{"id":"https://openalex.org/W3194003167","doi":"https://doi.org/10.1109/tnse.2021.3098443","title":"Preventive and Reactive Cyber Defense Dynamics With Ergodic Time-Dependent Parameters is Globally Attractive","display_name":"Preventive and Reactive Cyber Defense Dynamics With Ergodic Time-Dependent Parameters is Globally Attractive","publication_year":2021,"publication_date":"2021-07-01","ids":{"openalex":"https://openalex.org/W3194003167","doi":"https://doi.org/10.1109/tnse.2021.3098443","mag":"3194003167"},"language":"en","primary_location":{"id":"doi:10.1109/tnse.2021.3098443","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnse.2021.3098443","pdf_url":null,"source":{"id":"https://openalex.org/S2484352698","display_name":"IEEE Transactions on Network Science and Engineering","issn_l":"2327-4697","issn":["2327-4697","2334-329X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Network Science and Engineering","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029401729","display_name":"Yujuan Han","orcid":"https://orcid.org/0000-0002-1395-4247"},"institutions":[{"id":"https://openalex.org/I96733725","display_name":"Shanghai Maritime University","ror":"https://ror.org/04z7qrj66","country_code":"CN","type":"education","lineage":["https://openalex.org/I96733725"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yujuan Han","raw_affiliation_strings":["College of Information Engineering, Shanghai Maritime University, Shanghai, China"],"raw_orcid":"https://orcid.org/0000-0002-1395-4247","affiliations":[{"raw_affiliation_string":"College of Information Engineering, Shanghai Maritime University, Shanghai, China","institution_ids":["https://openalex.org/I96733725"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030103251","display_name":"Wenlian Lu","orcid":"https://orcid.org/0000-0003-1880-6240"},"institutions":[{"id":"https://openalex.org/I24943067","display_name":"Fudan University","ror":"https://ror.org/013q1eq08","country_code":"CN","type":"education","lineage":["https://openalex.org/I24943067"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wenlian Lu","raw_affiliation_strings":["School of Mathematical Sciences, Fudan University, Shanghai, China","Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China","Shanghai Key Laboratory for Contemporary Applied Mathematics, Shanghai, China"],"raw_orcid":"https://orcid.org/0000-0003-1880-6240","affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Fudan University, Shanghai, China","institution_ids":["https://openalex.org/I24943067"]},{"raw_affiliation_string":"Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, China","institution_ids":["https://openalex.org/I24943067"]},{"raw_affiliation_string":"Shanghai Key Laboratory for Contemporary Applied Mathematics, Shanghai, China","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019179799","display_name":"Shouhuai Xu","orcid":"https://orcid.org/0000-0001-8034-0942"},"institutions":[{"id":"https://openalex.org/I888729015","display_name":"University of Colorado Colorado Springs","ror":"https://ror.org/054spjc55","country_code":"US","type":"education","lineage":["https://openalex.org/I888729015"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Shouhuai Xu","raw_affiliation_strings":["Department of Computer Science, University of Colorado Colorado Springs, Colorado Springs, CO, USA"],"raw_orcid":"https://orcid.org/0000-0001-8034-0942","affiliations":[{"raw_affiliation_string":"Department of Computer Science, University of Colorado Colorado Springs, Colorado Springs, CO, USA","institution_ids":["https://openalex.org/I888729015"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.0634,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.7830323,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"8","issue":"3","first_page":"2517","last_page":"2532"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10400","display_name":"Network Security and Intrusion Detection","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10400","display_name":"Network Security and Intrusion Detection","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12592","display_name":"Opinion Dynamics and Social Influence","score":0.9958000183105469,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/ergodic-theory","display_name":"Ergodic theory","score":0.6450168490409851},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5121979117393494},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.24444526433944702},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.10960698127746582}],"concepts":[{"id":"https://openalex.org/C122044880","wikidata":"https://www.wikidata.org/wiki/Q5498822","display_name":"Ergodic theory","level":2,"score":0.6450168490409851},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5121979117393494},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.24444526433944702},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.10960698127746582}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tnse.2021.3098443","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tnse.2021.3098443","pdf_url":null,"source":{"id":"https://openalex.org/S2484352698","display_name":"IEEE Transactions on Network Science and Engineering","issn_l":"2327-4697","issn":["2327-4697","2334-329X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Network Science and Engineering","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions","score":0.41999998688697815}],"awards":[{"id":"https://openalex.org/G1285171503","display_name":null,"funder_award_id":"#1814825","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5277577257","display_name":"\u89e6\u53d1\u66f4\u65b0\u673a\u5236\u4e0b\u590d\u6742\u8026\u5408\u7cfb\u7edf\u7684\u62d3\u6251\u63a7\u5236\u7b97\u6cd5\u7684\u8bbe\u8ba1\u4e0e\u5206\u6790","funder_award_id":"61703271","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7771494911","display_name":null,"funder_award_id":"62072111","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8027550448","display_name":null,"funder_award_id":"51879156","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":60,"referenced_works":["https://openalex.org/W429124266","https://openalex.org/W1491237615","https://openalex.org/W1500921805","https://openalex.org/W1501825524","https://openalex.org/W1577634535","https://openalex.org/W1851248039","https://openalex.org/W1878853999","https://openalex.org/W1914027636","https://openalex.org/W1914370899","https://openalex.org/W1984047690","https://openalex.org/W1985782993","https://openalex.org/W1998754934","https://openalex.org/W2008711923","https://openalex.org/W2019229544","https://openalex.org/W2049529576","https://openalex.org/W2055605553","https://openalex.org/W2080026455","https://openalex.org/W2084618626","https://openalex.org/W2099440098","https://openalex.org/W2108050790","https://openalex.org/W2118820208","https://openalex.org/W2121385949","https://openalex.org/W2124486211","https://openalex.org/W2140341064","https://openalex.org/W2142949813","https://openalex.org/W2148301044","https://openalex.org/W2151071772","https://openalex.org/W2152742892","https://openalex.org/W2161499589","https://openalex.org/W2166127999","https://openalex.org/W2274964336","https://openalex.org/W2296181180","https://openalex.org/W2440001402","https://openalex.org/W2514025363","https://openalex.org/W2564147261","https://openalex.org/W2577900305","https://openalex.org/W2755855437","https://openalex.org/W2796990566","https://openalex.org/W2941527469","https://openalex.org/W2946688610","https://openalex.org/W2946917174","https://openalex.org/W2962817891","https://openalex.org/W2963120360","https://openalex.org/W2963218066","https://openalex.org/W2963790977","https://openalex.org/W2963815167","https://openalex.org/W2963979702","https://openalex.org/W2964074424","https://openalex.org/W2964313867","https://openalex.org/W3097146022","https://openalex.org/W3101837101","https://openalex.org/W3106753174","https://openalex.org/W3108055496","https://openalex.org/W3157294495","https://openalex.org/W4230015191","https://openalex.org/W4295016751","https://openalex.org/W4300645796","https://openalex.org/W6629287744","https://openalex.org/W6682044371","https://openalex.org/W6718465079"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W2390585021","https://openalex.org/W412641959","https://openalex.org/W50969306","https://openalex.org/W4292101436","https://openalex.org/W2341040961","https://openalex.org/W4287207389","https://openalex.org/W2779828239"],"abstract_inverted_index":{"Cybersecurity":[0],"dynamics":[1,56,116],"is":[2,41,66,81,126],"a":[3,42,135],"mathematical":[4],"approach":[5,142],"to":[6,143],"modeling":[7,146],"and":[8,34,80],"analyzing":[9],"cyber":[10,36],"attacker-defender":[11],"interactions":[12],"in":[13,22,57,117,138],"networks.":[14],"In":[15],"this":[16,53],"paper,":[17],"we":[18],"advance":[19,137],"the":[20,72,87,91,104,108,112,118,139,150,157],"state-of-the-art":[21,105],"characterizing":[23],"one":[24],"kind":[25],"of":[26,44,55,94,115,121],"cybersecurity":[27,145],"dynamics,":[28],"known":[29],"as":[30],"<italic":[31,62,67,76,82,96,122,127],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[32,63,68,77,83,97,123,128],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">preventive":[33],"reactive":[35],"defense":[37],"dynamics</i>":[38],",":[39,79],"which":[40],"family":[43,54,114],"highly":[45],"nonlinear":[46],"dynamical":[47],"system":[48],"models.":[49],"We":[50],"prove":[51],"that":[52,111],"its":[58],"general":[59],"form":[60],"with":[61,156],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">time-dependent</i>":[64],"parameters":[65,74,89,125],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">globally":[69,129],"attractive</i>":[70],"when":[71,86,154],"time-dependent":[73,88],"are":[75],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">ergodic</i>":[78],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">(almost)":[84,98],"periodic</i>":[85,99],"have":[90],"stronger":[92],"properties":[93],"being":[95],".":[100,131],"Our":[101],"results":[102],"supersede":[103],"ones,":[106],"including":[107],"recent":[109],"result":[110],"same":[113],"special":[119],"case":[120],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">time-independent</i>":[124],"convergent</i>":[130],"This":[132],"paper":[133],"represents":[134],"significant":[136],"network":[140],"science":[141],"theoretical":[144],"because":[147],"it":[148],"makes":[149],"so-far":[151],"weakest":[152],"assumptions":[153],"compared":[155],"literature":[158],"results.":[159]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}