{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,27]],"date-time":"2024-08-27T16:38:59Z","timestamp":1724776739268},"reference-count":41,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T00:00:00Z","timestamp":1682553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["42271481"],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the High-Performance Computing Platform of Central South University and HPC Central of Department of GIS, in providing HPC resources"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"Traffic flow forecasting is a basic function of intelligent transportation systems, and the accuracy of prediction is of great significance for traffic management and urban planning. The main difficulty of traffic flow predictions is that there is complex underlying spatiotemporal dependence in traffic flow; thus, the existing spatiotemporal graph neural network (STGNN) models need to model both temporal dependence and spatial dependence. Graph neural networks (GNNs) are adopted to capture the spatial dependence in traffic flow, which can model the symmetric or asymmetric spatial relations between nodes in the traffic network. The transmission process of traffic features in GNNs is guided by the node-to-node relationship (e.g., adjacency or spatial distance) between nodes, ignoring the spatial dependence caused by local topological constraints in the road network. To further consider the influence of local topology on the spatial dependence of road networks, in this paper, we introduce Ollivier\u2013Ricci curvature information between connected edges in the road network, which is based on optimal transport theory and makes comprehensive use of the neighborhood-to-neighborhood relationship to guide the transmission process of traffic features between nodes in STGNNs. Experiments on real-world traffic datasets show that the models with Ollivier\u2013Ricci curvature information outperforms those based on only node-to-node relationships between nodes by ten percent on average in the RMSE metric. This study indicates that by utilizing complex topological features in road networks, spatial dependence can be captured more sufficiently, further improving the predictive ability of traffic forecasting models.<\/jats:p>","DOI":"10.3390\/sym15050995","type":"journal-article","created":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T08:36:15Z","timestamp":1682670975000},"page":"995","source":"Crossref","is-referenced-by-count":4,"title":["Ollivier\u2013Ricci Curvature Based Spatio-Temporal Graph Neural Networks for Traffic Flow Forecasting"],"prefix":"10.3390","volume":"15","author":[{"given":"Xing","family":"Han","sequence":"first","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"}]},{"given":"Guowei","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-6103-1113","authenticated-orcid":false,"given":"Ling","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"}]},{"given":"Ronghua","family":"Du","sequence":"additional","affiliation":[{"name":"College of Automotive and Mechanical Engineering, Changsha University of Science & Technology, Changsha 410114, China"}]},{"given":"Yuhan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"}]},{"given":"Zhe","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Geographical Sciences, Hunan Normal University, Changsha 410083, China"}]},{"given":"Yang","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"},{"name":"The 27th Research Institute, China Electronic Technology Group Corporation, Zhengzhou 450047, China"}]},{"ORCID":"http:\/\/orcid.org\/0000-0003-0144-2524","authenticated-orcid":false,"given":"Silu","family":"He","sequence":"additional","affiliation":[{"name":"School of Geosciences and Info-Physics, Central South University, Changsha 410083, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,27]]},"reference":[{"key":"ref_1","first-page":"84","article-title":"Imagenet classification with deep convolutional neural networks","volume":"25","author":"Krizhevsky","year":"2012","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2763","DOI":"10.1007\/s10489-021-02587-w","article-title":"Spatial-temporal graph neural network for traffic forecasting: An overview and open research issues","volume":"52","author":"Bui","year":"2021","journal-title":"Appl. Intell."},{"key":"ref_3","unstructured":"Gilmer, J., Schoenholz, S.S., Riley, P.F., Vinyals, O., and Dahl, G.E. (2023, January 20\u201323). Neural message passing for quantum chemistry. Proceedings of the International Conference on Machine Learning, PMLR, Baltimore, MA, USA."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1109\/TITS.2019.2963722","article-title":"Optimized graph convolution recurrent neural network for traffic prediction","volume":"22","author":"Guo","year":"2020","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"103125","DOI":"10.1063\/1.5117180","article-title":"Road traffic state prediction based on a graph embedding recurrent neural network under the SCATS","volume":"29","author":"Xu","year":"2019","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Wu, T., Chen, F., and Wan, Y. (2018, January 20\u201322). Graph attention LSTM network: A new model for traffic flow forecasting. Proceedings of the 2018 5th International Conference on Information Science and Control Engineering (ICISCE), Zhengzhou, China.","DOI":"10.1109\/ICISCE.2018.00058"},{"key":"ref_7","unstructured":"Li, Y., Yu, R., Shahabi, C., and Liu, Y. (2017). Diffusion convolutional recurrent neural network: Data-driven traffic forecasting. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Kang, Z., Xu, H., Hu, J., and Pei, X. (2019, January 27\u201330). Learning dynamic graph embedding for traffic flow forecasting: A graph self-attentive method. Proceedings of the 2019 IEEE Intelligent Transportation Systems Conference (ITSC), Auckland, New Zealand.","DOI":"10.1109\/ITSC.2019.8917213"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"103466","DOI":"10.1016\/j.trc.2021.103466","article-title":"DDP-GCN: Multi-graph convolutional network for spatiotemporal traffic forecasting","volume":"134","author":"Lee","year":"2022","journal-title":"Transp. Res. Part C Emerg. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"810","DOI":"10.1016\/j.jfa.2008.11.001","article-title":"Ricci curvature of Markov chains on metric spaces","volume":"256","author":"Ollivier","year":"2009","journal-title":"J. Funct. Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1038\/nature14539","article-title":"Deep learning","volume":"521","author":"LeCun","year":"2015","journal-title":"Nature"},{"key":"ref_12","unstructured":"Oord, A.v.d., Dieleman, S., Zen, H., Simonyan, K., Vinyals, O., Graves, A., Kalchbrenner, N., Senior, A., and Kavukcuoglu, K. (2016). Wavenet: A generative model for raw audio. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Zhang, J., Zheng, Y., and Qi, D. (2016, January 12\u201317). Deep spatio-temporal residual networks for citywide crowd flows prediction. Proceedings of the AAAI Conference on Artificial Intelligence, Phoenix, AR, USA.","DOI":"10.1609\/aaai.v31i1.10735"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"31401","DOI":"10.1007\/s11042-020-10486-4","article-title":"A data aggregation based approach to exploit dynamic spatio-temporal correlations for citywide crowd flows prediction in fog computing","volume":"80","author":"Ali","year":"2021","journal-title":"Multimed. Tools Appl."},{"key":"ref_15","unstructured":"Kipf, T.N., and Welling, M. (2017, January 24\u201326). Semi-Supervised Classification with Graph Convolutional Networks. Proceedings of the 5th International Conference on Learning Representations, Toulon, France."},{"key":"ref_16","first-page":"5171","article-title":"Link prediction based on graph neural networks","volume":"31","author":"Zhang","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Li, H., Cao, J., Jun, J., Luo, Q., He, S., and Wang, X. (2023). Augmentation-Free Graph Contrastive Learning of Invariant-Discriminative Representations. IEEE Trans. Neural Networks Learn. Syst.","DOI":"10.1109\/TNNLS.2023.3248871"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3848","DOI":"10.1109\/TITS.2019.2935152","article-title":"T-GCN: A temporal graph convolutional network for traffic prediction","volume":"21","author":"Zhao","year":"2020","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"15055","DOI":"10.1109\/TITS.2021.3136287","article-title":"KST-GCN: A knowledge-driven spatial-temporal graph convolutional network for traffic forecasting","volume":"23","author":"Zhu","year":"2022","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/j.neunet.2021.10.021","article-title":"Exploiting dynamic spatio-temporal graph convolutional neural networks for citywide traffic flows prediction","volume":"145","author":"Ali","year":"2022","journal-title":"Neural Netw."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Yu, B., Yin, H., and Zhu, Z. (2017). Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting. arXiv.","DOI":"10.24963\/ijcai.2018\/505"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Wu, Z., Pan, S., Long, G., Jiang, J., and Zhang, C. (2019). Graph wavenet for deep spatial-temporal graph modeling. arXiv.","DOI":"10.24963\/ijcai.2019\/264"},{"key":"ref_23","first-page":"865","article-title":"Traffic flow prediction with big data: A deep learning approach","volume":"16","author":"Lv","year":"2014","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Li, M., and Zhu, Z. (2021, January 2\u20139). Spatial-temporal fusion graph neural networks for traffic flow forecasting. Proceedings of the AAAI Conference on Artificial Intelligence, Virtually.","DOI":"10.1609\/aaai.v35i5.16542"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"He, S., and Shin, K.G. (2020, January 20\u201324). Towards fine-grained flow forecasting: A graph attention approach for bike sharing systems. Proceedings of the Web Conference 2020, Taipei, Taiwan.","DOI":"10.1145\/3366423.3380097"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"8650","DOI":"10.1038\/s41598-018-27001-3","article-title":"Comparative analysis of two discretizations of Ricci curvature for complex networks","volume":"8","author":"Samal","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_27","first-page":"1","article-title":"Community detection on networks with Ricci flow","volume":"9","author":"Ni","year":"2019","journal-title":"Sci. Rep."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"121071","DOI":"10.1016\/j.physa.2019.121071","article-title":"Measuring road network topology vulnerability by Ricci curvature","volume":"527","author":"Gao","year":"2019","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_29","first-page":"102666","article-title":"Applying Ollivier-Ricci curvature to indicate the mismatch of travel demand and supply in urban transit network","volume":"106","author":"Wang","year":"2022","journal-title":"Int. J. Appl. Earth Obs. Geoinf."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Wang, C., Jonckheere, E., and Banirazi, R. (2014, January 4\u20136). Wireless network capacity versus Ollivier-Ricci curvature under Heat-Diffusion (HD) protocol. Proceedings of the 2014 American Control Conference, Portland, OR, USA.","DOI":"10.1109\/ACC.2014.6858912"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s00454-002-0743-x","article-title":"Bochner\u2019s method for cell complexes and combinatorial Ricci curvature","volume":"29","author":"Forman","year":"2003","journal-title":"Discret. Comput. Geom."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1093\/comnet\/cnw030","article-title":"Characterizing complex networks with Forman-Ricci curvature and associated geometric flows","volume":"5","author":"Weber","year":"2017","journal-title":"J. Complex Netw."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Saucan, E., Wolansky, G., Appleboim, E., and Zeevi, Y.Y. (2009, January 17\u201319). Combinatorial ricci curvature and laplacians for image processing. Proceedings of the 2009 2nd International Congress on Image and Signal Processing, Tianjin, China.","DOI":"10.1109\/CISP.2009.5304710"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1561\/2200000073","article-title":"Computational Optimal Transport","volume":"11","author":"Cuturi","year":"2019","journal-title":"Found. Trends Mach. Learn."},{"key":"ref_35","unstructured":"Arjovsky, M., Chintala, S., and Bottou, L. (2023, January 20\u201323). Wasserstein generative adversarial networks. Proceedings of the International Conference on Machine Learning, PMLR, Singapore."},{"key":"ref_36","unstructured":"Kusner, M., Sun, Y., Kolkin, N., and Weinberger, K. (2023, January 20\u201323). From word embeddings to document distances. Proceedings of the International Conference on Machine Learning, PMLR, Singapore."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.ins.2021.12.077","article-title":"Curvature graph neural network","volume":"592","author":"Li","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_38","first-page":"1","article-title":"Introduction to graph neural networks","volume":"14","author":"Liu","year":"2020","journal-title":"Synth. Lect. Artif. Intell. Mach. Learn."},{"key":"ref_39","first-page":"8026","article-title":"PyTorch: An imperative style, high-performance deep learning library","volume":"32","author":"Paszke","year":"2019","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Wang, J., Jiang, J., Jiang, W., Li, C., and Zhao, W.X. (2021, January 2\u20135). Libcity: An open library for traffic prediction. Proceedings of the 29th International Conference on Advances in Geographic Information Systems, Beijing, China.","DOI":"10.1145\/3474717.3483923"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Ni, C.C., Lin, Y.Y., Gao, J., Gu, X.D., and Saucan, E. (May, January 26). Ricci curvature of the internet topology. Proceedings of the 2015 IEEE conference on computer communications (INFOCOM), Hong Kong, China.","DOI":"10.1109\/INFOCOM.2015.7218668"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/995\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T10:19:20Z","timestamp":1682677160000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/995"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,27]]},"references-count":41,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["sym15050995"],"URL":"https:\/\/doi.org\/10.3390\/sym15050995","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,27]]}}}