{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T16:19:43Z","timestamp":1783095583759,"version":"3.54.6"},"reference-count":69,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T00:00:00Z","timestamp":1743638400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"STI 2030\u2013Major Projects","award":["2022ZD0211400"],"award-info":[{"award-number":["2022ZD0211400"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["T2293771"],"award-info":[{"award-number":["T2293771"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2022M710620"],"award-info":[{"award-number":["2022M710620"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Sichuan Science and Technology Program","award":["2023NSFSC1919"],"award-info":[{"award-number":["2023NSFSC1919"]}]},{"name":"Sichuan Science and Technology Program","award":["2023NSFSC1353"],"award-info":[{"award-number":["2023NSFSC1353"]}]},{"name":"Project of Huzhou Science and Technology Bureau","award":["2021YZ12"],"award-info":[{"award-number":["2021YZ12"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Identifying key nodes in networks is a fundamental problem in network science. This study proposes a quantum deep reinforcement learning (QDRL) framework that integrates reinforcement learning with a variational quantum graph neural network, effectively identifying distributed influential nodes while preserving the network\u2019s fundamental topological properties. By leveraging principles of quantum computing, our method is designed to reduce model parameters and computational complexity compared to traditional neural networks. Trained on small networks, it demonstrated strong generalization across diverse scenarios. We compared the proposed algorithm with some classical node ranking and network dismantling algorithms on various synthetical and empirical networks. The results suggest that the proposed algorithm outperforms existing baseline methods. Moreover, in synthetic networks based on Erd\u0151s\u2013R\u00e9nyi and Watts\u2013Strogatz models, QDRL demonstrated its capability to alleviate the issue of localization in network information propagation and node influence ranking. Our research provides insights into addressing fundamental problems in complex networks using quantum machine learning, demonstrating the potential of quantum approaches for network analysis tasks.<\/jats:p>","DOI":"10.3390\/e27040382","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T05:51:26Z","timestamp":1743659486000},"page":"382","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Finding Key Nodes in Complex Networks Through Quantum Deep Reinforcement Learning"],"prefix":"10.3390","volume":"27","author":[{"given":"Juechan","family":"Xiong","sequence":"first","affiliation":[{"name":"Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China"},{"name":"Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China, Huzhou 313001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2323-163X","authenticated-orcid":false,"given":"Xiao-Long","family":"Ren","sequence":"additional","affiliation":[{"name":"Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China, Huzhou 313001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Linyuan","family":"L\u00fc","sequence":"additional","affiliation":[{"name":"School of Cyber Science and Technology, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","article-title":"Statistical mechanics of complex networks","volume":"74","author":"Albert","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wasserman, S., and Faust, K. (1994). Social Network Analysis: Methods and Applications, Cambridge University Press.","DOI":"10.1017\/CBO9780511815478"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1073\/pnas.98.2.404","article-title":"The structure of scientific collaboration networks","volume":"98","author":"Newman","year":"2001","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1038\/43601","article-title":"Diameter of the world-wide web","volume":"401","author":"Albert","year":"1999","journal-title":"Nature"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/s100510050359","article-title":"How popular is your paper? An empirical study of the citation distribution","volume":"4","author":"Redner","year":"1998","journal-title":"Eur. Phys. J. B-Condens. Matter Complex Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1038\/35004572","article-title":"Simple rules yield complex food webs","volume":"404","author":"Williams","year":"2000","journal-title":"Nature"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.1038\/nature04209","article-title":"Towards a proteome-scale map of the human protein\u2013protein interaction network","volume":"437","author":"Rual","year":"2005","journal-title":"Nature"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"068702","DOI":"10.1103\/PhysRevLett.109.068702","article-title":"Locating the source of diffusion in large-scale networks","volume":"109","author":"Pinto","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1038\/ncomms1396","article-title":"Ranking stability and super-stable nodes in complex networks","volume":"2","author":"Ghoshal","year":"2011","journal-title":"Nat. Commun."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"128702","DOI":"10.1103\/PhysRevLett.109.128702","article-title":"Localization and spreading of diseases in complex networks","volume":"109","author":"Goltsev","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2016.05.004","article-title":"Vital nodes identification in complex networks","volume":"650","author":"Chen","year":"2016","journal-title":"Phys. Rep."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"29200","DOI":"10.1109\/ACCESS.2018.2843532","article-title":"Identification of vital nodes in complex network via belief propagation and node reinsertion","volume":"6","author":"Zhong","year":"2018","journal-title":"IEEE Access"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Xu, X., Zhu, C., Wang, Q., Zhu, X., and Zhou, Y. (2020). Identifying vital nodes in complex networks by adjacency information entropy. Sci. Rep., 10.","DOI":"10.1038\/s41598-020-59616-w"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1007\/s00355-023-01456-4","article-title":"Centrality measures in networks","volume":"61","author":"Bloch","year":"2023","journal-title":"Soc. Choice Welf."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1038\/35019019","article-title":"Error and attack tolerance of complex networks","volume":"406","author":"Albert","year":"2000","journal-title":"Nature"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4626","DOI":"10.1103\/PhysRevLett.85.4626","article-title":"Resilience of the internet to random breakdowns","volume":"85","author":"Cohen","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1109\/MSP.2017.2693418","article-title":"Geometric deep learning: Going beyond euclidean data","volume":"34","author":"Bronstein","year":"2017","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2319","DOI":"10.1126\/science.290.5500.2319","article-title":"A global geometric framework for nonlinear dimensionality reduction","volume":"290","author":"Tenenbaum","year":"2000","journal-title":"Science"},{"key":"ref_19","first-page":"585","article-title":"Laplacian eigenmaps and spectral techniques for embedding and clustering","volume":"14","author":"Belkin","year":"2001","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Grover, A., and Leskovec, J. (2016, January 13\u201317). node2vec: Scalable feature learning for networks. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA.","DOI":"10.1145\/2939672.2939754"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Perozzi, B., Al-Rfou, R., and Skiena, S. (2014, January 24\u201327). Deepwalk: Online learning of social representations. Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, New York, NY, USA.","DOI":"10.1145\/2623330.2623732"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.aiopen.2021.01.001","article-title":"Graph neural networks: A review of methods and applications","volume":"1","author":"Zhou","year":"2020","journal-title":"AI Open"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"12368","DOI":"10.1073\/pnas.1605083113","article-title":"Network dismantling","volume":"113","author":"Braunstein","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.cosrev.2018.02.002","article-title":"The critical node detection problem in networks: A survey","volume":"28","author":"Lalou","year":"2018","journal-title":"Comput. Sci. Rev."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"6554","DOI":"10.1073\/pnas.1806108116","article-title":"Generalized network dismantling","volume":"116","author":"Ren","year":"2019","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_26","first-page":"6348","article-title":"Learning combinatorial optimization algorithms over graphs","volume":"30","author":"Khalil","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_27","first-page":"9839","article-title":"Reinforcement learning for solving the vehicle routing problem","volume":"31","author":"Nazari","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_28","first-page":"539","article-title":"Combinatorial optimization with graph convolutional networks and guided tree search","volume":"31","author":"Li","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Sutton, R.S., and Barto, A.G. (1998). Reinforcement Learning: An Introduction, MIT Press.","DOI":"10.1109\/TNN.1998.712192"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1038\/nature14236","article-title":"Human-level control through deep reinforcement learning","volume":"518","author":"Mnih","year":"2015","journal-title":"Nature"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1207","DOI":"10.1109\/TSMCB.2008.925743","article-title":"Quantum reinforcement learning","volume":"38","author":"Dong","year":"2008","journal-title":"IEEE Trans. Syst. Man Cybern. Part B Cybern."},{"key":"ref_32","unstructured":"Meyer, N., Ufrecht, C., Periyasamy, M., Scherer, D.D., Plinge, A., and Mutschler, C. (2022). A survey on quantum reinforcement learning. arXiv."},{"key":"ref_33","unstructured":"Cheng, Z., Zhang, K., Shen, L., and Tao, D. (2023, January 7\u201314). Offline quantum reinforcement learning in a conservative manner. Proceedings of the AAAI Conference on Artificial Intelligence, Washington, DC, USA."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Eisenmann, S., Hein, D., Udluft, S., and Runkler, T.A. (2024, January 15\u201320). Model-based Offline Quantum Reinforcement Learning. Proceedings of the 2024 IEEE International Conference on Quantum Computing and Engineering (QCE), Montreal, QC, Canada.","DOI":"10.1109\/QCE60285.2024.00175"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1007\/BF00992698","article-title":"Q-learning","volume":"8","author":"Watkins","year":"1992","journal-title":"Mach. Learn."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1145\/203330.203343","article-title":"Temporal difference learning and TD-Gammon","volume":"38","author":"Tesauro","year":"1995","journal-title":"Commun. ACM"},{"key":"ref_37","unstructured":"Watkins, C.J.C.H. (1989). Learning from Delayed Rewards. [Ph.D. Thesis, University of Cambridge]."},{"key":"ref_38","unstructured":"Melo, F.S. (2001). Convergence of Q-learning: A simple proof. Technical Report, Institute for Systems and Robotics, Instituto Superior T\u00e9cnico."},{"key":"ref_39","unstructured":"Nielsen, M.A., and Chuang, I.L. (2010). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"79","DOI":"10.22331\/q-2018-08-06-79","article-title":"Quantum computing in the NISQ era and beyond","volume":"2","author":"Preskill","year":"2018","journal-title":"Quantum"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1038\/s42254-021-00348-9","article-title":"Variational quantum algorithms","volume":"3","author":"Cerezo","year":"2021","journal-title":"Nat. Rev. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"043001","DOI":"10.1088\/2058-9565\/ab4eb5","article-title":"Parameterized quantum circuits as machine learning models","volume":"4","author":"Benedetti","year":"2019","journal-title":"Quantum Sci. Technol."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"023023","DOI":"10.1088\/1367-2630\/18\/2\/023023","article-title":"The theory of variational hybrid quantum-classical algorithms","volume":"18","author":"McClean","year":"2016","journal-title":"New J. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"4213","DOI":"10.1038\/ncomms5213","article-title":"A variational eigenvalue solver on a photonic quantum processor","volume":"5","author":"Peruzzo","year":"2014","journal-title":"Nat. Commun."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"032309","DOI":"10.1103\/PhysRevA.98.032309","article-title":"Quantum circuit learning","volume":"98","author":"Mitarai","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"032430","DOI":"10.1103\/PhysRevA.103.032430","article-title":"Effect of data encoding on the expressive power of variational quantum-machine-learning models","volume":"103","author":"Schuld","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1791","DOI":"10.1038\/s41467-021-21728-w","article-title":"Cost function dependent barren plateaus in shallow parametrized quantum circuits","volume":"12","author":"Cerezo","year":"2021","journal-title":"Nat. Commun."},{"key":"ref_48","unstructured":"Crooks, G.E. (2019). Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition. arXiv."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1038\/s41586-019-0980-2","article-title":"Supervised learning with quantum-enhanced feature spaces","volume":"567","author":"Temme","year":"2019","journal-title":"Nature"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"040504","DOI":"10.1103\/PhysRevLett.122.040504","article-title":"Quantum machine learning in feature hilbert spaces","volume":"122","author":"Schuld","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/S0377-0427(99)00088-6","article-title":"Closed-form expressions for the finite difference approximations of first and higher derivatives based on Taylor series","volume":"107","author":"Khan","year":"1999","journal-title":"J. Comput. Appl. Math."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"032331","DOI":"10.1103\/PhysRevA.99.032331","article-title":"Evaluating analytic gradients on quantum hardware","volume":"99","author":"Schuld","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"341","DOI":"10.22331\/q-2020-10-11-341","article-title":"Yao. jl: Extensible, efficient framework for quantum algorithm design","volume":"4","author":"Luo","year":"2020","journal-title":"Quantum"},{"key":"ref_54","first-page":"1024","article-title":"Inductive representation learning on large graphs","volume":"30","author":"Hamilton","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_55","unstructured":"Kipf, T.N., and Welling, M. (2016). Semi-supervised classification with graph convolutional networks. arXiv."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Sch\u00fctze, H., Manning, C.D., and Raghavan, P. (2008). Introduction to Information Retrieval, Cambridge University Press.","DOI":"10.1017\/CBO9780511809071"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1007\/s101070100290","article-title":"UOBYQA: Unconstrained optimization by quadratic approximation","volume":"92","author":"Powell","year":"2002","journal-title":"Math. Program."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"3838","DOI":"10.1073\/pnas.1009440108","article-title":"Mitigation of malicious attacks on networks","volume":"108","author":"Schneider","year":"2011","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"226","DOI":"10.22331\/q-2020-02-06-226","article-title":"Data re-uploading for a universal quantum classifier","volume":"4","author":"Latorre","year":"2020","journal-title":"Quantum"},{"key":"ref_60","first-page":"17","article-title":"On the evolution of random graphs","volume":"5","year":"1960","journal-title":"Publ. Math. Inst. Hung. Acad. Sci."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Newman, M. (2018). Networks, Oxford University Press.","DOI":"10.1093\/oso\/9780198805090.001.0001"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1080\/0022250X.1972.9989806","article-title":"Factoring and weighting approaches to status scores and clique identification","volume":"2","author":"Bonacich","year":"1972","journal-title":"J. Math. Sociol."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"16","DOI":"10.17730\/humo.7.3.f4033344851gl053","article-title":"A mathematical model for group structures","volume":"7","author":"Bavelas","year":"1948","journal-title":"Hum. Organ."},{"key":"ref_65","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_66","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/0378-8733(83)90028-X","article-title":"Network structure and minimum degree","volume":"5","author":"Seidman","year":"1983","journal-title":"Soc. Netw."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"7821","DOI":"10.1073\/pnas.122653799","article-title":"Community structure in social and biological networks","volume":"99","author":"Girvan","year":"2002","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"7794","DOI":"10.1073\/pnas.0407994102","article-title":"The worldwide air transportation network: Anomalous centrality, community structure, and cities\u2019 global roles","volume":"102","author":"Guimera","year":"2005","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1086\/jar.33.4.3629752","article-title":"An information flow model for conflict and fission in small groups","volume":"33","author":"Zachary","year":"1977","journal-title":"J. Anthropol. Res."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/382\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:09:14Z","timestamp":1760029754000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/382"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,3]]},"references-count":69,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["e27040382"],"URL":"https:\/\/doi.org\/10.3390\/e27040382","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,3]]}}}