{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T17:16:46Z","timestamp":1773681406425,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,21]],"date-time":"2021-09-21T00:00:00Z","timestamp":1632182400000},"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":["61973143, 71974080, 71690242 and 11731014"],"award-info":[{"award-number":["61973143, 71974080, 71690242 and 11731014"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In real systems, some damaged nodes can spontaneously become active again when recovered from themselves or their active neighbours. However, the spontaneous dynamical recovery of complex networks that suffer a local failure has not yet been taken into consideration. To model this recovery process, we develop a framework to study the resilience behaviours of the network under a localised attack (LA). Since the nodes\u2019 state within the network affects the subsequent dynamic evolution, we study the dynamic behaviours of local failure propagation and node recoveries based on this memory characteristic. It can be found that the fraction of active nodes switches back and forth between high network activity and low network activity, which leads to the spontaneous emergence of phase-flipping phenomena. These behaviours can be found in a random regular network, Erd\u0151s-R\u00e9nyi network and Scale-free network, which shows that these three types of networks have the same or different resilience behaviours under an LA and random attack. These results will be helpful for studying the spontaneous recovery real systems under an LA. Our work provides insight into understanding the recovery process and a protection strategy of various complex systems from the perspective of damaged memory.<\/jats:p>","DOI":"10.3390\/a14090274","type":"journal-article","created":{"date-parts":[[2021,9,21]],"date-time":"2021-09-21T22:34:01Z","timestamp":1632263641000},"page":"274","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Dynamical Recovery of Complex Networks under a Localised Attack"],"prefix":"10.3390","volume":"14","author":[{"given":"Fan","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China"},{"name":"Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel"}]},{"given":"Gaogao","family":"Dong","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4817-8580","authenticated-orcid":false,"given":"Lixin","family":"Tian","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1126\/science.1165821","article-title":"Network analysis in the social sciences","volume":"323","author":"Borgatti","year":"2009","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2015","DOI":"10.1073\/pnas.0510525103","article-title":"The role of the airline transportation network in the prediction and predictability of global epidemics","volume":"103","author":"Colizza","year":"2006","journal-title":"Proc. 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