{"id":"https://openalex.org/W1965647971","doi":"https://doi.org/10.1512/iumj.2010.59.4008","title":"Giant components in biased graph processes","display_name":"Giant components in biased graph processes","publication_year":2010,"publication_date":"2010-01-01","ids":{"openalex":"https://openalex.org/W1965647971","doi":"https://doi.org/10.1512/iumj.2010.59.4008","mag":"1965647971"},"language":"en","primary_location":{"id":"doi:10.1512/iumj.2010.59.4008","doi":"https://doi.org/10.1512/iumj.2010.59.4008","is_oa":false,"landing_page_url":"https://doi.org/10.1512/iumj.2010.59.4008","pdf_url":null,"source":{"id":"https://openalex.org/S196210972","display_name":"Indiana University Mathematics Journal","issn_l":"0022-2518","issn":["0022-2518","1943-5258","1943-5266"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-b","jufo-2","norway-2","poland-140","russia-white-list-2"],"host_organization":"https://openalex.org/P4310315561","host_organization_name":"Indiana University","host_organization_lineage":["https://openalex.org/P4310315561"],"host_organization_lineage_names":["Indiana University"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Indiana University Mathematics Journal","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/math/0511526","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5036781157","display_name":"Gideon Amir","orcid":"https://orcid.org/0000-0001-7738-0441","observed_orcids":["https://orcid.org/0000-0001-7738-0441"]},"institutions":[{"id":"https://openalex.org/I185261750","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087","country_code":"CA","type":"education","lineage":["https://openalex.org/I185261750"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Gideon Amir","raw_affiliation_strings":["University of Toronto"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Toronto","institution_ids":["https://openalex.org/I185261750"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5060399988","display_name":"Ori Gurel-Gurevich","orcid":"https://orcid.org/0000-0002-5317-6371","observed_orcids":["https://orcid.org/0000-0002-5317-6371"]},"institutions":[{"id":"https://openalex.org/I1290206253","display_name":"Microsoft (United States)","ror":"https://ror.org/00d0nc645","country_code":"US","type":"company","lineage":["https://openalex.org/I1290206253"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ori Gurel-Gurevich","raw_affiliation_strings":["Microsoft , USA "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Microsoft , USA ","institution_ids":["https://openalex.org/I1290206253"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5047720470","display_name":"Eyal Lubetzky","orcid":"https://orcid.org/0000-0002-2281-3542","observed_orcids":["https://orcid.org/0000-0002-2281-3542"]},"institutions":[{"id":"https://openalex.org/I1290206253","display_name":"Microsoft (United States)","ror":"https://ror.org/00d0nc645","country_code":"US","type":"company","lineage":["https://openalex.org/I1290206253"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Eyal Lubetzky","raw_affiliation_strings":["Microsoft"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Microsoft","institution_ids":["https://openalex.org/I1290206253"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5062844820","display_name":"Amit Singer","orcid":"https://orcid.org/0000-0002-6975-7955","observed_orcids":["https://orcid.org/0000-0002-6975-7955"]},"institutions":[{"id":"https://openalex.org/I20089843","display_name":"Princeton University","ror":"https://ror.org/00hx57361","country_code":"US","type":"education","lineage":["https://openalex.org/I20089843"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Amit Singer","raw_affiliation_strings":["Princeton University#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Princeton University#TAB#","institution_ids":["https://openalex.org/I20089843"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.07539683,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"59","issue":"6","first_page":"1893","last_page":"1930"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.4950999915599823,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.4950999915599823,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10064","display_name":"Complex Network Analysis Techniques","score":0.42239999771118164,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.06520000100135803,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/random-graph-processes","display_name":"random graph processes","score":0.796999990940094},{"id":"https://openalex.org/keywords/giant-component","display_name":"giant component","score":0.7739999890327454},{"id":"https://openalex.org/keywords/epidemic-spreading","display_name":"epidemic spread","score":0.5740000009536743},{"id":"https://openalex.org/keywords/double-jump","display_name":"double jump","score":0.7940000295639038},{"id":"https://openalex.org/keywords/emergence-time","display_name":"emergence timing","score":0.5910000205039978},{"id":"https://openalex.org/keywords/differential-equations","display_name":"differential equations","score":0.6940000057220459}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7452964782714844},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4802282452583313},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2889649569988251}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1512/iumj.2010.59.4008","doi":"https://doi.org/10.1512/iumj.2010.59.4008","is_oa":false,"landing_page_url":"https://doi.org/10.1512/iumj.2010.59.4008","pdf_url":null,"source":{"id":"https://openalex.org/S196210972","display_name":"Indiana University Mathematics Journal","issn_l":"0022-2518","issn":["0022-2518","1943-5258","1943-5266"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-b","jufo-2","norway-2","poland-140","russia-white-list-2"],"host_organization":"https://openalex.org/P4310315561","host_organization_name":"Indiana University","host_organization_lineage":["https://openalex.org/P4310315561"],"host_organization_lineage_names":["Indiana University"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Indiana University Mathematics Journal","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:math/0511526","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/math/0511526","pdf_url":"https://arxiv.org/pdf/math/0511526","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.178.8850","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.178.8850","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://research.microsoft.com/%7Eeyal/papers/agls.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.188.1133","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.188.1133","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://research.microsoft.com/%7Eorigurel/papers/AGGLS07.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.240.4395","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.240.4395","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/0511526v1.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.241.4423","doi":null,"is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.241.4423","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/0511526v2.pdf","raw_type":"text"},{"id":"doi:10.48550/arxiv.math/0511526","doi":"https://doi.org/10.48550/arxiv.math/0511526","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.math/0511526","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:math/0511526","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/math/0511526","pdf_url":"https://arxiv.org/pdf/math/0511526","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1965647971.pdf","grobid_xml":"https://content.openalex.org/works/W1965647971.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W5090298","https://openalex.org/W1657347807","https://openalex.org/W1975999171","https://openalex.org/W2002387903","https://openalex.org/W2004427634","https://openalex.org/W2012690654","https://openalex.org/W2033468284","https://openalex.org/W2061826427","https://openalex.org/W2066652472","https://openalex.org/W2070589948","https://openalex.org/W2094803507","https://openalex.org/W2095403663","https://openalex.org/W2752885492","https://openalex.org/W2905110430","https://openalex.org/W3143219376"],"related_works":["https://openalex.org/W1656503067","https://openalex.org/W1989459250","https://openalex.org/W2343747071","https://openalex.org/W7095970991","https://openalex.org/W1970474336","https://openalex.org/W7097387014","https://openalex.org/W182885774","https://openalex.org/W2059507660","https://openalex.org/W7099786802","https://openalex.org/W160659757"],"abstract_inverted_index":{"A":[0],"random":[1],"graph":[2],"process,":[3,85,168],"$\\Gorg[1](n)$,":[4],"is":[5,27,39,135],"a":[6,24,31,45,47,71,81,89,103,114,129,163,175,178],"sequence":[7],"of":[8,73,83,91,99,105,128,151,162,174,180,193,224,233],"graphs":[9],"on":[10,34],"$n$":[11,121],"vertices":[12],"which":[13,87],"begins":[14],"with":[15,153,230,238],"the":[16,35,126,158,171,225,234,239],"edgeless":[17],"graph,":[18],"and":[19,102,190,195,215,237],"where":[20,116,125],"at":[21,66],"each":[22,77,133],"step":[23,134],"single":[25],"edge":[26],"added":[28,119],"according":[29,137],"to":[30,94,113,138,177,185,206,213],"uniform":[32],"distribution":[33],"missing":[36,95],"edges.":[37],"It":[38],"well":[40],"known":[41,60],"that":[42,200,216],"in":[43,76,132,166,196],"such":[44],"process":[46,235],"giant":[48,164],"component":[49,165],"(of":[50],"linear":[51],"size)":[52],"typically":[53],"emerges":[54],"after":[55],"$(1+o(1))\\frac{n}{2}$":[56],"edges":[57,75,96],"(a":[58],"phenomenon":[59],"as":[61,170,208],"``the":[62],"double":[63],"jump''),":[64],"i.e.,":[65],"time":[67,161],"$t=1$":[68],"when":[69],"using":[70],"timescale":[72],"$n/2$":[74],"step.":[78],"We":[79,183],"consider":[80],"generalization":[82],"this":[84,167],"$\\Gorg[K](n)$,":[86],"gives":[88],"weight":[90,104],"size":[92,106],"1":[93],"between":[97,120],"pairs":[98],"isolated":[100,123],"vertices,":[101],"$K":[107],"\\in":[108],"[0,\\infty)$":[109],"otherwise.":[110],"This":[111],"corresponds":[112],"case":[115],"links":[117],"are":[118,146],"initially":[122],"settlements,":[124],"probability":[127],"new":[130],"link":[131],"biased":[136],"whether":[139],"or":[140],"not":[141],"its":[142],"two":[143],"endpoint":[144],"settlements":[145],"still":[147],"isolated.":[148],"Combining":[149],"methods":[150],"\\cite{SpencerWormald}":[152],"analytical":[154,240],"techniques,":[155],"we":[156,198],"describe":[157],"typical":[159],"emerging":[160],"$t_c(K)$,":[169],"singularity":[172],"point":[173],"solution":[176],"set":[179],"differential":[181,188,226],"equations.":[182],"proceed":[184],"analyze":[186],"these":[187],"equations":[189,227],"obtain":[191],"properties":[192],"$\\Gorg$,":[194],"particular,":[197],"show":[199],"$t_c(K)$":[201],"strictly":[202],"decreases":[203],"from":[204,211],"3/2":[205],"0":[207,212],"$K$":[209],"increases":[210],"$\\infty$,":[214],"$t_c(K)":[217],"=":[218],"\\frac{4}{\\sqrt{3K}}(1":[219],"+":[220],"o(1))$.":[221],"Numerical":[222],"approximations":[223],"agree":[228],"both":[229],"computer":[231],"simulations":[232],"$\\Gorg(n)$":[236],"results.":[241]},"counts_by_year":[{"year":2016,"cited_by_count":1}],"updated_date":"2026-10-12T09:40:04.860316","created_date":"2016-06-24T00:00:00"}