Abstract
The identification of influential nodes has extensive applications in complex network research. To address the challenge of balancing accuracy and computational efficiency in existing methods, this paper proposes an algorithm named HKEN that integrates hierarchical k-shell decomposition with extended neighborhood information. The approach incorporates both global structural features and local topological information of nodes. First, the coarse-grained hierarchical mechanism of the k-shell algorithm is optimized, and the initial weight of nodes is dynamically computed based on their degree and k-shell values. Second, the neighborhood range of nodes is extended, and a local clustering coefficient is introduced to establish a threshold for information transmission distance. Finally, an influence aggregation strategy based on the Jaccard similarity coefficient is proposed. The performance of HKEN was validated through comparative experiments on 10 real-world networks against 12 benchmark methods. Experimental results demonstrate that the proposed method achieves higher consistency with SIR model outcomes and enhances the propagation capability of top-ranked nodes, confirming its improved identification accuracy.
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Data availability
The datasets generated and analysed during the current study, as well as the source code for all algorithms, are available in the GitHub repository: https://github.com/wffei123/HKEN.
References
Dong, C., Xu, G., Yang, P. & Meng, L. TSIFIM: A three-stage iterative framework for influence maximization in complex networks. Expert Syst. Appl. 212, 118702. https://doi.org/10.1016/j.eswa.2022.118702 (2023).
Li, M. et al. Percolation on complex networks: Theory and application. Phys. Rep. 907, 1–68. https://doi.org/10.1016/j.physrep.2020.12.003 (2021).
Sun, Z. et al. Finding critical nodes in a complex network from information diffusion and Matthew effect aggregation. Expert Syst. Appl. 233, 120927. https://doi.org/10.1016/j.eswa.2023.120927 (2023).
Al-Garadi, M. A. et al. Analysis of online social network connections for identification of influential users: Survey and open research issues. ACM Comput. Surv. 51, 1–37. https://doi.org/10.1145/3155897 (2018).
Bahutair, M., Al Aghbari, Z. & Kamel, I. NodeRank: Finding influential nodes in social networks based on interests. J. Supercomput. 78, 2098–2124. https://doi.org/10.1007/s11227-021-03947-6 (2022).
Das, K., Samanta, S. & Pal, M. Study on centrality measures in social networks: A survey. Soc. Netw. Anal. Min. 8, 13. https://doi.org/10.1007/s13278-018-0493-2 (2018).
Gupta, M. & Mishra, R. Spreading the information in complex networks: Identifying a set of top-N influential nodes using network structure. Decis. Support Syst. 149, 113608. https://doi.org/10.1016/j.dss.2021.113608 (2021).
Xu, W. et al. Identifying structural hole spanners to maximally block information propagation. Inf. Sci. 505, 100–126. https://doi.org/10.1016/j.ins.2019.07.072 (2019).
Albert, R., Albert, I. & Nakarado, G. L. Structural vulnerability of the North American power grid. Phys. Rev. E 69, 025103. https://doi.org/10.1103/PhysRevE.69.025103 (2004).
Zhao, M., Ye, J., Li, J. & Wu, M. NRD: A node importance evaluation algorithm based on neighborhood reliance degree for power networks. Phys. A Stat. Mech. Appl. https://doi.org/10.1016/j.physa.2023.128941 (2023).
Chaharborj, S. S., Nabi, K. N., Feng, K. L., Chaharborj, S. S. & Phang, P. S. Controlling COVID-19 transmission with isolation of influential nodes. Chaos, Solitons Fract. 159, 112035. https://doi.org/10.1016/j.chaos.2022.112035 (2022).
Bauer, F. & Lizier, J. T. Identifying influential spreaders and efficiently estimating infection numbers in epidemic models: A walk counting approach. Europhys. Lett. 99, 68007. https://doi.org/10.1209/0295-5075/99/68007 (2012).
Du, Z. et al. Identifying critical nodes in metro network considering topological potential: A case study in Shenzhen city—China. Phys. A 539, 122926. https://doi.org/10.1016/j.physa.2019.122926 (2020).
Porta, S., Crucitti, P. & Latora, V. The network analysis of urban streets: A dual approach. Phys. A 369, 853–866. https://doi.org/10.1016/j.physa.2005.12.063 (2006).
Liu, W., Li, X., Liu, T. & Liu, B. Approximating betweenness centrality to identify key nodes in a weighted urban complex transportation network. J. Adv. Transp. 2019, 9024745. https://doi.org/10.1155/2019/9024745 (2019).
Wu, Y., Dong, A., Ren, Y. & Jiang, Q. Identify influential nodes in complex networks: A k-orders entropy-based method. Phys. A 632, 129302. https://doi.org/10.1016/j.physa.2023.129302 (2023).
Namtirtha, A., Dutta, A. & Dutta, B. Weighted kshell degree neighborhood: A new method for identifying the influential spreaders from a variety of complex network connectivity structures. Expert Syst. Appl. 139, 112859. https://doi.org/10.1016/j.eswa.2019.112859 (2020).
Sheng, J. et al. Identifying influential nodes in complex networks based on global and local structure. Physica A 541, 123262. https://doi.org/10.1016/j.physa.2019.123262 (2020).
Ullah, A. et al. Identifying vital nodes from local and global perspectives in complex networks. Expert Syst. Appl. 186, 115778. https://doi.org/10.1016/j.eswa.2021.115778 (2021).
Li, Z. & Huang, X. Identifying influential spreaders in complex networks by an improved gravity model. Sci. Rep. 11, 22194. https://doi.org/10.1038/s41598-021-01218-1 (2021).
Liu, F., Wang, Z. & Deng, Y. GMM: A generalized mechanics model for identifying the importance of nodes in complex networks. Knowl. Based Syst. 193, 105464. https://doi.org/10.1016/j.knosys.2019.105464 (2020).
Maji, G. Influential spreaders identification in complex networks with potential edge weight based k-shell degree neighborhood method. J. Comput. Sci. 39, 101055. https://doi.org/10.1016/j.jocs.2019.101055 (2020).
Li, Z. et al. Identifying influential spreaders by gravity model. Sci. Rep. 9, 8387. https://doi.org/10.1038/s41598-019-44930-9 (2019).
Tong, T., Dong, Q., Sun, J. & Jiang, Y. Vital spreaders identification synthesizing cross entropy and information entropy with kshell method. Expert Syst. Appl. 224, 119928. https://doi.org/10.1016/j.eswa.2023.119928 (2023).
Zhao, Z., Li, D., Sun, Y., Zhang, R. & Liu, J. Ranking influential spreaders based on both node k-shell and structural hole. Knowl. Based Syst. 260, 110163. https://doi.org/10.1016/j.knosys.2022.110163 (2023).
Bonacich, P. Factoring and weighting approaches to status scores and clique identification. J. Math. Soc. 2, 113–120. https://doi.org/10.1080/0022250X.1972.9989806 (1972).
Lü, L., Zhou, T., Zhang, Q. M. & Stanley, H. E. The H-index of a network node and its relation to degree and coreness. Nat. Commun. 7, 10168. https://doi.org/10.1038/ncomms10168 (2016).
Chen, D. B., Gao, H., Lü, L. & Zhou, T. Identifying influential nodes in large-scale directed networks: The role of clustering. PLoS ONE 8, e77455. https://doi.org/10.1371/journal.pone.0077455 (2013).
Ivchenko, G. I. & Honov, S. A. On the jaccard similarity test. J. Math. Sci. 88, 789–794. https://doi.org/10.1007/BF02365362 (1998).
Freeman, L. C. A set of measures of centrality based on betweenness. Sociometry https://doi.org/10.2307/3033543 (1977).
Sabidussi, G. The centrality index of a graph. Psychometrika 31, 581–603. https://doi.org/10.1007/BF02289527 (1966).
Kitsak, M. et al. Identification of influential spreaders in complex networks. Nat. Phys. 6, 888–893. https://doi.org/10.1038/nphys1746 (2010).
Bonacich, P. Power and centrality: A family of measures. Am. J. Sociol. 92, 1170–1182. https://doi.org/10.1086/228631 (1987).
Yang, F. et al. Identifying the most influential spreaders in complex networks by an extended local K - shell sum. Int. J. Mod. Phys. C 28, 1750014. https://doi.org/10.1142/S0129183117500140 (2017).
Yu, Z., Shao, J., Yang, Q. & Sun, Z. Profitleader: Identifying leaders in networks with profit capacity. World Wide Web 22, 533–553. https://doi.org/10.1007/s11280-018-0537-6 (2019).
Fu, L., Ma, X., Dou, Z., Bai, Y. & Zhao, X. Key node identification method based on multilayer neighbor node gravity and information entropy. Entropy 26, 1041. https://doi.org/10.3390/e26121041 (2024).
Wang, G. et al. Influential nodes identification method based on adaptive adjustment of voting ability. Heliyon https://doi.org/10.1016/j.heliyon.2023.e16112 (2023).
Zhao, J., Wang, Y. & Deng, Y. Identifying influential nodes in complex networks from global perspective. Chaos, Solitons Fract. 133, 109637. https://doi.org/10.1016/j.chaos.2020.109637 (2020).
Zareie, A., Sheikhahmadi, A., Jalili, M. & Fasaei, M. S. K. Finding influential nodes in social networks based on neighborhood correlation coefficient. Knowl. Based Syst. 194, 105580. https://doi.org/10.1016/j.knosys.2020.105580 (2020).
Esfandiari, S. & Moosavi, M. R. Identifying influential nodes in complex networks through the k - shell index and neighborhood information. J. Comput. Sci. 84, 102473. https://doi.org/10.1016/j.jocs.2024.102473 (2025).
Nandi, S., Curado Malta, M., Maji, G. & Dutta, A. IC - SNI: Measuring nodes’ influential capability in complex networks through structural and neighboring information. Knowl. Inf. Syst. 67, 1309–1350. https://doi.org/10.1007/s10115-024-02262-9 (2025).
Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small - world’ networks. Nature 393, 440–442. https://doi.org/10.1038/30918 (1998).
Tuljapurkar, S. Infectious diseases of humans: Dynamics and control. Science 254, 591–593 (1991).
Kendall, M. G. A new measure of rank correlation. Biometrika 30, 81–93. https://doi.org/10.2307/2332226 (1938).
Lü, L. et al. Vital nodes identification in complex networks. Phys. Rep. 650, 1–63. https://doi.org/10.1016/j.physrep.2016.06.007 (2016).
Albert, R., Jeong, H. & Barabási, A. L. Error and attack tolerance of complex networks. Nature 406, 378–382. https://doi.org/10.1038/35019019 (2000).
Holme, P., Kim, B. J., Yoon, C. N. & Han, S. K. Attack vulnerability of complex networks. Phys. Rev. E 65, 056109. https://doi.org/10.1103/PhysRevE.65.056109 (2002).
Morone, F. & Makse, H. A. Influence maximization in complex networks through optimal percolation. Nature 524, 65–68. https://doi.org/10.1038/nature14604 (2015).
Funding
The research is supported by the Key Scientific Research Projects of Colleges and Universities in Henan Province under Grant Nos. 25A520040, 26A520032, and 26B520037, as well as the Science and Technology Research Project of the Department of Science and Technology in Henan Province under Grant No. 252102210028.
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F. W. proposed the methodology and drafted the original manuscript; Z. S. curated the data; G. W. designed the visualizations; H. H. reviewed and edited the manuscript; X. S. performed validation; and S. Z. conducted the investigation. All authors reviewed the final manuscript.
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Wang, F., Sun, Z., Wang, G. et al. Identifying influential nodes through hierarchical k-shell and extended neighborhood integration. Sci Rep (2026). https://doi.org/10.1038/s41598-026-40209-y
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DOI: https://doi.org/10.1038/s41598-026-40209-y
