{"id":"https://openalex.org/W2097670287","doi":"https://doi.org/10.37236/91","title":"Minimal Percolating Sets in Bootstrap Percolation","display_name":"Minimal Percolating Sets in Bootstrap Percolation","publication_year":2009,"publication_date":"2009-01-07","ids":{"openalex":"https://openalex.org/W2097670287","doi":"https://doi.org/10.37236/91","mag":"2097670287"},"language":"en","primary_location":{"id":"doi:10.37236/91","is_oa":true,"landing_page_url":"https://doi.org/10.37236/91","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r2/pdf","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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r2/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029881267","display_name":"Robert Morris","orcid":"https://orcid.org/0000-0002-7826-7907"},"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/I2800402874","display_name":"Edwards (United Kingdom)","ror":"https://ror.org/05avn2m80","country_code":"GB","type":"company","lineage":["https://openalex.org/I2800402874"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Robert Morris","raw_affiliation_strings":["Murray Edwards College, The University of Cambridge, Cambridge CB3 0DF, England"],"affiliations":[{"raw_affiliation_string":"Murray Edwards College, The University of Cambridge, Cambridge CB3 0DF, England","institution_ids":["https://openalex.org/I2800402874","https://openalex.org/I241749"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5029881267"],"corresponding_institution_ids":["https://openalex.org/I241749","https://openalex.org/I2800402874"],"apc_list":null,"apc_paid":null,"fwci":4.5611,"has_fulltext":true,"cited_by_count":45,"citation_normalized_percentile":{"value":0.95014073,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"16","issue":"1","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":0.9998000264167786,"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":0.9998000264167786,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9990000128746033,"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":0.9976000189781189,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8034479022026062},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7486501336097717},{"id":"https://openalex.org/keywords/percolation","display_name":"Percolation (cognitive psychology)","score":0.7044461369514465},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.6177880764007568},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.5775611996650696},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3837912678718567},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15059569478034973},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10805156826972961}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8034479022026062},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7486501336097717},{"id":"https://openalex.org/C2780457167","wikidata":"https://www.wikidata.org/wiki/Q17156484","display_name":"Percolation (cognitive psychology)","level":2,"score":0.7044461369514465},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.6177880764007568},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.5775611996650696},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3837912678718567},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15059569478034973},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10805156826972961},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C169760540","wikidata":"https://www.wikidata.org/wiki/Q207011","display_name":"Neuroscience","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.37236/91","is_oa":true,"landing_page_url":"https://doi.org/10.37236/91","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r2/pdf","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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.214.5539","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.214.5539","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/Volume_16/PDF/v16i1r2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.239.5010","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.239.5010","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/0702370v2.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.240.2209","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.240.2209","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/math/0702370v1.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/91","is_oa":true,"landing_page_url":"https://doi.org/10.37236/91","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r2/pdf","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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2097670287.pdf","grobid_xml":"https://content.openalex.org/works/W2097670287.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W1898632731","https://openalex.org/W1966508910","https://openalex.org/W1972354222","https://openalex.org/W2016526545","https://openalex.org/W2017452431","https://openalex.org/W2027223596","https://openalex.org/W2027508917","https://openalex.org/W2035529977","https://openalex.org/W2045453971","https://openalex.org/W2048774399","https://openalex.org/W2055616644","https://openalex.org/W2064465458","https://openalex.org/W2067253751","https://openalex.org/W2093333397","https://openalex.org/W2107074561","https://openalex.org/W2108704847","https://openalex.org/W2116818317","https://openalex.org/W2125271247","https://openalex.org/W2276192920","https://openalex.org/W4230129974","https://openalex.org/W6632600758"],"related_works":["https://openalex.org/W2980963558","https://openalex.org/W1882246857","https://openalex.org/W3103304230","https://openalex.org/W2185360082","https://openalex.org/W1992365905","https://openalex.org/W4287664878","https://openalex.org/W3212023493","https://openalex.org/W4288278856","https://openalex.org/W2572370460","https://openalex.org/W1976691234"],"abstract_inverted_index":{"In":[0],"standard":[1],"bootstrap":[2],"percolation,":[3],"a":[4,65,74],"subset":[5],"$A$":[6,38],"of":[7,24,60,67,78],"the":[8,45],"grid":[9,47],"$[n]^2$":[10],"is":[11,17,39,48,53],"initially":[12],"infected.":[13,49],"A":[14,50],"new":[15],"site":[16,32],"then":[18],"infected":[19,31,34],"if":[20,43,58],"at":[21],"least":[22],"two":[23],"its":[25,61],"neighbours":[26],"are":[27],"infected,":[28],"and":[29],"an":[30],"stays":[33],"forever.":[35],"The":[36],"set":[37,52,77],"said":[40,54],"to":[41,55],"percolate":[42],"eventually":[44],"entire":[46],"percolating":[51,76],"be":[56],"minimal":[57,75],"none":[59],"subsets":[62],"percolate.":[63],"Answering":[64],"question":[66],"Bollob\u00e1s,":[68],"we":[69],"show":[70],"that":[71],"there":[72,84],"exists":[73],"size":[79],"$4n^2/33":[80],"+":[81,92],"o(n^2)$,":[82],"but":[83],"does":[85],"not":[86],"exist":[87],"one":[88],"larger":[89],"than":[90],"$(n":[91],"2)^2/6$.":[93]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":5},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":3},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":5},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":7}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}