{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:39:50Z","timestamp":1760146790397,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,16]],"date-time":"2024-12-16T00:00:00Z","timestamp":1734307200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12472032","12272295"],"award-info":[{"award-number":["12472032","12272295"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In light of the fact that existing centrality indexes disregard the influence of dynamic characteristics and lack generalizability due to standard diversification, this study investigates dynamic survivability centrality, which enables quantification of oscillators\u2019 capacity to impact the dynamic survivability of nonlinear oscillator systems. Taking an Erd\u0151s\u2013R\u00e9nyi random graph system consisting of Stuart\u2013Landau oscillators as an illustrative example, the typical symmetry synchronization is considered as the key mission to be accomplished in light of the study and the dynamic survivability centrality value is found to be dependent on both the system size and connection density. Starting with a small scale system, the correctness of the theoretical results and the superiority in comparison to traditional indexes are verified. Further, we present the quantitative results by means of error analysis, distribution comparison of various indexes and relationship with system structure exploration, and give the position of the key oscillator. The results demonstrate a negligible error between the theoretical and numerical outcomes, and highlighting that the distribution of dynamic survivability centrality closely resembles the distribution of system state changes. The conclusions serve as evidence for the accuracy and validity of the proposed index. The findings provide an effective approach to protect systems to improve dynamic survivability.<\/jats:p>","DOI":"10.3390\/sym16121661","type":"journal-article","created":{"date-parts":[[2024,12,16]],"date-time":"2024-12-16T06:48:39Z","timestamp":1734331719000},"page":"1661","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Dynamic Survivability Centrality in Nonlinear Oscillator Systems"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6481-4359","authenticated-orcid":false,"given":"Yuexin","family":"Wang","sequence":"first","affiliation":[{"name":"Xi\u2019an Modern Control Technology Research Institute, Xi\u2019an 710065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhongkui","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an 710129, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sijun","family":"Ye","sequence":"additional","affiliation":[{"name":"Xi\u2019an Modern Control Technology Research Institute, Xi\u2019an 710065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tao","family":"Zhao","sequence":"additional","affiliation":[{"name":"Xi\u2019an Modern Control Technology Research Institute, Xi\u2019an 710065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinshuai","family":"Zhang","sequence":"additional","affiliation":[{"name":"Xi\u2019an Modern Control Technology Research Institute, Xi\u2019an 710065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northwestern Polytechnical University, Xi\u2019an 710129, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1145\/1629607.1629613","article-title":"A brief history of the Internet","volume":"39","author":"Leiner","year":"2009","journal-title":"Comput. Commun. Rev."},{"key":"ref_2","unstructured":"Sun, K. (2005, January 18). Complex networks theory: A new method of research in power grid. Proceedings of the 2005 IEEE\/PES Transmission & Distribution Conference & Exposition: Asia and Pacific, Dalian, China."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1109\/81.974874","article-title":"Synchronization in scale-free dynamical networks: Robustness and fragility","volume":"49","author":"Wang","year":"2002","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1142\/S0218127402004292","article-title":"Synchronization in small-world dynamical networks","volume":"12","author":"Wang","year":"2002","journal-title":"Int. J. Bifurcat. Chaos"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"114298","DOI":"10.1016\/j.chaos.2023.114298","article-title":"Consensus formation among mobile agents in networks of heterogeneous interaction venues","volume":"178","author":"Mikaberidze","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1038\/nature16948","article-title":"Universal resilience patterns in complex networks","volume":"530","author":"Gao","year":"2016","journal-title":"Nature"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"095208","DOI":"10.1088\/1402-4896\/aceadd","article-title":"Dynamic survivability in oscillator systems","volume":"98","author":"Sun","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1109\/52.776952","article-title":"Survivable network system analysis: A case study","volume":"16","author":"Ellison","year":"1999","journal-title":"IEEE Softw."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"056109","DOI":"10.1103\/PhysRevE.65.056109","article-title":"Attack vulnerability of complex networks","volume":"65","author":"Holme","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"120957","DOI":"10.1016\/j.physa.2019.04.193","article-title":"Survivability analysis of weighted-edge attacks on complex networks with incomplete information","volume":"531","author":"Yin","year":"2019","journal-title":"Phys. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"062304","DOI":"10.1103\/PhysRevE.102.062304","article-title":"Resource dependency and survivability in complex networks","volume":"102","author":"Ingale","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2350049","DOI":"10.1142\/S0218127423500499","article-title":"Dynamic Survivability in Nonlinear Oscillation Systems with Attractive\u2013Repulsive Interaction","volume":"33","author":"Wang","year":"2023","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1140\/epjp\/s13360-024-04906-9","article-title":"Dynamic survivability of two-layer networks with different topologies","volume":"139","author":"Wang","year":"2024","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"114571","DOI":"10.1016\/j.chaos.2024.114571","article-title":"Dynamic survivability of two-layer networks: The role of interlayer coupling","volume":"180","author":"Wang","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"098701","DOI":"10.1103\/PhysRevLett.93.098701","article-title":"Cascade control and defense in complex networks","volume":"93","author":"Motter","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1072","DOI":"10.1088\/0256-307X\/22\/5\/012","article-title":"Catastrophes in scale-free networks","volume":"22","author":"Zhou","year":"2005","journal-title":"Chin. Phys. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"036104","DOI":"10.1103\/PhysRevE.65.036104","article-title":"Immunization of complex networks","volume":"65","author":"Pastor","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"057102","DOI":"10.1103\/PhysRevE.72.057102","article-title":"Enhanced synchronizability by structural perturbations","volume":"72","author":"Zhao","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.physd.2006.09.008","article-title":"Structural and functional clusters of complex brain networks","volume":"224","author":"Zhou","year":"2006","journal-title":"Phys. D"},{"key":"ref_21","first-page":"613","article-title":"Cortical hubs form a module for multisensory integration on top of the hierarchy of cortical networks","volume":"4","author":"Zamora","year":"2010","journal-title":"Front. Neuroinform."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Knoke, D., and Yang, S. (2008). Social Network Analysis, Sage. Number 154.","DOI":"10.4135\/9781412985864"},{"key":"ref_23","unstructured":"David, K., and BDaniel J, B. (1994). Intraorganizational networks. Advances in Social Network Analysis, Sage Publishing."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"316","DOI":"10.2307\/3069458","article-title":"Social networks and the performance of individuals and groups","volume":"44","author":"Sparrowe","year":"2001","journal-title":"Acad. Manag. J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1111\/j.1540-6261.2007.01207.x","article-title":"Whom you know matters: Venture capital networks and investment performance","volume":"62","author":"Hochberg","year":"2007","journal-title":"J. Financ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/0378-8733(88)90014-7","article-title":"Sorting out centrality: An analysis of the performance of four centrality models in real and simulated networks","volume":"10","author":"Bolland","year":"1988","journal-title":"Soc. Netw."},{"key":"ref_27","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_28","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1002\/bs.3830100205","article-title":"An improved index of centrality","volume":"10","author":"Beauchamp","year":"1965","journal-title":"Behav. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1007\/BF02289527","article-title":"The centrality index of a graph","volume":"31","author":"Sabidussi","year":"1966","journal-title":"Psychometrika"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"35","DOI":"10.2307\/3033543","article-title":"A set of measures of centrality based on betweenness","volume":"40","author":"Freeman","year":"1977","journal-title":"Sociometry"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/0378-8733(94)00248-9","article-title":"Eccentricity and centrality in networks","volume":"17","author":"Hage","year":"1995","journal-title":"Soc. Netw."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1170","DOI":"10.1086\/228631","article-title":"Power and centrality: A family of measures","volume":"92","author":"Bonacich","year":"1987","journal-title":"Am. J. Sociol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/S0378-8733(01)00038-7","article-title":"Eigenvector-like measures of centrality for asymmetric relations","volume":"23","author":"Bonacich","year":"2001","journal-title":"Soc. Netw."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"052808","DOI":"10.1103\/PhysRevE.90.052808","article-title":"Localization and centrality in networks","volume":"90","author":"Martin","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"022304","DOI":"10.1103\/PhysRevE.101.022304","article-title":"Resilience centrality in complex networks","volume":"101","author":"Zhang","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"107030","DOI":"10.1016\/j.rinp.2023.107030","article-title":"Existence, uniqueness and synchronization of a fractional tumor growth model in discrete time with numerical results","volume":"54","author":"Alzabut","year":"2023","journal-title":"Results Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"034313","DOI":"10.1103\/PhysRevE.107.034313","article-title":"Interlayer antisynchronization in degree-biased duplex networks","volume":"107","author":"Rakshit","year":"2023","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Pikovsky, A., Rosenblum, M., and Kurths, J. (2001). Synchronization, Camb. Univ. Press.","DOI":"10.1017\/CBO9780511755743"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"012904","DOI":"10.1103\/PhysRevE.92.012904","article-title":"Explosive synchronization is discontinuous","volume":"92","author":"Vlasov","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2250018","DOI":"10.1142\/S0218127422500183","article-title":"Explosive transition in coupled oscillators through mixed attractive-repulsive interactions","volume":"32","author":"Liu","year":"2022","journal-title":"Int. J. Bifurcat. Chaos"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2741","DOI":"10.1007\/s11071-022-07712-z","article-title":"Explosive behaviors on coupled fractional-order system","volume":"110","author":"Liu","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2016.10.004","article-title":"Explosive transitions in complex networks\u2019 structure and dynamics: Percolation and synchronization","volume":"660","author":"Boccaletti","year":"2016","journal-title":"Phys. Rep."},{"key":"ref_43","first-page":"126384","article-title":"Dynamical robustness and firing modes in multilayer memristive neural networks of nonidentical neurons","volume":"409","author":"Liu","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2109","DOI":"10.1103\/PhysRevLett.80.2109","article-title":"Master stability functions for synchronized coupled systems","volume":"80","author":"Pecora","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1007\/s12346-024-01041-9","article-title":"Modeling and Qualitative Dynamics of the Effects of Internal and External Storage device in a Discrete Fractional Computer Virus","volume":"23","author":"Dhineshbabu","year":"2024","journal-title":"Qual. Theor. Dyn. Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"325","DOI":"10.2307\/1970008","article-title":"On the distribution of the roots of certain symmetric matrices","volume":"67","author":"Wigner","year":"1958","journal-title":"Ann. Math."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Wigner, E.P. (1993). Characteristic vectors of bordered matrices with infinite dimensions II. The Collected Works of Eugene Paul Wigner: Part A: The Scientific Papers, Springer.","DOI":"10.1007\/978-3-662-02781-3_36"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Wigner, E.P. (1993). Characteristic vectors of bordered matrices with infinite dimensions i. The Collected Works of Eugene Paul Wigner: Part A: The Scientific Papers, Springer.","DOI":"10.1007\/978-3-662-02781-3_35"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1661\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:52:53Z","timestamp":1760115173000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1661"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,16]]},"references-count":48,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["sym16121661"],"URL":"https:\/\/doi.org\/10.3390\/sym16121661","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,12,16]]}}}