured":"Cartwright, D., & Harary, F. (1956). Structural balance: A generalization of Heider\u2019s theory. Psychological Review, 63(5), 277.","journal-title":"Psychological Review"},{"key":"6794_CR6","unstructured":"Chamberlain, B., Rowbottom, J., Gorinova, M. I., Bronstein, M., Webb, S., & Rossi, E. (2021). Grand: Graph neural diffusion. In International conference on machine learning (pp. 1407\u20131418)."},{"key":"6794_CR7","doi-asserted-by":"crossref","unstructured":"Chen, Y., Qian, T., Liu, H., & Sun, K. (2018). \"Bridge\" enhanced signed directed network embedding. In Proceedings of the 27th ACM international conference on information and knowledge management (pp. 773\u2013782).","DOI":"10.1145\/3269206.3271738"},{"key":"6794_CR8","unstructured":"Chen, R. T., Rubanova, Y., Bettencourt, J., & Duvenaud, D. K. (2018). Neural ordinary differential equations. In Advances in neural information processing systems (Vol.\u00a031)."},{"key":"6794_CR9","unstructured":"Choi, J., Hong, S., Park, N., & Cho, S.-B. (2023). Gread: Graph neural reaction\u2013diffusion networks. In International conference on machine learning (pp. 5722\u20135747)."},{"key":"6794_CR10","doi-asserted-by":"crossref","unstructured":"Chung, C., & Whang, J. J. (2023). Learning representations of bi-level knowledge graphs for reasoning beyond link prediction. In Proceedings of the AAAI conference on artificial intelligence (Vol.\u00a037, pp. 4208\u20134216).","DOI":"10.1609\/aaai.v37i4.25538"},{"key":"6794_CR11","doi-asserted-by":"crossref","unstructured":"Derr, T., Ma, Y., & Tang, J. (2018). Signed graph convolutional networks. In 2018 IEEE international conference on data mining (ICDM) (pp. 929\u2013934).","DOI":"10.1109\/ICDM.2018.00113"},{"key":"6794_CR12","doi-asserted-by":"crossref","unstructured":"Fang, Z., Tan, S., & Wang, Y. (2023). A signed subgraph encoding approach via linear optimization for link sign prediction. IEEE Transactions on Neural Networks and Learning Systems.","DOI":"10.1109\/TNNLS.2023.3280924"},{"issue":"11","key":"6794_CR13","doi-asserted-by":"publisher","first-page":"1129","DOI":"10.1002\/spe.4380211102","volume":"21","author":"TM Fruchterman","year":"1991","unstructured":"Fruchterman, T. M., & Reingold, E. M. (1991). Graph drawing by force-directed placement. Practice and Experience, 21(11), 1129\u20131164.","journal-title":"Practice and Experience"},{"key":"6794_CR14","doi-asserted-by":"crossref","unstructured":"Grover, A., & Leskovec, J. (2016). node2vec: Scalable feature learning for networks. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining (pp. 855\u2013864).","DOI":"10.1145\/2939672.2939754"},{"key":"6794_CR15","doi-asserted-by":"crossref","unstructured":"Guha, R., Kumar, R., Raghavan, P., & Tomkins, A. (2004). Propagation of trust and distrust. In Proceedings of the 13th international conference on World Wide Web (pp. 403\u2013412).","DOI":"10.1145\/988672.988727"},{"issue":"1","key":"6794_CR16","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1016\/S0166-218X(00)00389-9","volume":"113","author":"R Hadany","year":"2001","unstructured":"Hadany, R., & Harel, D. (2001). A multi-scale algorithm for drawing graphs nicely. Discrete Applied Mathematics, 113(1), 3\u201321.","journal-title":"Discrete Applied Mathematics"},{"key":"6794_CR17","doi-asserted-by":"crossref","unstructured":"Huang, J., Shen, H., Hou, L., & Cheng, X. (2019). Signed graph attention networks. In Artificial neural networks and machine learning\u2013ICANN 2019: Workshop and special sessions: 28th international conference on artificial neural networks, Munich, Germany, September 17\u201319, 2019, Proceedings 28 (pp. 566\u2013577).","DOI":"10.1007\/978-3-030-30493-5_53"},{"key":"6794_CR18","doi-asserted-by":"crossref","unstructured":"Huang, J., Shen, H., Hou, L., & Cheng, X. (2021). Sdgnn: Learning node representation for signed directed networks. In Proceedings of the AAAI conference on artificial intelligence (Vol.\u00a035, pp. 196\u2013203).","DOI":"10.1609\/aaai.v35i1.16093"},{"key":"6794_CR19","first-page":"16177","volume":"33","author":"Z Huang","year":"2020","unstructured":"Huang, Z., Sun, Y., & Wang, W. (2020). Learning continuous system dynamics from irregularly-sampled partial observations. Advances in Neural Information Processing Systems, 33, 16177\u201316187.","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"4","key":"6794_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipm.2023.103403","volume":"60","author":"J Huang","year":"2023","unstructured":"Huang, J., Xie, R., Cao, Q., Shen, H., Zhang, S., Xia, F., & Cheng, X. (2023). Negative can be positive: Signed graph neural networks for recommendation. Information Processing and Management, 60(4), Article 103403.","journal-title":"Information Processing and Management"},{"key":"6794_CR21","doi-asserted-by":"crossref","unstructured":"Islam, M. R., Aditya\u00a0Prakash, B., & Ramakrishnan, N. (2018). Signet: Scalable embeddings for signed networks. In Advances in knowledge discovery and data mining: 22nd Pacific-Asia Conference, PAKDD 2018, Melbourne, VIC, Australia, June 3-6, 2018, Proceedings, Part II 22 (pp. 157\u2013169).","DOI":"10.1007\/978-3-319-93037-4_13"},{"key":"6794_CR22","unstructured":"Jung, J., Yoo, J., & Kang, U. (2020). Signed graph diffusion network. Preprint at https:\/\/arxiv.org\/abs\/2012.14191"},{"issue":"1","key":"6794_CR23","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1016\/0020-0190(89)90102-6","volume":"31","author":"T Kamada","year":"1989","unstructured":"Kamada, T., Kawai, S., et al. (1989). An algorithm for drawing general undirected graphs. Information Processing Letters, 31(1), 7\u201315.","journal-title":"Information Processing Letters"},{"key":"6794_CR24","unstructured":"Kang, Q., Zhao, K., Song, Y., Wang, S., & Tay, W. P. (2023). Node embedding from neural Hamiltonian orbits in graph neural networks. In International conference on machine learning (pp. 15786\u201315808)."},{"key":"6794_CR25","unstructured":"Kingma, D. P., & Ba, J. (2014). Adam: A method for stochastic optimization. Preprint at https:\/\/arxiv.org\/abs\/1412.6980"},{"key":"6794_CR26","doi-asserted-by":"crossref","unstructured":"Kumar, S., Spezzano, F., Subrahmanian, V., & Faloutsos, C. (2016). Edge weight prediction in weighted signed networks. In 2016 IEEE 16th international conference on data mining (ICDM) (pp. 221\u2013230).","DOI":"10.1109\/ICDM.2016.0033"},{"key":"6794_CR27","doi-asserted-by":"crossref","unstructured":"Lee, D., Lee, J., & Shin, K. (2024). Spear and shield: Adversarial attacks and defense methods for model-based link prediction on continuous-time dynamic graphs. In Proceedings of the AAAI Conference on artificial intelligence (Vol. 38, pp. 13374\u201313382).","DOI":"10.1609\/aaai.v38i12.29239"},{"key":"6794_CR28","doi-asserted-by":"crossref","unstructured":"Leskovec, J., Huttenlocher, D., & Kleinberg, J. (2010). Signed networks in social media. In Proceedings of the SIGCHI conference on human factors in computing systems (pp. 1361\u20131370).","DOI":"10.1145\/1753326.1753532"},{"issue":"1","key":"6794_CR29","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1080\/15427951.2009.10129177","volume":"6","author":"J Leskovec","year":"2009","unstructured":"Leskovec, J., Lang, K. J., Dasgupta, A., & Mahoney, M. W. (2009). Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics, 6(1), 29\u2013123.","journal-title":"Internet Mathematics"},{"key":"6794_CR30","doi-asserted-by":"crossref","unstructured":"Li, Y., Qu, M., Tang, J., & Chang, Y. (2023). Signed laplacian graph neural networks. In Proceedings of the AAAI conference on Artificial Intelligence (Vol. 37, pp. 4444\u20134452).","DOI":"10.1609\/aaai.v37i4.25565"},{"key":"6794_CR31","doi-asserted-by":"crossref","unstructured":"Li, Y., Tian, Y., Zhang, J., & Chang, Y. (2020). Learning signed network embedding via graph attention. In Proceedings of the AAAI conference on artificial intelligence (Vol. 34, pp. 4772\u20134779).","DOI":"10.1609\/aaai.v34i04.5911"},{"key":"6794_CR32","doi-asserted-by":"crossref","unstructured":"Liu, H. (2022). Lightsgcn: Powering signed graph convolution network for link sign prediction with simplified architecture design. In Proceedings of the 45th international ACM SIGIR conference on research and development in information retrieval (pp. 2680\u20132685).","DOI":"10.1145\/3477495.3531917"},{"key":"6794_CR33","doi-asserted-by":"crossref","unstructured":"Lizurej, T., Michalak, T., & Dziembowski, S. (2023). On manipulating weight predictions in signed weighted networks. In Proceedings of the AAAI conference on artificial intelligence (Vol. 37, pp. 5222\u20135229).","DOI":"10.1609\/aaai.v37i4.25652"},{"key":"6794_CR34","doi-asserted-by":"crossref","unstructured":"Lotfalizadeh, H., & Al\u00a0Hasan, M. (2023). Force-directed graph embedding with hops distance. In 2023 IEEE international conference on big data (BigData) (pp. 2946\u20132953).","DOI":"10.1109\/BigData59044.2023.10386461"},{"key":"6794_CR35","first-page":"2825","volume":"12","author":"F Pedregosa","year":"2011","unstructured":"Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., & Duchesnay, E. (2011). Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 12, 2825\u20132830.","journal-title":"Journal of Machine Learning Research"},{"key":"6794_CR36","doi-asserted-by":"crossref","unstructured":"Perozzi, B., Al-Rfou, R., & Skiena, S. (2014). Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD international conference on knowledge discovery and data mining (pp. 701\u2013710).","DOI":"10.1145\/2623330.2623732"},{"key":"6794_CR37","unstructured":"Poli, M., Massaroli, S., Park, J., Yamashita, A., Asama, H., & Park, J. (2019). Graph neural ordinary differential equations. Preprint at https:\/\/arxiv.org\/abs\/1911.07532"},{"key":"6794_CR38","doi-asserted-by":"crossref","unstructured":"Rahman, M. K., Sujon, M. H., & Azad, A. (2020). Force2vec: Parallel force-directed graph embedding. In 2020 IEEE international conference on data mining (ICDM) (pp. 442\u2013451).","DOI":"10.1109\/ICDM50108.2020.00053"},{"key":"6794_CR39","doi-asserted-by":"crossref","unstructured":"Ren, G., Ding, X., Xu, X.-K., & Zhang, H.-F. (2024). Link prediction in multilayer networks via cross-network embedding. In Proceedings of the AAAI conference on artificial intelligence (Vol. 38, pp. 8939\u20138947).","DOI":"10.1609\/aaai.v38i8.28742"},{"key":"6794_CR40","doi-asserted-by":"crossref","unstructured":"Richardson, M., Agrawal, R., spsampsps Domingos, P. (2003). Trust management for the semantic web. In International semantic web conference (pp. 351\u2013368).","DOI":"10.1007\/978-3-540-39718-2_23"},{"key":"6794_CR41","unstructured":"Rusch, T. K., Chamberlain, B., Rowbottom, J., Mishra, S., & Bronstein, M. (2022). Graph-coupled oscillator networks. In International conference on machine learning (pp. 18888\u201318909)."},{"key":"6794_CR42","unstructured":"Sanchez-Gonzalez, A., Bapst, V., Cranmer, K., & Battaglia, P. (2019). Hamiltonian graph networks with ode integrators. Preprint at https:\/\/arxiv.org\/abs\/1909.12790"},{"key":"6794_CR43","doi-asserted-by":"crossref","unstructured":"Shi, L., Hu, B., Zhao, D., He, J., Zhang, Z., & Zhou, J. (2024). Structural information enhanced graph representation for link prediction. In Proceedings of the AAAI conference on artificial intelligence (Vol. 38, pp. 14964\u201314972).","DOI":"10.1609\/aaai.v38i13.29417"},{"key":"6794_CR44","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s13278-015-0288-7","volume":"5","author":"D Song","year":"2015","unstructured":"Song, D., & Meyer, D. A. (2015). Link sign prediction and ranking in signed directed social networks. Social Network Analysis and Mining, 5, 1\u201314.","journal-title":"Social Network Analysis and Mining"},{"issue":"7","key":"6794_CR45","doi-asserted-by":"publisher","first-page":"4037","DOI":"10.1007\/s10994-023-06473-z","volume":"113","author":"H Sun","year":"2024","unstructured":"Sun, H., Tian, P., Xiong, Y., Zhang, Y., Xiang, Y., Jia, X., & Wang, H. (2024). Dynamise: Dynamic signed network embedding for link prediction. Machine Learning, 113(7), 4037\u20134053.","journal-title":"Machine Learning"},{"key":"6794_CR46","unstructured":"Thuerey, N., Holl, P., Mueller, M., Schnell, P., Trost, F., & Um, K. (2021). Physics-based deep learning. Preprint at https:\/\/arxiv.org\/abs\/2109.05237"},{"key":"6794_CR47","unstructured":"Veli\u010dkovi\u0107, P., Cucurull, G., Casanova, A., Romero, A., Lio, P., & Bengio, Y. (2017). Graph attention networks. Preprint at https:\/\/arxiv.org\/abs\/1710.10903"},{"key":"6794_CR48","doi-asserted-by":"crossref","unstructured":"Wang, S., Tang, J., Aggarwal, C., Chang, Y., spsampsps Liu, H. (2017). Signed network embedding in social media. In Proceedings of the 2017 SIAM international conference on data mining (pp. 327\u2013335).","DOI":"10.1137\/1.9781611974973.37"},{"key":"6794_CR49","unstructured":"Xhonneux, L.-P., Qu, M., & Tang, J. (2020). Continuous graph neural networks. In International conference on machine learning (pp. 10432\u201310441)."},{"key":"6794_CR50","unstructured":"Zhang, J., He, T., Sra, S., & Jadbabaie, A. (2019). Why gradient clipping accelerates training: A theoretical justification for adaptivity. Preprint at https:\/\/arxiv.org\/abs\/1905.11881"}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06794-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10994-025-06794-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06794-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T16:12:50Z","timestamp":1757175170000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10994-025-06794-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,28]]},"references-count":50,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["6794"],"URL":"https:\/\/doi.org\/10.1007\/s10994-025-06794-1","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"type":"print","value":"0885-6125"},{"type":"electronic","value":"1573-0565"}],"subject":[],"published":{"date-parts":[[2025,5,28]]},"assertion":[{"value":"16 January 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 April 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 May 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 that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"All authors have given their consent for publication.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}],"article-number":"152"}}