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Research Funds for Undergraduate Universities in Liaoning Province, China","award":["LJ212410144003"],"award-info":[{"award-number":["LJ212410144003"]}]},{"name":"Basic Research Funds for Undergraduate Universities in Liaoning Province, China","award":["LJKZZ20220038"],"award-info":[{"award-number":["LJKZZ20220038"]}]},{"name":"Basic Research Funds for Undergraduate Universities in Liaoning Province, China","award":["SYLUGXXZ202204"],"award-info":[{"award-number":["SYLUGXXZ202204"]}]},{"name":"Basic Research Funds for Undergraduate Universities in Liaoning Province, China","award":["2023JH26\/10300007"],"award-info":[{"award-number":["2023JH26\/10300007"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Identifying influential nodes in complex networks is essential for a wide range of applications, from social network analysis to enhancing infrastructure resilience. While quantum walk-based methods offer potential advantages, existing approaches face challenges in dimensionality, computational efficiency, and accuracy. To address these limitations, this study proposes a novel method inspired by the one-dimensional discrete-time quantum walk (IOQW). This design enables the development of a simplified shift operator that leverages both self-loops and the network\u2019s structural connectivity. Furthermore, degree centrality and path-based features are integrated into the coin operator, enhancing the accuracy and scalability of the IOQW framework. Comparative evaluations against state-of-the-art quantum and classical methods demonstrate that IOQW excels in capturing both local and global topological properties while maintaining a low computational complexity of O(N\u27e8k\u27e9), where \u27e8k\u27e9 denotes the average degree. These advancements establish IOQW as a powerful and practical tool for influential node identification in complex networks, bridging quantum-inspired techniques with real-world network science applications.<\/jats:p>","DOI":"10.3390\/e27060634","type":"journal-article","created":{"date-parts":[[2025,6,16]],"date-time":"2025-06-16T10:47:22Z","timestamp":1750070842000},"page":"634","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Method Inspired by One-Dimensional Discrete-Time Quantum Walks for Influential Node Identification"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1938-0149","authenticated-orcid":false,"given":"Wen","family":"Liang","sequence":"first","affiliation":[{"name":"College of Information Science and Engineering, Shenyang Ligong University, Shenyang 110159, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yifan","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Information Science and Engineering, Shenyang Ligong University, Shenyang 110159, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-2171-9770","authenticated-orcid":false,"given":"Qiwei","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Public Administration and Policy, Dalian University of Technology, Dalian 116081, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenbo","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Information Science and Engineering, Shenyang Ligong University, Shenyang 110159, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1038\/s42254-023-00676-y","article-title":"Robustness and resilience of complex networks","volume":"6","author":"Artime","year":"2024","journal-title":"Nat. Rev. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1038\/s41467-023-44257-0","article-title":"DomiRank Centrality reveals structural fragility of complex networks via node dominance","volume":"15","author":"Engsig","year":"2024","journal-title":"Nat. Commun."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Nandini, Y., Lakshmi, T.J., Enduri, M.K., and Sharma, H. (2024). Link prediction in complex networks using average centrality-based similarity score. Entropy, 26.","DOI":"10.3390\/e26060433"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"10186","DOI":"10.1109\/ACCESS.2024.3355096","article-title":"Node significance analysis in complex networks using machine learning and centrality measures","volume":"12","author":"Hajarathaiah","year":"2024","journal-title":"IEEE Access"},{"key":"ref_5","first-page":"021007","article-title":"Information propagation in multilayer systems with higher-order interactions across timescales","volume":"14","author":"Nicoletti","year":"2024","journal-title":"Phys. Rev. X"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"109766","DOI":"10.1016\/j.ress.2023.109766","article-title":"Node importance identification of unweighted urban rail transit network: An Adjacency Information Entropy based approach","volume":"242","author":"Huang","year":"2024","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"124946","DOI":"10.1016\/j.eswa.2024.124946","article-title":"Source localization in complex networks with optimal observers based on maximum entropy sampling","volume":"256","author":"Hu","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_8","first-page":"437","article-title":"Node Importance in Controlled Complex Networks","volume":"66","author":"Zhou","year":"2019","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3597","DOI":"10.1109\/TSC.2024.3433580","article-title":"Enhanced Network Traffic Anomaly Detection: Integration of Tensor Eigenvector Centrality with Low-Rank Recovery Models","volume":"17","author":"Lin","year":"2024","journal-title":"IEEE Trans. Serv. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"111638","DOI":"10.1016\/j.automatica.2024.111638","article-title":"Control of multiconsensus in multi-agent systems based on eigenvector centrality","volume":"164","author":"Tomaselli","year":"2024","journal-title":"Automatica"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"122819","DOI":"10.1016\/j.eswa.2023.122819","article-title":"DeepWalk with Reinforcement Learning (DWRL) for node embedding","volume":"243","author":"Jeyaraj","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106207","DOI":"10.1016\/j.neunet.2024.106207","article-title":"A Comprehensive Survey on Deep Graph Representation Learning","volume":"173","author":"Ju","year":"2024","journal-title":"Neural Netw."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2574","DOI":"10.1007\/s10489-020-01970-3","article-title":"QSIM: A novel approach to node proximity estimation based on Discrete-time quantum walk","volume":"51","author":"Wang","year":"2021","journal-title":"Appl. Intell."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1493","DOI":"10.1007\/s10489-021-02342-1","article-title":"RED: Learning the role embedding in networks via Discrete-time quantum walk","volume":"52","author":"Wang","year":"2022","journal-title":"Appl. Intell."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"23163","DOI":"10.1109\/ACCESS.2024.3363635","article-title":"An Enhanced Gravity Model for Determining Crucial Nodes in Social Networks Based on Degree K-Shell Eigenvector Index","volume":"12","author":"Singh","year":"2024","journal-title":"IEEE Access"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"e2313708120","DOI":"10.1073\/pnas.2313708120","article-title":"The probability of epidemic burnout in the stochastic SIR model with vital dynamics","volume":"121","author":"Parsons","year":"2024","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Alazman, I., Mishra, M.N., Alkahtani, B.S., and Dubey, R.S. (2024). Analysis of infection and diffusion coefficient in an sir model by using generalized fractional derivative. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8090537"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s11128-016-1456-z","article-title":"Comparing classical and quantum PageRanks","volume":"16","author":"Loke","year":"2017","journal-title":"Quantum Inf. Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"032305","DOI":"10.1103\/PhysRevA.96.032305","article-title":"Quantum centrality testing on directed graphs via PT-symmetric quantum walks","volume":"96","author":"Izaac","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"062448","DOI":"10.1103\/PhysRevA.105.062448","article-title":"Experimental investigation of equivalent Laplacian and adjacency quantum walks on irregular graphs","volume":"105","author":"Qu","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Liang, W., Yan, F., Iliyasu, A.M., Salama, A.S., and Hirota, K. (2022). A Simplified Quantum Walk Model for Predicting Missing Links of Complex Networks. Entropy, 24.","DOI":"10.3390\/e24111547"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"13455","DOI":"10.1007\/s00521-022-07168-7","article-title":"An information propagation model for social networks based on continuous-time quantum walk","volume":"34","author":"Yan","year":"2022","journal-title":"Neural Comput. Appl."},{"key":"ref_23","first-page":"1","article-title":"Three Degrees of Influence Rule-Based Grover Walk Model with Applications in Identifying Significant Nodes of Complex Networks","volume":"13","author":"Liang","year":"2023","journal-title":"Hum.-Centric Comput. Inf. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1016\/j.comcom.2022.07.045","article-title":"A Hadamard walk model and its application in identification of important edges in complex networks","volume":"193","author":"Liang","year":"2022","journal-title":"Comput. Commun."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"074501","DOI":"10.1103\/PhysRevD.109.074501","article-title":"New basis for Hamiltonian SU (2) simulations","volume":"109","author":"Bauer","year":"2024","journal-title":"Phys. Rev. D"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"023378","DOI":"10.1103\/PhysRevResearch.2.023378","article-title":"Discrete-time quantum walk on complex networks for community detection","volume":"2","author":"Mukai","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Rak, R., and Rak, E. (2020). The Fractional Preferential Attachment Scale-Free Network Model. Entropy, 22.","DOI":"10.3390\/e22050509"},{"key":"ref_28","unstructured":"Erd6s, P., and R\u00e9nyi, A. (1960). On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci., 5, 17\u201361."},{"key":"ref_29","first-page":"103","article-title":"Node Centrality for Continuous-Time Quantum Walks","volume":"Volume 8621","author":"Rossi","year":"2014","journal-title":"Proceedings of the Structural, Syntactic, and Statistical Pattern Recognition\u2014Joint IAPR International Workshop"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"125532","DOI":"10.1016\/j.physa.2020.125532","article-title":"Experimental analyses on 2-hop-based and 3-hop-based link prediction algorithms","volume":"564","author":"Zhou","year":"2021","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Wang, S., Yang, J., Ding, X., Zhang, J., and Zhao, M. (2023). A local community detection algorithm based on potential community exploration. Front. Phys., 11.","DOI":"10.3389\/fphy.2023.1114296"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"121154","DOI":"10.1016\/j.eswa.2023.121154","article-title":"CAGM: A communicability-based adaptive gravity model for influential nodes identification in complex networks","volume":"235","author":"Xu","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Bendahman, N., and Lotfi, D. (2024). Unveiling Influence in Networks: A Novel Centrality Metric and Comparative Analysis through Graph-Based Models. Entropy, 26.","DOI":"10.20944\/preprints202402.1161.v1"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"114487","DOI":"10.1016\/j.chaos.2024.114487","article-title":"Identifying critical nodes in complex networks based on distance Laplacian energy","volume":"180","author":"Yin","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"93711","DOI":"10.1109\/ACCESS.2024.3424834","article-title":"Quantifying Node Influence in Networks: Isolating-Betweenness Centrality for Improved Ranking","volume":"12","author":"Chiranjeevi","year":"2024","journal-title":"IEEE Access"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3260","DOI":"10.1007\/s10489-024-05336-x","article-title":"Learning to rank influential nodes in complex networks via convolutional neural networks","volume":"54","author":"Ahmad","year":"2024","journal-title":"Appl. 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