{"id":"https://openalex.org/W2007626053","doi":"https://doi.org/10.1002/1098-2418(200103)18:2<164::aid-rsa1004>3.0.co;2-h","title":"A phase transition phenomenon in a random directed acyclic graph","display_name":"A phase transition phenomenon in a random directed acyclic graph","publication_year":2001,"publication_date":"2001-01-01","ids":{"openalex":"https://openalex.org/W2007626053","doi":"https://doi.org/10.1002/1098-2418(200103)18:2<164::aid-rsa1004>3.0.co;2-h","mag":"2007626053"},"language":"en","primary_location":{"id":"doi:10.1002/1098-2418(200103)18:2<164::aid-rsa1004>3.0.co;2-h","is_oa":false,"landing_page_url":"https://doi.org/10.1002/1098-2418(200103)18:2<164::aid-rsa1004>3.0.co;2-h","pdf_url":null,"source":{"id":"https://openalex.org/S59667848","display_name":"Random Structures and Algorithms","issn_l":"1042-9832","issn":["1042-9832","1098-2418"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Random Structures and Algorithms","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/A5084113050","display_name":"Boris Pittel","orcid":"https://orcid.org/0000-0003-1875-9317"},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"B. Pittel","raw_affiliation_strings":["Department of Mathematics, Ohio State University, Columbus, OH 43210"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Ohio State University, Columbus, OH 43210","institution_ids":["https://openalex.org/I52357470"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053605316","display_name":"Ronald M. Tungol","orcid":null},"institutions":[{"id":"https://openalex.org/I87074743","display_name":"University of the Philippines Diliman","ror":"https://ror.org/03tbh6y23","country_code":"PH","type":"education","lineage":["https://openalex.org/I103911934","https://openalex.org/I87074743"]}],"countries":["PH"],"is_corresponding":false,"raw_author_name":"R. Tungol","raw_affiliation_strings":["Computer Science Department, University of the Philippines-Diliman, Quezon City, Philippines","Computer Science Department, University of the Philippines\u2010Diliman, Quezon City, Philippines"],"affiliations":[{"raw_affiliation_string":"Computer Science Department, University of the Philippines-Diliman, Quezon City, Philippines","institution_ids":["https://openalex.org/I87074743"]},{"raw_affiliation_string":"Computer Science Department, University of the Philippines\u2010Diliman, Quezon City, Philippines","institution_ids":["https://openalex.org/I87074743"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5084113050"],"corresponding_institution_ids":["https://openalex.org/I52357470"],"apc_list":{"value":4330,"currency":"USD","value_usd":4330},"apc_paid":null,"fwci":1.477,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.77917468,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"18","issue":"2","first_page":"164","last_page":"184"},"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9958999752998352,"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/T10591","display_name":"Theoretical and Computational Physics","score":0.9901999831199646,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter 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/combinatorics","display_name":"Combinatorics","score":0.788640558719635},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6862416863441467},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.5551780462265015},{"id":"https://openalex.org/keywords/random-graph","display_name":"Random graph","score":0.5287727117538452},{"id":"https://openalex.org/keywords/transitive-closure","display_name":"Transitive closure","score":0.5273711681365967},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.5252075791358948},{"id":"https://openalex.org/keywords/limiting","display_name":"Limiting","score":0.45402792096138},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.41859757900238037},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4062657952308655}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.788640558719635},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6862416863441467},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.5551780462265015},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.5287727117538452},{"id":"https://openalex.org/C128896722","wikidata":"https://www.wikidata.org/wiki/Q1501387","display_name":"Transitive closure","level":2,"score":0.5273711681365967},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5252075791358948},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.45402792096138},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.41859757900238037},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4062657952308655},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1002/1098-2418(200103)18:2<164::aid-rsa1004>3.0.co;2-h","is_oa":false,"landing_page_url":"https://doi.org/10.1002/1098-2418(200103)18:2<164::aid-rsa1004>3.0.co;2-h","pdf_url":null,"source":{"id":"https://openalex.org/S59667848","display_name":"Random Structures and Algorithms","issn_l":"1042-9832","issn":["1042-9832","1098-2418"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Random Structures and Algorithms","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":8,"referenced_works":["https://openalex.org/W1513610428","https://openalex.org/W1970543429","https://openalex.org/W1974575606","https://openalex.org/W2017139167","https://openalex.org/W2076597612","https://openalex.org/W2085227596","https://openalex.org/W2091049001","https://openalex.org/W2109767783"],"related_works":["https://openalex.org/W2359423766","https://openalex.org/W1600485007","https://openalex.org/W2364216866","https://openalex.org/W2362009423","https://openalex.org/W2614633957","https://openalex.org/W2391894074","https://openalex.org/W2070131906","https://openalex.org/W1546475804","https://openalex.org/W2365750584","https://openalex.org/W2070329106"],"abstract_inverted_index":{"Let":[0,27],"a":[1,39],"random":[2,13],"directed":[3],"acyclic":[4],"graph":[5,14],"be":[6,29],"defined":[7],"as":[8],"being":[9],"obtained":[10],"from":[11],"the":[12,19,23,30,33,73,77,93],"Gn,":[15],"p":[16],"by":[17],"orienting":[18],"edges":[20],"according":[21],"to":[22,53,101],"ordering":[24],"of":[25,32,38,76,85,96],"vertices.":[26],"\u03b3n*":[28,50],"size":[31],"largest":[34],"(reflexive,":[35],"transitive)":[36],"closure":[37,80],"vertex.":[40],"For":[41],"p=c(log":[42],"n)/n,":[43],"we":[44,90],"prove":[45],"that,":[46],"with":[47],"high":[48],"probability,":[49],"is":[51,99],"asymptotic":[52,100],"nc":[54],"log":[55,58,103,106],"n,":[56,60,104],"2n(log":[57],"n)/log":[59,107],"and":[61,109],"n(1\u22121/c)":[62],"depending":[63],"on":[64],"whether":[65],"c<1,":[66],"c=1,":[67],"or":[68],"c>1.":[69],"We":[70],"also":[71],"determine":[72],"limiting":[74],"distribution":[75],"first":[78],"vertex":[79],"in":[81],"all":[82],"three":[83],"ranges":[84],"c.":[86],"As":[87],"an":[88],"application,":[89],"show":[91],"that":[92],"expected":[94],"number":[95],"comparable":[97],"pairs":[98],"n1+c/c":[102],"\u00bd(n(log":[105],"n)2,":[108],"\u00bd(n(1\u22121/c))2,":[110],"respectively.":[111],"\u00a9":[112],"2001":[113,124],"John":[114],"Wiley":[115],"&":[116],"Sons,":[117],"Inc.":[118],"Random":[119],"Struct.":[120],"Alg.,":[121],"18:":[122],"164\u2013184,":[123]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}