{"id":"https://openalex.org/W1839626290","doi":"https://doi.org/10.37236/5771","title":"On the Maximum Running Time in Graph Bootstrap Percolation","display_name":"On the Maximum Running Time in Graph Bootstrap Percolation","publication_year":2017,"publication_date":"2017-05-05","ids":{"openalex":"https://openalex.org/W1839626290","doi":"https://doi.org/10.37236/5771","mag":"1839626290"},"language":"en","primary_location":{"id":"doi:10.37236/5771","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5771","pdf_url":null,"source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.37236/5771","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5109915650","display_name":"B\u00e9la Bollob\u00e1s","orcid":"https://orcid.org/0000-0002-9819-7213"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]},{"id":"https://openalex.org/I94658018","display_name":"University of Memphis","ror":"https://ror.org/01cq23130","country_code":"US","type":"education","lineage":["https://openalex.org/I94658018"]},{"id":"https://openalex.org/I4210124601","display_name":"Department of Mathematical Sciences","ror":"https://ror.org/02xkdev90","country_code":"RU","type":"government","lineage":["https://openalex.org/I1313323035","https://openalex.org/I4210124601"]},{"id":"https://openalex.org/I4210153325","display_name":"Institute of Mathematical Statistics","ror":"https://ror.org/043f04882","country_code":"US","type":"other","lineage":["https://openalex.org/I4210153325"]}],"countries":["GB","RU","US"],"is_corresponding":true,"raw_author_name":"B\u00e9la Bollob\u00e1s","raw_affiliation_strings":["Department of Mathematics and Mathematical Statistics, Cambridge University and \r\nDepartment of Mathematical Sciences, University of Memphis","Department of Mathematics and Mathematical Statistics, Cambridge University and #R##N#Department of Mathematical Sciences, University of Memphis"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Mathematical Statistics, Cambridge University and \r\nDepartment of Mathematical Sciences, University of Memphis","institution_ids":["https://openalex.org/I4210124601","https://openalex.org/I4210153325","https://openalex.org/I94658018"]},{"raw_affiliation_string":"Department of Mathematics and Mathematical Statistics, Cambridge University and #R##N#Department of Mathematical Sciences, University of Memphis","institution_ids":["https://openalex.org/I241749","https://openalex.org/I94658018"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013300408","display_name":"Micha\u0142 Przykucki","orcid":"https://orcid.org/0000-0002-1565-6841"},"institutions":[{"id":"https://openalex.org/I4210120676","display_name":"Mathematical Institute of the Slovak Academy of Sciences","ror":"https://ror.org/02fhy7464","country_code":"SK","type":"facility","lineage":["https://openalex.org/I207624831","https://openalex.org/I4210120676"]},{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB","SK"],"is_corresponding":false,"raw_author_name":"Micha\u0142 Przykucki","raw_affiliation_strings":["Mathematical Institute, University of Oxford","MATHEMATICAL INSTITUTE, UNIVERSITY OF OXFORD"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Oxford","institution_ids":["https://openalex.org/I4210120676","https://openalex.org/I40120149"]},{"raw_affiliation_string":"MATHEMATICAL INSTITUTE, UNIVERSITY OF OXFORD","institution_ids":["https://openalex.org/I4210120676"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050759286","display_name":"Oliver Riordan","orcid":"https://orcid.org/0000-0002-2934-8065"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Oliver Riordan","raw_affiliation_strings":["Mathematics Institute, University of Oxford"],"affiliations":[{"raw_affiliation_string":"Mathematics Institute, University of Oxford","institution_ids":["https://openalex.org/I40120149"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5074703575","display_name":"Julian Sahasrabudhe","orcid":null},"institutions":[{"id":"https://openalex.org/I94658018","display_name":"University of Memphis","ror":"https://ror.org/01cq23130","country_code":"US","type":"education","lineage":["https://openalex.org/I94658018"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Julian Sahasrabudhe","raw_affiliation_strings":["Department of Mathematical Sciences, University of Memphis","Dept. of Mathematical Sciences, University of Memphis"],"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, University of Memphis","institution_ids":["https://openalex.org/I94658018"]},{"raw_affiliation_string":"Dept. of Mathematical Sciences, University of Memphis","institution_ids":["https://openalex.org/I94658018"]}]}],"institutions":[],"countries_distinct_count":4,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5109915650"],"corresponding_institution_ids":["https://openalex.org/I241749","https://openalex.org/I4210124601","https://openalex.org/I4210153325","https://openalex.org/I94658018"],"apc_list":null,"apc_paid":null,"fwci":0.3749,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.47814786,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"24","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12162","display_name":"Cellular Automata and Applications","score":0.9961000084877014,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8042891621589661},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6187604069709778},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.537497341632843},{"id":"https://openalex.org/keywords/cellular-automaton","display_name":"Cellular automaton","score":0.49579867720603943},{"id":"https://openalex.org/keywords/percolation","display_name":"Percolation (cognitive psychology)","score":0.4787999093532562},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.4247400462627411},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39965903759002686},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.20600739121437073}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8042891621589661},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6187604069709778},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.537497341632843},{"id":"https://openalex.org/C35527583","wikidata":"https://www.wikidata.org/wiki/Q189156","display_name":"Cellular automaton","level":2,"score":0.49579867720603943},{"id":"https://openalex.org/C2780457167","wikidata":"https://www.wikidata.org/wiki/Q17156484","display_name":"Percolation (cognitive psychology)","level":2,"score":0.4787999093532562},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.4247400462627411},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39965903759002686},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.20600739121437073},{"id":"https://openalex.org/C169760540","wikidata":"https://www.wikidata.org/wiki/Q207011","display_name":"Neuroscience","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.37236/5771","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5771","pdf_url":null,"source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1510.07096","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1510.07096","pdf_url":"https://arxiv.org/pdf/1510.07096","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"mag:1839626290","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/1510.07096","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:ora.ox.ac.uk:uuid:17b69740-0d82-4ebd-b5ed-0400419f305b","is_oa":false,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:17b69740-0d82-4ebd-b5ed-0400419f305b","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symplectic Elements at Oxford","raw_type":"Journal article"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/7576fef8-8e19-470e-b055-e0fae5341806","is_oa":true,"landing_page_url":"https://research.birmingham.ac.uk/en/publications/7576fef8-8e19-470e-b055-e0fae5341806","pdf_url":null,"source":{"id":"https://openalex.org/S4306402634","display_name":"University of Birmingham Research Portal (University of Birmingham)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79619799","host_organization_name":"University of Birmingham","host_organization_lineage":["https://openalex.org/I79619799"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Bollob\u00e1s, B, Przykucki, M, Riordan, O & Sahasrabudhe, J 2017, 'On the maximum running time in graph bootstrap percolation', Electronic Journal of Combinatorics, vol. 24, no. 2, P2.16. < https://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i2p16 >","raw_type":"article"},{"id":"doi:10.48550/arxiv.1510.07096","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1510.07096","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.37236/5771","is_oa":true,"landing_page_url":"https://doi.org/10.37236/5771","pdf_url":null,"source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G312817341","display_name":null,"funder_award_id":"DMS-1301614","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G608326898","display_name":"Random Geometric Graphs","funder_award_id":"1301614","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G848032724","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":19,"referenced_works":["https://openalex.org/W1502213986","https://openalex.org/W1540294500","https://openalex.org/W1551639281","https://openalex.org/W1966508910","https://openalex.org/W2007804480","https://openalex.org/W2027223596","https://openalex.org/W2027508917","https://openalex.org/W2034962669","https://openalex.org/W2048774399","https://openalex.org/W2055616644","https://openalex.org/W2063472176","https://openalex.org/W2091222603","https://openalex.org/W2093333397","https://openalex.org/W2097670287","https://openalex.org/W2148333157","https://openalex.org/W2156051808","https://openalex.org/W2156275851","https://openalex.org/W2964202470","https://openalex.org/W2967936998"],"related_works":["https://openalex.org/W2963865169","https://openalex.org/W3028781085","https://openalex.org/W2950813788","https://openalex.org/W2933595851","https://openalex.org/W3115851297","https://openalex.org/W2963168794","https://openalex.org/W2619130080","https://openalex.org/W3112610719","https://openalex.org/W3158280387","https://openalex.org/W3042354966","https://openalex.org/W2962848126","https://openalex.org/W2804545386","https://openalex.org/W1916689895","https://openalex.org/W2002045419","https://openalex.org/W3082690547","https://openalex.org/W2042818642","https://openalex.org/W3102409779","https://openalex.org/W2183714943","https://openalex.org/W2987836604","https://openalex.org/W2789066782"],"abstract_inverted_index":{"Graph":[0],"bootstrap":[1],"percolation":[2],"is":[3,62,99,111,120,143,154],"a":[4,14,18,43,90,127],"simple":[5],"cellular":[6],"automaton":[7],"introduced":[8],"by":[9,82,96,161],"Bollob\u00e1s":[10],"in":[11,28,32,49,60],"1968.":[12],"Given":[13],"graph":[15],"$H$":[16,48],"and":[17,30,85],"set":[19],"$G":[20],"\\subseteq":[21],"E(K_n)$":[22],"we":[23,35,88,135,159],"initially":[24],"`infect'":[25],"all":[26],"edges":[27],"$G$":[29,54],"then,":[31],"consecutive":[33],"steps,":[34],"infect":[36],"every":[37,58],"$e":[38],"\\in":[39],"K_n$":[40],"that":[41,53,115,163,178],"completes":[42],"new":[44,128],"infected":[45],"copy":[46],"of":[47,70,140,151],"$K_n$.":[50],"We":[51],"say":[52],"percolates":[55],"if":[56],"eventually":[57],"edge":[59],"$K_n$":[61],"infected.":[63],"The":[64],"extremal":[65],"question":[66,92],"about":[67],"the":[68,71,100,103,137,141,149,152,164],"size":[69],"smallest":[72],"percolating":[73],"sets":[74],"when":[75],"$H":[76],"=":[77],"K_r$":[78],"was":[79],"answered":[80],"independently":[81],"Alon,":[83],"Kalai":[84],"Frankl.":[86],"Here":[87],"consider":[89],"different":[91],"raised":[93],"more":[94,156],"recently":[95],"Bollob\u00e1s:":[97],"what":[98],"maximum":[101,119,138],"time":[102,139,173],"process":[104,142,166],"can":[105,167],"run":[106,168],"before":[107],"it":[108],"stabilizes?":[109],"It":[110],"an":[112],"easy":[113],"observation":[114],"for":[116,131,169,175],"$r=3$":[117],"this":[118],"$\\lceil":[121],"\\log_2":[122],"(n-1)":[123],"\\rceil":[124],"$.":[125],"However,":[126],"phenomenon":[129],"occurs":[130],"$r=4$":[132],"when,":[133],"as":[134,182],"show,":[136],"$n-3$.":[144],"For":[145],"$r":[146,183],"\\geq":[147],"5$":[148],"behaviour":[150],"dynamics":[153],"even":[155],"complex,":[157],"which":[158],"demonstrate":[160],"showing":[162],"$K_r$-bootstrap":[165],"at":[170],"least":[171],"$n^{2-\\varepsilon_r}$":[172],"steps":[174],"some":[176],"$\\varepsilon_r$":[177],"tends":[179],"to":[180],"$0$":[181],"\\to":[184],"\\infty$.":[185]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2018,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
