{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,25]],"date-time":"2026-08-25T15:14:57Z","timestamp":1787670897818,"version":"build-2736575974"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,3,13]],"date-time":"2024-03-13T00:00:00Z","timestamp":1710288000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,3,13]],"date-time":"2024-03-13T00:00:00Z","timestamp":1710288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Innovation Program for Quantum Science and Technology","award":["2021ZD0300703"],"award-info":[{"award-number":["2021ZD0300703"]}]},{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["61971146"],"award-info":[{"award-number":["61971146"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"the University-Industry Collaborative Education Program","award":["230805384035416"],"award-info":[{"award-number":["230805384035416"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["EURASIP J. Adv. Signal Process."],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Learning graph structure from observed signals over graph is a crucial task in many graph signal processing (GSP) applications. Existing approaches focus on inferring static graph, typically assuming that all nodes are available. However, these approaches ignore the situation where only a subset of nodes are available from spatiotemporal measurements, and the remaining nodes are never observed due to application-specific constraints, resulting in time-varying graph estimation accuracy declines dramatically. To handle this problem, we propose a framework that consider the presence of hidden nodes to identify time-varying graph. Specifically, we assume that the graph signals are smooth and stationary on the graphs and only a small number of edges are allowed to change between two consecutive graphs. With these assumptions, we present a challenging time-varying graph inference problem, which models the influence of hidden nodes in terms of estimating the graph-shift operator matrices that have a form of graph Laplacian. Moreover, we emphasize similar edge pattern (column-sparsity) between different graphs. Finally, our method is evaluated on both synthetic and real-world data. The experimental results demonstrate the advantage of our method when compared to existing benchmarking methods.<\/jats:p>","DOI":"10.1186\/s13634-024-01128-0","type":"journal-article","created":{"date-parts":[[2024,3,13]],"date-time":"2024-03-13T08:33:15Z","timestamp":1710318795000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Time-varying graph learning from smooth and stationary graph signals with hidden nodes"],"prefix":"10.1186","volume":"2024","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-8704-4925","authenticated-orcid":false,"given":"Rong","family":"Ye","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xue-Qin","family":"Jiang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hui","family":"Feng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jian","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Runhe","family":"Qiu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinxin","family":"Hou","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,3,13]]},"reference":[{"key":"1128_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-88146-1","volume-title":"Statistical Analysis of Network Data: Methods and Models","author":"ED Kolaczyk","year":"2009","unstructured":"E.D. Kolaczyk, Statistical Analysis of Network Data: Methods and Models (Springer, New York, 2009)"},{"key":"1128_CR2","doi-asserted-by":"publisher","DOI":"10.7551\/mitpress\/9266.001.0001","volume-title":"Discovering the Human Connectome","author":"O Sporns","year":"2012","unstructured":"O. Sporns, Discovering the Human Connectome (MIT Press, Boston, 2012)"},{"key":"1128_CR3","unstructured":"S. Myers, J. Leskovec, On the convexity of latent social network inference. NIPS. 23 (2010)"},{"issue":"21","key":"1128_CR4","doi-asserted-by":"publisher","first-page":"3835","DOI":"10.1016\/j.physa.2011.06.033","volume":"390","author":"A Namaki","year":"2011","unstructured":"A. Namaki, A. Shirazi, R. Raei, G. Jafari, Network analysis of a financial market based on genuine correlation and threshold method. Phys. A 390(21), 3835\u20133841 (2011)","journal-title":"Phys. A"},{"issue":"3","key":"1128_CR5","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1109\/MSP.2012.2235192","volume":"30","author":"DI Shuman","year":"2013","unstructured":"D.I. Shuman, S.K. Narang, P. Frossard, A. Ortega, P. Vandergheynst, The emerging field of signal processing on graphs: extending high-dimensional data analysis to networks and other irregular domains. IEEE Sign. Process. Mag. 30(3), 83\u201398 (2013). https:\/\/doi.org\/10.1109\/MSP.2012.2235192","journal-title":"IEEE Sign. Process. Mag."},{"issue":"5","key":"1128_CR6","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1109\/MSP.2014.2329213","volume":"31","author":"A Sandryhaila","year":"2014","unstructured":"A. Sandryhaila, J.M.F. Moura, Big data analysis with signal processing on graphs: representation and processing of massive data sets with irregular structure. IEEE Sign. Process. Mag. 31(5), 80\u201390 (2014). https:\/\/doi.org\/10.1109\/MSP.2014.2329213","journal-title":"IEEE Sign. Process. Mag."},{"issue":"6","key":"1128_CR7","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1109\/MSP.2020.3020715","volume":"37","author":"AG Marques","year":"2020","unstructured":"A.G. Marques, N. Kiyavash, J.M.F. Moura, D. Van De Ville, R. Willett, Graph signal processing: foundations and emerging directions [from the guest editors]. IEEE Sign. Process. Mag. 37(6), 11\u201313 (2020). https:\/\/doi.org\/10.1109\/MSP.2020.3020715","journal-title":"IEEE Sign. Process. Mag."},{"key":"1128_CR8","doi-asserted-by":"publisher","unstructured":"E. Pavez, A. Ortega, Generalized Laplacian precision matrix estimation for graph signal processing. in 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 6350\u20136354 (2016). https:\/\/doi.org\/10.1109\/ICASSP.2016.7472899","DOI":"10.1109\/ICASSP.2016.7472899"},{"issue":"3","key":"1128_CR9","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1109\/TSIPN.2017.2731051","volume":"3","author":"S Segarra","year":"2017","unstructured":"S. Segarra, A.G. Marques, G. Mateos, A. Ribeiro, Network topology inference from spectral templates. IEEE Trans. Sign. Inf. Process. Netw. 3(3), 467\u2013483 (2017). https:\/\/doi.org\/10.1109\/TSIPN.2017.2731051","journal-title":"IEEE Trans. Sign. Inf. Process. Netw."},{"key":"1128_CR10","doi-asserted-by":"publisher","unstructured":"S. Segarra, A.G. Marques, M. Goyal, S. Rey, Network topology inference from input-output diffusion pairs. in 2018 IEEE Statistical Signal Processing Workshop (SSP), pp. 508\u2013512 (2018). https:\/\/doi.org\/10.1109\/SSP.2018.8450838","DOI":"10.1109\/SSP.2018.8450838"},{"issue":"3","key":"1128_CR11","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1109\/MSP.2018.2887284","volume":"36","author":"X Dong","year":"2019","unstructured":"X. Dong, D. Thanou, M. Rabbat, P. Frossard, Learning graphs from data: a signal representation perspective. IEEE Sign. Process. Mag. 36(3), 44\u201363 (2019). https:\/\/doi.org\/10.1109\/MSP.2018.2887284","journal-title":"IEEE Sign. Process. Mag."},{"issue":"3","key":"1128_CR12","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1109\/MSP.2018.2890143","volume":"36","author":"G Mateos","year":"2019","unstructured":"G. Mateos, S. Segarra, A.G. Marques, A. Ribeiro, Connecting the dots: identifying network structure via graph signal processing. IEEE Sign. Process. Mag. 36(3), 16\u201343 (2019). https:\/\/doi.org\/10.1109\/MSP.2018.2890143","journal-title":"IEEE Sign. Process. Mag."},{"issue":"2","key":"1128_CR13","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1109\/TAI.2021.3076021","volume":"2","author":"F Xia","year":"2021","unstructured":"F. Xia, K. Sun, S. Yu, A. Aziz, L. Wan, S. Pan, H. Liu, Graph learning: a survey. IEEE Trans. Artif. Intell. 2(2), 109\u2013127 (2021). https:\/\/doi.org\/10.1109\/TAI.2021.3076021","journal-title":"IEEE Trans. Artif. Intell."},{"key":"1128_CR14","unstructured":"V. Kalofolias, How to learn a graph from smooth signals. in Proceedings of International Conference on Artificial Intelligence and Statistics, vol. 51, pp. 920\u2013929 (2016)"},{"issue":"23","key":"1128_CR15","doi-asserted-by":"publisher","first-page":"6160","DOI":"10.1109\/TSP.2016.2602809","volume":"64","author":"X Dong","year":"2016","unstructured":"X. Dong, D. Thanou, P. Frossard, P. Vandergheynst, Learning Laplacian matrix in smooth graph signal representations. IEEE Trans Sign. Process. 64(23), 6160\u20136173 (2016). https:\/\/doi.org\/10.1109\/TSP.2016.2602809","journal-title":"IEEE Trans Sign. Process."},{"key":"1128_CR16","doi-asserted-by":"publisher","unstructured":"S.P. Chepuri, S. Liu, G. Leus, A.O. Hero, Learning sparse graphs under smoothness prior. in 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 6508\u20136512 (2017). https:\/\/doi.org\/10.1109\/ICASSP.2017.7953410","DOI":"10.1109\/ICASSP.2017.7953410"},{"issue":"6","key":"1128_CR17","doi-asserted-by":"publisher","first-page":"825","DOI":"10.1109\/JSTSP.2017.2726975","volume":"11","author":"HE Egilmez","year":"2017","unstructured":"H.E. Egilmez, E. Pavez, A. Ortega, Graph learning from data under Laplacian and structural constraints. IEEE J. Sel. Top. Sign. Process. 11(6), 825\u2013841 (2017). https:\/\/doi.org\/10.1109\/JSTSP.2017.2726975","journal-title":"IEEE J. Sel. Top. Sign. Process."},{"key":"1128_CR18","first-page":"11647","volume":"32","author":"S Kumar","year":"2019","unstructured":"S. Kumar, J. Ying, J.V. Miranda Cardoso, D. Palomar, Structured graph learning via Laplacian spectral constraints. Adv. Neural Inf. Process. Syst. 32, 11647\u201311658 (2019)","journal-title":"Adv. Neural Inf. Process. Syst."},{"issue":"3","key":"1128_CR19","doi-asserted-by":"publisher","first-page":"484","DOI":"10.1109\/TSIPN.2017.2731164","volume":"3","author":"D Thanou","year":"2017","unstructured":"D. Thanou, X. Dong, D. Kressner, P. Frossard, Learning heat diffusion graphs. IEEE Trans. Sign. Inf. Process. Netw. 3(3), 484\u2013499 (2017). https:\/\/doi.org\/10.1109\/TSIPN.2017.2731164","journal-title":"IEEE Trans. Sign. Inf. Process. Netw."},{"key":"1128_CR20","doi-asserted-by":"publisher","unstructured":"V. Chandrasekaran, P.A. Parrilo, A.S. Willsky, Latent variable graphical model selection via convex optimization. in 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 1610\u20131613 (2010). https:\/\/doi.org\/10.1109\/ALLERTON.2010.5707106","DOI":"10.1109\/ALLERTON.2010.5707106"},{"key":"1128_CR21","doi-asserted-by":"publisher","unstructured":"A. Chang, T. Yao, G.I. Allen, Graphical models and dynamic latent factors for modeling functional brain connectivity. in 2019 IEEE Data Science Workshop (DSW), pp. 57\u201363 (2019). https:\/\/doi.org\/10.1109\/DSW.2019.8755783","DOI":"10.1109\/DSW.2019.8755783"},{"key":"1128_CR22","doi-asserted-by":"publisher","first-page":"314","DOI":"10.1109\/TSP.2020.3039880","volume":"69","author":"X Yang","year":"2021","unstructured":"X. Yang, M. Sheng, Y. Yuan, T.Q.S. Quek, Network topology inference from heterogeneous incomplete graph signals. IEEE Trans. Sign. Process. 69, 314\u2013327 (2021). https:\/\/doi.org\/10.1109\/TSP.2020.3039880","journal-title":"IEEE Trans. Sign. Process."},{"key":"1128_CR23","unstructured":"A. Anandkumar, D. Hsu, S. A. Javanmard, Kakade, Learning linear bayesian networks with latent variables. in International Conference on Machine Learning, pp. 249\u2013257 (2013)"},{"issue":"11","key":"1128_CR24","doi-asserted-by":"publisher","first-page":"2790","DOI":"10.1109\/TSP.2018.2818075","volume":"66","author":"J Mei","year":"2018","unstructured":"J. Mei, M.F. Moura, Silvar: single index latent variable models. IEEE Trans. Sign. Process. 66(11), 2790\u20132803 (2018). https:\/\/doi.org\/10.1109\/TSP.2018.2818075","journal-title":"IEEE Trans. Sign. Process."},{"key":"1128_CR25","doi-asserted-by":"publisher","unstructured":"A. Buciulea, S. Rey, C. Cabrera, A.G. Marques, Network reconstruction from graph-stationary signals with hidden variables. in 2019 53rd Asilomar Conference on Signals, Systems, and Computers, pp. 56\u201360 (2019). https:\/\/doi.org\/10.1109\/IEEECONF44664.2019.9048913","DOI":"10.1109\/IEEECONF44664.2019.9048913"},{"key":"1128_CR26","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1109\/TSIPN.2022.3161079","volume":"8","author":"A Buciulea","year":"2022","unstructured":"A. Buciulea, S. Rey, A.G. Marques, Learning graphs from smooth and graph-stationary signals with hidden variables. IEEE Trans. Sign. Inf. Process. Netw. 8, 273\u2013287 (2022). https:\/\/doi.org\/10.1109\/TSIPN.2022.3161079","journal-title":"IEEE Trans. Sign. Inf. Process. Netw."},{"key":"1128_CR27","doi-asserted-by":"publisher","unstructured":"S. Rey, A. Buciulea, M. Navarro, S. Segarra, A.G. Marques, Joint inference of multiple graphs with hidden variables from stationary graph signals. in 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5817\u20135821 (2022). https:\/\/doi.org\/10.1109\/ICASSP43922.2022.9747524","DOI":"10.1109\/ICASSP43922.2022.9747524"},{"issue":"22","key":"1128_CR28","doi-asserted-by":"publisher","first-page":"5911","DOI":"10.1109\/TSP.2017.2739099","volume":"65","author":"AG Marques","year":"2017","unstructured":"A.G. Marques, S. Segarra, G. Leus, A. Ribeiro, Stationary graph processes and spectral estimation. IEEE Trans. Sign. Process. 65(22), 5911\u20135926 (2017). https:\/\/doi.org\/10.1109\/TSP.2017.2739099","journal-title":"IEEE Trans. Sign. Process."},{"issue":"13","key":"1128_CR29","doi-asserted-by":"publisher","first-page":"3462","DOI":"10.1109\/TSP.2017.2690388","volume":"65","author":"N Perraudin","year":"2017","unstructured":"N. Perraudin, P. Vandergheynst, Stationary signal processing on graphs. IEEE Trans. Sign. Process. 65(13), 3462\u20133477 (2017). https:\/\/doi.org\/10.1109\/TSP.2017.2690388","journal-title":"IEEE Trans. Sign. Process."},{"key":"1128_CR30","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1016\/j.neuroimage.2016.12.061","volume":"160","author":"MG Preti","year":"2017","unstructured":"M.G. Preti, T.A. Bolton, D. Van De Ville, The dynamic functional connectome: state-of-the-art and perspectives. Neuroimage 160, 41\u201354 (2017)","journal-title":"Neuroimage"},{"issue":"2","key":"1128_CR31","doi-asserted-by":"publisher","first-page":"212","DOI":"10.1093\/bib\/bbt028","volume":"15","author":"Y Kim","year":"2014","unstructured":"Y. Kim, S. Han, S. Choi, D. Hwang, Inference of dynamic networks using time-course data. Brief. Bioinform. 15(2), 212\u2013228 (2014)","journal-title":"Brief. Bioinform."},{"issue":"1","key":"1128_CR32","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/s100510050929","volume":"11","author":"RN Mantegna","year":"1999","unstructured":"R.N. Mantegna, Hierarchical structure in financial markets. Eur. Phys. J. B- Condens. Matter Complex Syst. 11(1), 193\u2013197 (1999)","journal-title":"Eur. Phys. J. B- Condens. Matter Complex Syst."},{"key":"1128_CR33","doi-asserted-by":"publisher","unstructured":"V. Kalofolias, A. Loukas, D. Thanou, P. Frossard, Learning time varying graphs. in 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 2826\u20132830 (2017). https:\/\/doi.org\/10.1109\/ICASSP.2017.7952672","DOI":"10.1109\/ICASSP.2017.7952672"},{"key":"1128_CR34","unstructured":"K. Yamada, Y. Tanaka, A. Ortega, Time-varying graph learning with constraints on graph temporal variation (2020). Preprint at https:\/\/arxiv.org\/abs\/2001.03346"},{"key":"1128_CR35","doi-asserted-by":"crossref","unstructured":"D. Hallac, Y. Park, S. Boyd, J. Leskovec, Network inference via the time-varying graphical lasso. in Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 205\u2013213 (2017)","DOI":"10.1145\/3097983.3098037"},{"issue":"3","key":"1128_CR36","doi-asserted-by":"publisher","first-page":"432","DOI":"10.1093\/biostatistics\/kxm045","volume":"9","author":"J Friedman","year":"2008","unstructured":"J. Friedman, T. Hastie, R. Tibshirani, Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9(3), 432\u2013441 (2008)","journal-title":"Biostatistics"},{"issue":"7","key":"1128_CR37","doi-asserted-by":"publisher","first-page":"1644","DOI":"10.1109\/TSP.2013.2238935","volume":"61","author":"A Sandryhaila","year":"2013","unstructured":"A. Sandryhaila, J. Moura, Discrete signal processing on graphs. IEEE Trans. Sign. Process. 61(7), 1644\u20131656 (2013)","journal-title":"IEEE Trans. Sign. Process."},{"key":"1128_CR38","doi-asserted-by":"crossref","unstructured":"B. Girault, Stationary graph signals using an isometric graph translation. in 2015 23rd European Signal Processing Conference (EUSIPCO), pp. 1516\u20131520 (2015)","DOI":"10.1109\/EUSIPCO.2015.7362637"},{"key":"1128_CR39","unstructured":"M. Grant, S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1 beta. http:\/\/cvxr.com\/cvx (2013)"},{"key":"1128_CR40","unstructured":"Air data: Air quality data collected at outdoor monitors across the US. https:\/\/www.epa.gov\/outdoor-air-quality-data"},{"issue":"5","key":"1128_CR41","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1016\/S1473-3099(20)30120-1","volume":"20","author":"E Dong","year":"2020","unstructured":"E. Dong, H. Du, L. Gardner, An interactive web-based dashboard to track Covid-19 in real time. Lancet Infect. 20(5), 533\u2013534 (2020)","journal-title":"Lancet Infect."}],"container-title":["EURASIP Journal on Advances in Signal Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s13634-024-01128-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1186\/s13634-024-01128-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s13634-024-01128-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,13]],"date-time":"2024-03-13T08:39:34Z","timestamp":1710319174000},"score":1,"resource":{"primary":{"URL":"https:\/\/asp-eurasipjournals.springeropen.com\/articles\/10.1186\/s13634-024-01128-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,13]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["1128"],"URL":"https:\/\/doi.org\/10.1186\/s13634-024-01128-0","relation":{},"ISSN":["1687-6180"],"issn-type":[{"value":"1687-6180","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,13]]},"assertion":[{"value":"23 October 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 March 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 March 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author\u2019s declared that they have no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"33"}}