{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T08:10:41Z","timestamp":1767773441517,"version":"3.41.2"},"reference-count":32,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2022,6,29]],"date-time":"2022-06-29T00:00:00Z","timestamp":1656460800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"name":"Engineering and Physical Sciences Research Council (EPRSC) CoTRE - Complexity Twin for Resilient Ecosystems","award":["EP\/R041725\/1"],"award-info":[{"award-number":["EP\/R041725\/1"]}]},{"name":"Mengbang Zou is a PhD student supported by China Scholarship Council"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,6,29]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Many complex engineering systems network together functional elements to balance demand spikes but suffer from stability issues due to cascades. The research challenge is to prove the stability conditions for any arbitrarily large and dynamic network topology with any complex balancing function. Most current analyses linearize the system around fixed equilibrium solutions. This approach is insufficient for dynamic networks with multiple equilibria, for example, with different initial conditions or perturbations. Region of attraction (ROA) estimation is needed in order to ensure that the desirable equilibria are reached. This is challenging because a networked system of non-linear dynamics requires compression to obtain a tractable ROA analysis. Here, we employ master stability-inspired method to reveal that the extreme eigenvalues of the Laplacian are explicitly linked to the ROA. This novel relationship between the ROA and the largest eigenvalue in turn provides a pathway to augmenting the network structure to improve stability. We demonstrate using a case study on how the network with multiple equilibria can be optimized to ensure stability.<\/jats:p>","DOI":"10.1093\/comnet\/cnac025","type":"journal-article","created":{"date-parts":[[2022,6,29]],"date-time":"2022-06-29T15:33:11Z","timestamp":1656516791000},"source":"Crossref","is-referenced-by-count":4,"title":["Analysing region of attraction of load balancing on complex network"],"prefix":"10.1093","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0496-630X","authenticated-orcid":false,"given":"Mengbang","family":"Zou","sequence":"first","affiliation":[{"name":"Cranfield University Department of Aerospace, , Cranfield MK43 0AL, UK"}]},{"given":"Weisi","family":"Guo","sequence":"additional","affiliation":[{"name":"Cranfield University Department of Aerospace, , Cranfield MK43 0AL, UK"}]}],"member":"286","published-online":{"date-parts":[[2022,6,29]]},"reference":[{"key":"2022062915305240800_B1","doi-asserted-by":"crossref","first-page":"2551","DOI":"10.1109\/JSYST.2019.2956060","article-title":"Probabilistic stability of traffic load balancing on wireless complex networks","volume":"14","author":"Moutsinas,","year":"2019","journal-title":"IEEE Syst. J."},{"key":"2022062915305240800_B2","first-page":"1","article-title":"An intelligent UAV deployment scheme for load balance in small cell networks using machine learning","volume-title":"Proceedings of IEEE Wireless Communications and Networking Conference (WCNC)","author":"Hu,","year":"2019"},{"key":"2022062915305240800_B3","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1109\/JSAC.2013.130504","article-title":"Dynamic cell expansion with self-organizing cooperation","volume":"31","author":"Guo,","year":"2013","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"2022062915305240800_B4","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":"2022062915305240800_B5","first-page":"1","article-title":"Uncertainty of resilience in complex networks with nonlinear dynamics","volume":"15","author":"Moutsinas,","year":"2020","journal-title":"IEEE Syst. J."},{"key":"2022062915305240800_B6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-020-60501-9","article-title":"Node-level resilience loss in dynamic complex networks","volume":"10","author":"Moutsinas,","year":"2020","journal-title":"Sci. Rep."},{"article-title":"Uncertainty quantification of multi-scale resilience in nonlinear complex networks using arbitrary polynomial chaos","year":"2020","author":"Zou,","key":"2022062915305240800_B7"},{"key":"2022062915305240800_B8","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.simpat.2017.07.001","article-title":"Energy and mobility conscious multipath routing scheme for route stability and load balancing in MANETs","volume":"77","author":"Jabbar,","year":"2017","journal-title":"Simul. Model. Pract. Theory"},{"key":"2022062915305240800_B9","doi-asserted-by":"crossref","first-page":"1143","DOI":"10.1016\/j.apm.2020.10.010","article-title":"Estimating the region of attraction based on a polynomial Lyapunov function","volume":"90","author":"Awrejcewicz,","year":"2021","journal-title":"Appl. Math. Model."},{"key":"2022062915305240800_B10","article-title":"Exact asymptotic stability analysis and region-of-attraction estimation for nonlinear systems","volume-title":"Abstr. Appl. Anal.,","author":"Wu,","year":"2013"},{"key":"2022062915305240800_B11","doi-asserted-by":"crossref","first-page":"10056","DOI":"10.1109\/TVT.2019.2933560","article-title":"Location optimization and user association for unmanned aerial vehicles assisted mobile networks","volume":"68","author":"Sun,","year":"2019","journal-title":"IEEE Trans. Veh. Technol."},{"key":"2022062915305240800_B12","doi-asserted-by":"crossref","first-page":"2533","DOI":"10.1109\/TCSI.2013.2246233","article-title":"Algorithmic construction of Lyapunov functions for power system stability analysis","volume":"60","author":"Anghel,","year":"2013","journal-title":"IEEE Trans. Circuits Syst. I, Reg. Papers"},{"key":"2022062915305240800_B13","first-page":"1700","article-title":"Region of attraction estimation of biological continuous Boolean models","volume-title":"Proceedings of IEEE International Conference on Systems, Man, and Cybernetics (SMC)","author":"Matthews,","year":"2012"},{"key":"2022062915305240800_B14","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/j.conengprac.2010.12.001","article-title":"Nonlinear region of attraction analysis for flight control verification and validation","volume":"19","author":"Chakraborty,","year":"2011","journal-title":"Control Eng. Pract."},{"key":"2022062915305240800_B15","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/0005-1098(85)90099-8","article-title":"Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems","volume":"21","author":"Vannelli,","year":"1985","journal-title":"Automatica"},{"key":"2022062915305240800_B16","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1016\/j.mcm.2010.04.001","article-title":"Estimation of domains of attraction: a global optimization approach","volume":"52","author":"Matallana,","year":"2010","journal-title":"Math. Comput. Model."},{"key":"2022062915305240800_B17","first-page":"157","article-title":"Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach","volume-title":"2004 5th Asian Control Conference (IEEE Cat. No. 04EX904)","author":"Prajna,","year":"2004"},{"key":"2022062915305240800_B18","doi-asserted-by":"crossref","first-page":"2500","DOI":"10.1109\/TAC.2010.2046926","article-title":"LMI techniques for optimization over polynomials in control: a survey","volume":"55","author":"Chesi,","year":"2010","journal-title":"IEEE Trans. Automatic Control"},{"key":"2022062915305240800_B19","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1002\/rnc.967","article-title":"LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems","volume":"15","author":"Chesi,","year":"2005","journal-title":"Int. J. Robust Nonlinear Control."},{"key":"2022062915305240800_B20","doi-asserted-by":"crossref","first-page":"1263","DOI":"10.23919\/ACC.2017.7963126","article-title":"Estimating the region of attraction via forward reachable sets","volume-title":"2017 American Control Conference (ACC)","author":"El-Guindy,","year":"2017"},{"key":"2022062915305240800_B21","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1177\/0142331217752799","article-title":"Estimation of the regions of attraction for autonomous nonlinear systems","volume":"41","author":"Yuan,","year":"2019","journal-title":"Trans. Inst. Meas. Control."},{"key":"2022062915305240800_B22","doi-asserted-by":"crossref","first-page":"4171","DOI":"10.23919\/ACC.2017.7963596","article-title":"An algorithm for enlarging the region of attraction using trajectory reversing","volume-title":"2017 American Control Conference (ACC)","author":"Kant,","year":"2017"},{"key":"2022062915305240800_B23","doi-asserted-by":"crossref","first-page":"1209","DOI":"10.1177\/0278364914528059","article-title":"Convex optimization of nonlinear feedback controllers via occupation measures","volume":"33","author":"Majumdar,","year":"2014","journal-title":"Int. J. Robot. Res."},{"key":"2022062915305240800_B24","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1137\/18M1182498","article-title":"Estimating the region of attraction on a complex dynamical network","volume":"57","author":"Zhu,","year":"2019","journal-title":"SIAM J. Control Optim."},{"key":"2022062915305240800_B25","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":"2022062915305240800_B26","volume-title":"Nonlinear Systems","author":"Khalil,","year":"2002","edition":"3rd edn"},{"volume-title":"Linear Systems","year":"1997","author":"Antsaklis,","key":"2022062915305240800_B27"},{"key":"2022062915305240800_B28","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.physrep.2008.09.002","article-title":"Synchronization in complex networks","volume":"469","author":"Arenas,","year":"2008","journal-title":"Phys. Rep."},{"key":"2022062915305240800_B29","doi-asserted-by":"crossref","first-page":"054101","DOI":"10.1103\/PhysRevLett.89.054101","article-title":"Synchronization in small-world systems","volume":"89","author":"Barahona,","year":"2002","journal-title":"Phys. Rev. Lett"},{"key":"2022062915305240800_B30","doi-asserted-by":"crossref","first-page":"248701","DOI":"10.1103\/PhysRevLett.98.248701","article-title":"Ensemble averageability in network spectra","volume":"98","author":"Kim,","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"2022062915305240800_B31","doi-asserted-by":"crossref","first-page":"052207","DOI":"10.1103\/PhysRevE.99.052207","article-title":"Perturbation analysis and comparison of network synchronization methods","volume":"99","author":"He,","year":"2019","journal-title":"Phys. Rev. E"},{"key":"2022062915305240800_B32","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u201csmall-world\u201d networks","volume":"393","author":"Watts,","year":"1998","journal-title":"Nature"}],"container-title":["Journal of Complex Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/comnet\/article-pdf\/10\/4\/cnac025\/44282110\/cnac025.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/comnet\/article-pdf\/10\/4\/cnac025\/44282110\/cnac025.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,29]],"date-time":"2022-06-29T15:33:54Z","timestamp":1656516834000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/comnet\/article\/doi\/10.1093\/comnet\/cnac025\/6619624"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,29]]},"references-count":32,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,6,29]]}},"URL":"https:\/\/doi.org\/10.1093\/comnet\/cnac025","relation":{},"ISSN":["2051-1329"],"issn-type":[{"type":"electronic","value":"2051-1329"}],"subject":[],"published-other":{"date-parts":[[2022,8,1]]},"published":{"date-parts":[[2022,6,29]]},"article-number":"cnac025"}}