{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T17:42:19Z","timestamp":1781545339917,"version":"3.54.5"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T00:00:00Z","timestamp":1707436800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T00:00:00Z","timestamp":1707436800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province","doi-asserted-by":"publisher","award":["2023J01922"],"award-info":[{"award-number":["2023J01922"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61762036"],"award-info":[{"award-number":["61762036"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Headmaster Fund of Minnan Normal University","award":["KJ19009"],"award-info":[{"award-number":["KJ19009"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Process Lett"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Graph neural networks (GNNs) have emerged as a powerful tool in graph representation learning. However, they are increasingly challenged by over-smoothing as network depth grows, compromising their ability to capture and represent complex graph structures. Additionally, some popular GNN variants only consider local neighbor information during node updating, ignoring the global structural information and leading to inadequate learning and differentiation of graph structures. To address these challenges, we introduce a novel graph neural network framework, GraphSAGE++. Our model extracts the representation of the target node at each layer and then concatenates all layer weighted representations to obtain the final result. In addition, the strategies combining double aggregations with weighted concatenation are proposed, which significantly enhance the model\u2019s discernment and preservation of structural information. Empirical results on various datasets demonstrate that GraphSAGE++ excels in vertex classification, link prediction, and visualization tasks, surpassing existing methods in effectiveness.<\/jats:p>","DOI":"10.1007\/s11063-024-11496-1","type":"journal-article","created":{"date-parts":[[2024,2,9]],"date-time":"2024-02-09T10:02:12Z","timestamp":1707472932000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["GraphSAGE++: Weighted Multi-scale GNN for Graph Representation Learning"],"prefix":"10.1007","volume":"56","author":[{"given":"E.","family":"Jiawei","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yinglong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shangying","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xuewen","family":"Xia","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xing","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,2,9]]},"reference":[{"key":"11496_CR1","doi-asserted-by":"crossref","unstructured":"Tang J, Qu M, Wang M, Zhang M, Yan J, Mei Q (2015) Line: large-scale information network embedding. In: Proceedings of the 24th international conference on world wide web, pp 1067\u20131077","DOI":"10.1145\/2736277.2741093"},{"key":"11496_CR2","doi-asserted-by":"crossref","unstructured":"Wang X, Salim FD, Ren Y, Koniusz P (2020) Relation embedding for personalised translation-based poi recommendation. In: Advances in knowledge discovery and data mining: 24th Pacific-Asia conference, PAKDD 2020, Singapore, May 11\u201314, 2020, Proceedings, Part I 24, pp 53\u201364. Springer","DOI":"10.1007\/978-3-030-47426-3_5"},{"key":"11496_CR3","unstructured":"Yu PS, Han J, Faloutsos C (2014) Link mining: models, algorithms, and applications. In: Link mining"},{"key":"11496_CR4","doi-asserted-by":"publisher","first-page":"2227","DOI":"10.1007\/s00607-021-00982-2","volume":"103","author":"K Berahmand","year":"2021","unstructured":"Berahmand K, Nasiri E, Rostami M, Forouzandeh S (2021) A modified deepwalk method for link prediction in attributed social network. Computing 103:2227\u20132249","journal-title":"Computing"},{"issue":"8","key":"11496_CR5","first-page":"5375","volume":"34","author":"K Berahmand","year":"2022","unstructured":"Berahmand K, Nasiri E, Forouzandeh S, Li Y (2022) A preference random walk algorithm for link prediction through mutual influence nodes in complex networks. J King Saud Univ Comput Inf Sci 34(8):5375\u20135387","journal-title":"J King Saud Univ Comput Inf Sci"},{"issue":"1","key":"11496_CR6","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1109\/TNNLS.2020.2978386","volume":"32","author":"Z Wu","year":"2020","unstructured":"Wu Z, Pan S, Chen F, Long G, Zhang C, Philip SY (2020) A comprehensive survey on graph neural networks. IEEE Trans Neural Netw Learn Syst 32(1):4\u201324","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"key":"11496_CR7","doi-asserted-by":"crossref","unstructured":"Liu M, Gao H, Ji S (2020) Towards deeper graph neural networks. In: Proceedings of the 26th ACM SIGKDD international conference on knowledge discovery & data mining, pp 338\u2013348","DOI":"10.1145\/3394486.3403076"},{"key":"11496_CR8","doi-asserted-by":"crossref","unstructured":"Chen D, Lin Y, Li W, Li P, Zhou J, Sun X (2020) Measuring and relieving the over-smoothing problem for graph neural networks from the topological view. In: Proceedings of the AAAI conference on artificial intelligence, vol 34, pp 3438\u20133445","DOI":"10.1609\/aaai.v34i04.5747"},{"key":"11496_CR9","first-page":"2268","volume":"35","author":"N Keriven","year":"2022","unstructured":"Keriven N (2022) Not too little, not too much: a theoretical analysis of graph (over) smoothing. Adv Neural Inf Process Syst 35:2268\u20132281","journal-title":"Adv Neural Inf Process Syst"},{"key":"11496_CR10","first-page":"14498","volume":"33","author":"Y Min","year":"2020","unstructured":"Min Y, Wenkel F, Wolf G (2020) Scattering gcn: overcoming oversmoothness in graph convolutional networks. Adv Neural Inf Process Syst 33:14498\u201314508","journal-title":"Adv Neural Inf Process Syst"},{"key":"11496_CR11","unstructured":"Chen M, Wei Z, Huang Z, Ding B, Li Y (2020) Simple and deep graph convolutional networks, pp 1725\u20131735. PMLR"},{"key":"11496_CR12","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, pp 9267\u20139276","DOI":"10.1109\/ICCV.2019.00936"},{"key":"11496_CR13","unstructured":"Xu K, Li C, Tian Y, Sonobe T, Kawarabayashi K-i, Jegelka S (2018) Representation learning on graphs with jumping knowledge networks. In: International conference on machine learning, pp 5453\u20135462. PMLR"},{"key":"11496_CR14","unstructured":"Xu K, Hu W, Leskovec J, Jegelka S (2019) How powerful are graph neural networks? In: International conference on learning representations"},{"key":"11496_CR15","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.aiopen.2021.01.001","volume":"1","author":"J Zhou","year":"2020","unstructured":"Zhou J, Cui G, Hu S, Zhang Z, Yang C, Liu Z, Wang L, Li C, Sun M (2020) Graph neural networks: a review of methods and applications. AI Open 1:57\u201381","journal-title":"AI Open"},{"key":"11496_CR16","doi-asserted-by":"crossref","unstructured":"Peng S, Sugiyama K, Mine T (2022) Svd-gcn: a simplified graph convolution paradigm for recommendation. In: Proceedings of the 31st ACM international conference on information & knowledge management, pp 1625\u20131634","DOI":"10.1145\/3511808.3557462"},{"key":"11496_CR17","doi-asserted-by":"crossref","unstructured":"Liu Y, Zhang J, Dou R, Zhou X, Xu X, Wang S, Qi L (2022) Vehicle check-in data-driven poi recommendation based on improved svd and graph convolutional network. In: 2022 IEEE smartworld, ubiquitous intelligence & computing, scalable computing & communications, digital twin, privacy computing, metaverse, autonomous & trusted vehicles (SmartWorld\/UIC\/ScalCom\/DigitalTwin\/PriComp\/Meta), pp 2040\u20132047","DOI":"10.1109\/SmartWorld-UIC-ATC-ScalCom-DigitalTwin-PriComp-Metaverse56740.2022.00295"},{"key":"11496_CR18","doi-asserted-by":"crossref","unstructured":"Qiu J, Dong Y, Ma H, Li J, Wang K, Tang J (2018) Network embedding as matrix factorization: unifying deepwalk, line, pte, and node2vec. In: Proceedings of the eleventh ACM international conference on web search and data mining, pp 459\u2013467","DOI":"10.1145\/3159652.3159706"},{"key":"11496_CR19","unstructured":"Shanmugam\u00a0Sakthivadivel S (2019) Fast-netmf: graph embedding generation on single gpu and multi-core cpus with netmf. PhD thesis, The Ohio State University"},{"key":"11496_CR20","doi-asserted-by":"crossref","unstructured":"Cao S, Lu W, Xu Q (2015) Grarep: learning graph representations with global structural information. In: Proceedings of the 24th ACM international on conference on information and knowledge management, pp 891\u2013900","DOI":"10.1145\/2806416.2806512"},{"key":"11496_CR21","unstructured":"Feng Q, Liu N, Yang F, Tang R, Du M, Hu X (2022) DEGREE: decomposition based explanation for graph neural networks. In: International conference on learning representations"},{"issue":"1","key":"11496_CR22","doi-asserted-by":"publisher","first-page":"657","DOI":"10.1109\/TPAMI.2022.3154319","volume":"45","author":"G Bouritsas","year":"2022","unstructured":"Bouritsas G, Frasca F, Zafeiriou S, Bronstein MM (2022) Improving graph neural network expressivity via subgraph isomorphism counting. IEEE Trans Pattern Anal Mach Intell 45(1):657\u2013668","journal-title":"IEEE Trans Pattern Anal Mach Intell"},{"key":"11496_CR23","unstructured":"Shervashidze N, Schweitzer P, Van\u00a0Leeuwen EJ, Mehlhorn K, Borgwardt KM (2011) Weisfeiler-lehman graph kernels. J Mach Learn Res 12(9)"},{"key":"11496_CR24","unstructured":"Wijesinghe A, Wang Q (2021) A new perspective on\" how graph neural networks go beyond weisfeiler-lehman?\". In: International conference on learning representations"},{"key":"11496_CR25","unstructured":"Kipf TN, Welling M (2017) Semi-supervised classification with graph convolutional networks. In: International conference on learning representations"},{"issue":"1","key":"11496_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s40649-019-0069-y","volume":"6","author":"S Zhang","year":"2019","unstructured":"Zhang S, Tong H, Xu J, Maciejewski R (2019) Graph convolutional networks: a comprehensive review. Comput Soc Netw 6(1):1\u201323","journal-title":"Comput Soc Netw"},{"key":"11496_CR27","unstructured":"Wu F, Souza A, Zhang T, Fifty C, Yu T, Weinberger K (2019) Simplifying graph convolutional networks. In: International conference on machine learning, pp 6861\u20136871 . PMLR"},{"key":"11496_CR28","doi-asserted-by":"publisher","first-page":"108696","DOI":"10.1016\/j.patcog.2022.108696","volume":"128","author":"T Zhang","year":"2022","unstructured":"Zhang T, Shan H-R, Little MA (2022) Causal graphsage: a robust graph method for classification based on causal sampling. Pattern Recogn 128:108696","journal-title":"Pattern Recogn"},{"key":"11496_CR29","unstructured":"Hamilton W, Ying Z, Leskovec J (2017) Inductive representation learning on large graphs. Adv Neural Inf Process Syst 30"},{"key":"11496_CR30","unstructured":"Velickovic P, Cucurull G, Casanova A, Romero A, Li\u00f2 P, Bengio Y (2018) Graph attention networks. In: International conference on learning representations"},{"key":"11496_CR31","unstructured":"Brody S, Alon U, Yahav E (2022) How attentive are graph attention networks? In: International conference on learning representations"},{"key":"11496_CR32","doi-asserted-by":"crossref","unstructured":"Ouyang M, Zhang Y, Xia X, Xu X (2023) Grarep++: flexible learning graph representations with weighted global structural information. IEEE Access","DOI":"10.1109\/ACCESS.2023.3313411"},{"issue":"3","key":"11496_CR33","first-page":"93","volume":"29","author":"P Sen","year":"2008","unstructured":"Sen P, Namata G, Bilgic M, Getoor L, Galligher B, Eliassi-Rad T (2008) Collective classification in network data. AI Mag 29(3):93\u201393","journal-title":"AI Mag"},{"key":"11496_CR34","first-page":"4465","volume":"33","author":"P Li","year":"2020","unstructured":"Li P, Wang Y, Wang H, Leskovec J (2020) Distance encoding: design provably more powerful neural networks for graph representation learning. Adv Neural Inf Process Syst 33:4465\u20134478","journal-title":"Adv Neural Inf Process Syst"},{"key":"11496_CR35","doi-asserted-by":"crossref","unstructured":"Menezes RA, Nievola JC (2015) Predicting the function of proteins using differential evolution. In: IADIS international conference information systems","DOI":"10.1109\/FSKD.2014.6980888"},{"issue":"3","key":"11496_CR36","first-page":"469","volume":"86","author":"H Crane","year":"2015","unstructured":"Crane H, Dempsey W (2015) Community detection for interaction networks. Immunology 86(3):469\u201374","journal-title":"Immunology"},{"key":"11496_CR37","first-page":"9936","volume":"34","author":"W Cong","year":"2021","unstructured":"Cong W, Ramezani M, Mahdavi M (2021) On provable benefits of depth in training graph convolutional networks. Adv Neural Inf Process Syst 34:9936\u20139949","journal-title":"Adv Neural Inf Process Syst"},{"key":"11496_CR38","unstructured":"Maaten L, Hinton G (2008) Visualizing data using t-SNE. J Mach Learn Res 9(11)"},{"issue":"3","key":"11496_CR39","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1137\/140976649","volume":"57","author":"DF Gleich","year":"2015","unstructured":"Gleich DF (2015) Pagerank beyond the web. siam Rev 57(3):321\u2013363","journal-title":"siam Rev"},{"key":"11496_CR40","unstructured":"Klicpera J, Bojchevski A, G\u00fcnnemann S (2018) Predict then propagate: graph neural networks meet personalized pagerank. In: ICLR"}],"container-title":["Neural Processing Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11063-024-11496-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11063-024-11496-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11063-024-11496-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T20:11:20Z","timestamp":1709237480000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11063-024-11496-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,9]]},"references-count":40,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["11496"],"URL":"https:\/\/doi.org\/10.1007\/s11063-024-11496-1","relation":{},"ISSN":["1573-773X"],"issn-type":[{"value":"1573-773X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,9]]},"assertion":[{"value":"8 December 2023","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 February 2024","order":2,"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 competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"24"}}