{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T11:48:23Z","timestamp":1781264903207,"version":"3.54.1"},"reference-count":62,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T00:00:00Z","timestamp":1682640000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Emmy.network foundation","award":["336810"],"award-info":[{"award-number":["336810"]}]},{"name":"Emmy.network foundation","award":["336814"],"award-info":[{"award-number":["336814"]}]},{"name":"Emmy.network foundation","award":["341985"],"award-info":[{"award-number":["341985"]}]},{"DOI":"10.13039\/501100002341","name":"Academy of Finland via the Centre of Excellence program","doi-asserted-by":"publisher","award":["336810"],"award-info":[{"award-number":["336810"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002341","name":"Academy of Finland via the Centre of Excellence program","doi-asserted-by":"publisher","award":["336814"],"award-info":[{"award-number":["336814"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002341","name":"Academy of Finland via the Centre of Excellence program","doi-asserted-by":"publisher","award":["341985"],"award-info":[{"award-number":["341985"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002341","name":"Academy of Finland via the Postdoctoral Researcher program","doi-asserted-by":"publisher","award":["336810"],"award-info":[{"award-number":["336810"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002341","name":"Academy of Finland via the Postdoctoral Researcher program","doi-asserted-by":"publisher","award":["336814"],"award-info":[{"award-number":["336814"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002341","name":"Academy of Finland via the Postdoctoral Researcher program","doi-asserted-by":"publisher","award":["341985"],"award-info":[{"award-number":["341985"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Academy of Finland","award":["336810"],"award-info":[{"award-number":["336810"]}]},{"name":"Academy of Finland","award":["336814"],"award-info":[{"award-number":["336814"]}]},{"name":"Academy of Finland","award":["341985"],"award-info":[{"award-number":["341985"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Protein\u2013protein interaction (PPI) networks consist of the physical and\/or functional interactions between the proteins of an organism, and they form the basis for the field of network medicine. Since the biophysical and high-throughput methods used to form PPI networks are expensive, time-consuming, and often contain inaccuracies, the resulting networks are usually incomplete. In order to infer missing interactions in these networks, we propose a novel class of link prediction methods based on continuous-time classical and quantum walks. In the case of quantum walks, we examine the usage of both the network adjacency and Laplacian matrices for specifying the walk dynamics. We define a score function based on the corresponding transition probabilities and perform tests on six real-world PPI datasets. Our results show that continuous-time classical random walks and quantum walks using the network adjacency matrix can successfully predict missing protein\u2013protein interactions, with performance rivalling the state-of-the-art.<\/jats:p>","DOI":"10.3390\/e25050730","type":"journal-article","created":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T04:36:15Z","timestamp":1682656575000},"page":"730","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Link Prediction with Continuous-Time Classical and Quantum Walks"],"prefix":"10.3390","volume":"25","author":[{"given":"Mark","family":"Goldsmith","sequence":"first","affiliation":[{"name":"Algorithmiq Ltd., Kanavakatu 3 C, FI-00160 Helsinki, Finland"},{"name":"Complex Systems Research Group, Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Harto","family":"Saarinen","sequence":"additional","affiliation":[{"name":"Algorithmiq Ltd., Kanavakatu 3 C, FI-00160 Helsinki, Finland"},{"name":"Complex Systems Research Group, Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guillermo","family":"Garc\u00eda-P\u00e9rez","sequence":"additional","affiliation":[{"name":"Algorithmiq Ltd., Kanavakatu 3 C, FI-00160 Helsinki, Finland"},{"name":"Complex Systems Research Group, Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland"},{"name":"QTF Centre of Excellence, Department of Physics, Faculty of Science, University of Helsinki, FI-00014 Helsinki, Finland"},{"name":"InstituteQ-The Finnish Quantum Institute, University of Helsinki, FI-00014 Helsinki, Finland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joonas","family":"Malmi","sequence":"additional","affiliation":[{"name":"Algorithmiq Ltd., Kanavakatu 3 C, FI-00160 Helsinki, Finland"},{"name":"QTF Centre of Excellence, Department of Physics, Faculty of Science, University of Helsinki, FI-00014 Helsinki, Finland"},{"name":"InstituteQ-The Finnish Quantum Institute, University of Helsinki, FI-00014 Helsinki, Finland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4665-9284","authenticated-orcid":false,"given":"Matteo A. C.","family":"Rossi","sequence":"additional","affiliation":[{"name":"Algorithmiq Ltd., Kanavakatu 3 C, FI-00160 Helsinki, Finland"},{"name":"QTF Centre of Excellence, Department of Physics, Faculty of Science, University of Helsinki, FI-00014 Helsinki, Finland"},{"name":"QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland"},{"name":"InstituteQ-The Finnish Quantum Institute, Aalto University, FI-00076 Aalto, Finland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sabrina","family":"Maniscalco","sequence":"additional","affiliation":[{"name":"Algorithmiq Ltd., Kanavakatu 3 C, FI-00160 Helsinki, Finland"},{"name":"Complex Systems Research Group, Department of Mathematics and Statistics, University of Turku, FI-20014 Turku, Finland"},{"name":"QTF Centre of Excellence, Department of Physics, Faculty of Science, University of Helsinki, FI-00014 Helsinki, Finland"},{"name":"InstituteQ-The Finnish Quantum Institute, University of Helsinki, FI-00014 Helsinki, Finland"},{"name":"QTF Centre of Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland"},{"name":"InstituteQ-The Finnish Quantum Institute, Aalto University, FI-00076 Aalto, Finland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/S0378-8733(03)00009-1","article-title":"Friends and neighbors on the web","volume":"25","author":"Adamic","year":"2003","journal-title":"Soc. Netw."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Murata, T., and Moriyasu, S. (2007, January 2\u20135). Link prediction of social networks based on weighted proximity measures. Proceedings of the IEEE\/WIC\/ACM International Conference on Web Intelligence (WI\u201907), Fremont, CA, USA.","DOI":"10.1109\/WI.2007.52"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Leskovec, J., Huttenlocher, D., and Kleinberg, J. (2010, January 26\u201330). Predicting positive and negative links in online social networks. Proceedings of the 19th International Conference on World Wide Web, Raleigh, CA, USA.","DOI":"10.1145\/1772690.1772756"},{"key":"ref_4","first-page":"1","article-title":"Network-based prediction of protein interactions","volume":"10","author":"Luck","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"58007","DOI":"10.1209\/0295-5075\/89\/58007","article-title":"Link prediction based on local random walk","volume":"89","author":"Liu","year":"2010","journal-title":"EPL Europhys. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"124289","DOI":"10.1016\/j.physa.2020.124289","article-title":"Link prediction techniques, applications, and performance: A survey","volume":"553","author":"Kumar","year":"2020","journal-title":"Phys. Stat. Mech. Its Appl."},{"key":"ref_7","first-page":"69:1","article-title":"A Survey of Link Prediction in Complex Networks","volume":"49","author":"Berzal","year":"2017","journal-title":"ACM Comput. Surv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"103217","DOI":"10.1016\/j.isci.2021.103217","article-title":"Progresses and challenges in link prediction","volume":"24","author":"Zhou","year":"2021","journal-title":"iScience"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4597391","DOI":"10.1155\/2021\/4597391","article-title":"A Random Walk with Restart Model Based on Common Neighbors for Predicting the Clinical Drug Combinations on Coronary Heart Disease","volume":"2021","author":"Che","year":"2021","journal-title":"J. Healthc. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"125783","DOI":"10.1016\/j.physa.2021.125783","article-title":"Biased random walk with restart for link prediction with graph embedding method","volume":"570","author":"Zhou","year":"2021","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0169-7552(98)00110-X","article-title":"The anatomy of a large-scale hypertextual web search engine","volume":"30","author":"Brin","year":"1998","journal-title":"Comput. Netw. Isdn Syst."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Das Sarma, A., Molla, A.R., Pandurangan, G., and Upfal, E. (2013, January 3\u20136). Fast distributed pagerank computation. Proceedings of the International Conference on Distributed Computing and Networking, Mumbai, India.","DOI":"10.1007\/978-3-642-35668-1_2"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1109\/TKDE.2007.46","article-title":"Random-walk computation of similarities between nodes of a graph with application to collaborative recommendation","volume":"19","author":"Fouss","year":"2007","journal-title":"IEEE Trans. Knowl. Data Eng."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Pan, J.Y., Yang, H.J., Faloutsos, C., and Duygulu, P. (2004, January 22\u201325). Automatic multimedia cross-modal correlation discovery. Proceedings of the 10th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Seattle, WA, USA.","DOI":"10.1145\/1014052.1014135"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Tong, H., Faloutsos, C., and Pan, J.Y. (2006, January 18\u201322). Fast random walk with restart and its applications. Proceedings of the Sixth International Conference on Data Mining (ICDM\u201906), Hong Kong, China.","DOI":"10.1109\/ICDM.2006.70"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1103\/PhysRevA.58.915","article-title":"Quantum computation and decision trees","volume":"58","author":"Farhi","year":"1998","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1103\/PhysRevA.48.1687","article-title":"Quantum random walks","volume":"48","author":"Aharonov","year":"1993","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1080\/00107151031000110776","article-title":"Quantum random walks: An introductory overview","volume":"44","author":"Kempe","year":"2003","journal-title":"Contemp. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1007\/s11128-012-0432-5","article-title":"Quantum walks: A comprehensive review","volume":"11","year":"2012","journal-title":"Quantum Inf. Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"180501","DOI":"10.1103\/PhysRevLett.102.180501","article-title":"Universal computation by quantum walk","volume":"102","author":"Childs","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.physrep.2011.01.002","article-title":"Continuous-time quantum walks: Models for coherent transport on complex networks","volume":"502","author":"Blumen","year":"2011","journal-title":"Phys. Rep."},{"key":"ref_22","first-page":"1329","article-title":"Link prediction using discrete-time quantum walk","volume":"24","author":"Qian","year":"2017","journal-title":"Teh. Vjesn."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"032605","DOI":"10.1103\/PhysRevA.107.032605","article-title":"Quantum link prediction in complex networks","volume":"107","author":"Moutinho","year":"2023","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Manouchehri, K., and Wang, J. (2014). Physical Implementation of Quantum Walks, Springer.","DOI":"10.1007\/978-3-642-36014-5"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1126\/science.abo0608","article-title":"Tweezer-programmable 2D quantum walks in a Hubbard-regime lattice","volume":"377","author":"Young","year":"2022","journal-title":"Science"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1524","DOI":"10.1364\/OPTICA.396228","article-title":"Experimental realization of continuous-time quantum walks on directed graphs and their application in PageRank","volume":"7","author":"Wang","year":"2020","journal-title":"Optica"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"eaat3174","DOI":"10.1126\/sciadv.aat3174","article-title":"Experimental two-dimensional quantum walk on a photonic chip","volume":"4","author":"Tang","year":"2018","journal-title":"Sci. Adv."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1500","DOI":"10.1126\/science.1193515","article-title":"Quantum Walks of Correlated Photons","volume":"329","author":"Peruzzo","year":"2010","journal-title":"Science"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1229","DOI":"10.1126\/science.1260364","article-title":"Strongly correlated quantum walks in optical lattices","volume":"347","author":"Preiss","year":"2015","journal-title":"Science"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"948","DOI":"10.1126\/science.abg7812","article-title":"Quantum walks on a programmable two-dimensional 62-qubit superconducting processor","volume":"372","author":"Gong","year":"2021","journal-title":"Science"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1126\/science.aaw1611","article-title":"Strongly correlated quantum walks with a 12-qubit superconducting processor","volume":"364","author":"Yan","year":"2019","journal-title":"Science"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"055303","DOI":"10.1088\/1751-8121\/aa53a9","article-title":"Efficient quantum circuits for continuous-time quantum walks on composite graphs","volume":"50","author":"Loke","year":"2017","journal-title":"J. Phys. Math. Theor."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"11511","DOI":"10.1038\/ncomms11511","article-title":"Efficient quantum walk on a quantum processor","volume":"7","author":"Qiang","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"986","DOI":"10.1016\/j.cell.2011.02.016","article-title":"Interactome networks and human disease","volume":"144","author":"Vidal","year":"2011","journal-title":"Cell"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1016\/j.cell.2005.08.029","article-title":"A human protein\u2013protein interaction network: A resource for annotating the proteome","volume":"122","author":"Stelzl","year":"2005","journal-title":"Cell"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1212","DOI":"10.1016\/j.cell.2014.10.050","article-title":"A proteome-scale map of the human interactome network","volume":"159","author":"Rolland","year":"2014","journal-title":"Cell"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1038\/s41586-020-2188-x","article-title":"A reference map of the human binary protein interactome","volume":"580","author":"Luck","year":"2020","journal-title":"Nature"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Yuen, H.Y., and Jansson, J. (2020, January 26\u201328). Better Link Prediction for Protein-Protein Interaction Networks. Proceedings of the 2020 IEEE 20th International Conference on Bioinformatics and Bioengineering (BIBE), Cincinnati, OH, USA.","DOI":"10.1109\/BIBE50027.2020.00017"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Yuen, H.Y., and Jansson, J. (2023). Normalized L3-based link prediction in protein protein interaction networks. BMC Bioinform., 24.","DOI":"10.1186\/s12859-023-05178-3"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1002\/asi.20591","article-title":"The link-prediction problem for social networks","volume":"58","author":"Kleinberg","year":"2007","journal-title":"J. Am. Soc. Inf. Sci. Technol."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1016\/S0378-4371(02)00736-7","article-title":"Evolution of the social network of scientific collaborations","volume":"311","author":"Jeong","year":"2002","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2017.07.007","article-title":"Random walks and diffusion on networks","volume":"716","author":"Masuda","year":"2017","journal-title":"Phys. Rep."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1023\/A:1019609420309","article-title":"An Example of the Difference Between Quantum and Classical Random Walks","volume":"1","author":"Childs","year":"2002","journal-title":"Quantum Inf. Process."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"4029","DOI":"10.1007\/s11128-016-1373-1","article-title":"Wong, L.T.; Nahimov, N. Laplacian versus adjacency matrix in quantum walk search","volume":"15","author":"Thomas","year":"2016","journal-title":"Quantum Inf. Process."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"022314","DOI":"10.1103\/PhysRevA.70.022314","article-title":"Spatial search by quantum walk","volume":"70","author":"Childs","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"D535","DOI":"10.1093\/nar\/gkj109","article-title":"BioGRID: A general repository for interaction datasets","volume":"34","author":"Stark","year":"2006","journal-title":"Nucleic Acids Res."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1752-0509-6-92","article-title":"HINT: High-quality protein interactomes and their applications in understanding human disease","volume":"6","author":"Das","year":"2012","journal-title":"BMC Syst. Biol."},{"key":"ref_48","first-page":"baz005","article-title":"APID database: Redefining protein\u2013protein interaction experimental evidences and binary interactomes","volume":"2019","author":"Lambourne","year":"2019","journal-title":"Database"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"W529","DOI":"10.1093\/nar\/gkw363","article-title":"APID interactomes: Providing proteome-based interactomes with controlled quality for multiple species and derived networks","volume":"44","author":"Lopes","year":"2016","journal-title":"Nucleic Acids Res."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"D536","DOI":"10.1093\/nar\/gkv1115","article-title":"Integrated interactions database: Tissue-specific view of the human and model organism interactomes","volume":"44","author":"Kotlyar","year":"2016","journal-title":"Nucleic Acids Res."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","author":"Albert","year":"1999","journal-title":"Science"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1148\/radiology.143.1.7063747","article-title":"The meaning and use of the area under a receiver operating characteristic (ROC) curve","volume":"143","author":"Hanley","year":"1982","journal-title":"Radiology"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"861","DOI":"10.1016\/j.patrec.2005.10.010","article-title":"An introduction to ROC analysis","volume":"27","author":"Fawcett","year":"2006","journal-title":"Pattern Recognit. Lett."},{"key":"ref_54","unstructured":"Armengol, E., Boixader, D., and Grimaldo, F. (2015). Artificial Intelligence Research and Development: Proceedings of the 18th International Conference of the Catalan Association for Artificial Intelligence, IOS Press."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Saito, T., and Rehmsmeier, M. (2015). The precision\u2013recall plot is more informative than the ROC plot when evaluating binary classifiers on imbalanced datasets. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0118432"},{"key":"ref_56","first-page":"201424644","article-title":"Toward link predictability of complex networks","volume":"112","author":"Pan","year":"2015","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"2425","DOI":"10.1093\/bioinformatics\/bty112","article-title":"Prediction of potential disease-associated microRNAs using structural perturbation method","volume":"34","author":"Zeng","year":"2018","journal-title":"Bioinformatics"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"026107","DOI":"10.1103\/PhysRevE.65.026107","article-title":"Growing scale-free networks with tunable clustering","volume":"65","author":"Holme","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1137\/S00361445024180","article-title":"Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later","volume":"45","author":"Moler","year":"2003","journal-title":"SIAM Rev."},{"key":"ref_60","unstructured":"Wallach, H., Larochelle, H., Beygelzimer, A., d\u2019Alch\u00e9-Buc, F., Fox, E., and Garnett, R. (2019). Advances in Neural Information Processing Systems 32, Curran Associates, Inc."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Bader, P., Blanes, S., and Casas, F. (2019). Computing the matrix exponential with an optimized Taylor polynomial approximation. Mathematics, 7.","DOI":"10.3390\/math7121174"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1007\/BF02165411","article-title":"Gaussian Elimination is not Optimal","volume":"13","author":"Strassen","year":"1969","journal-title":"Numer. Math."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/5\/730\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:25:20Z","timestamp":1760124320000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/5\/730"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,28]]},"references-count":62,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["e25050730"],"URL":"https:\/\/doi.org\/10.3390\/e25050730","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,28]]}}}