{"id":"https://openalex.org/W7223209936","doi":"https://doi.org/10.1093/pnasnexus/pgag364","title":"A phase transition for the strength of weak ties","display_name":"A phase transition for the strength of weak ties","publication_year":2026,"publication_date":"2026-10-22","ids":{"openalex":"https://openalex.org/W7223209936","doi":"https://doi.org/10.1093/pnasnexus/pgag364"},"language":"en","primary_location":{"id":"doi:10.1093/pnasnexus/pgag364","doi":"https://doi.org/10.1093/pnasnexus/pgag364","is_oa":true,"landing_page_url":"https://doi.org/10.1093/pnasnexus/pgag364","pdf_url":"https://academic.oup.com/pnasnexus/advance-article-pdf/doi/10.1093/pnasnexus/pgag364/72371457/pgag364.pdf","source":{"id":"https://openalex.org/S4220651275","display_name":"PNAS Nexus","issn_l":"2752-6542","issn":["2752-6542"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-c","jufo-1","ki-jl-1","norway-1","vabb-shw"],"host_organization":"https://openalex.org/P4310311648","host_organization_name":"Oxford University Press","host_organization_lineage":["https://openalex.org/P4310311648","https://openalex.org/P4310311647"],"host_organization_lineage_names":["Oxford University Press","University of Oxford"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"PNAS Nexus","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://academic.oup.com/pnasnexus/advance-article-pdf/doi/10.1093/pnasnexus/pgag364/72371457/pgag364.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5141019604","display_name":"Dr. Esha Chatterjee","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I16269868","display_name":"Santa Clara University","ror":"https://ror.org/03ypqe447","country_code":"US","type":"education","lineage":["https://openalex.org/I16269868"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Esha Chatterjee","raw_affiliation_strings":["Office of Educational Assessment, Santa Clara University , 500 El Camino Real, Santa Clara, 95053, CA ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Office of Educational Assessment, Santa Clara University , 500 El Camino Real, Santa Clara, 95053, CA ,","institution_ids":["https://openalex.org/I16269868"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5025655425","display_name":"Sourav Chatterjee","orcid":"https://orcid.org/0000-0003-4460-209X","observed_orcids":["https://orcid.org/0000-0003-4460-209X"]},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Sourav Chatterjee","raw_affiliation_strings":["Department of Statistics, Stanford University , 390 Jane Stanford Way, Stanford, 94305, CA ,"],"raw_orcid":"https://orcid.org/0000-0003-4460-209X","affiliations":[{"raw_affiliation_string":"Department of Statistics, Stanford University , 390 Jane Stanford Way, Stanford, 94305, CA ,","institution_ids":["https://openalex.org/I97018004"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5025655425"],"corresponding_institution_ids":["https://openalex.org/I97018004"],"apc_list":{"value":4202,"currency":"USD","value_usd":4202},"apc_paid":{"value":4202,"currency":"USD","value_usd":4202},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.75057062,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.9916999936103821,"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"}},"topics":[{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.9916999936103821,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.006500000134110451,"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/T11476","display_name":"Graph theory and applications","score":0.00019999999494757503,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/strength-of-weak-ties","display_name":"strength of weak ties","score":0.9869999885559082},{"id":"https://openalex.org/keywords/strong-ties","display_name":"strong ties","score":0.734000027179718},{"id":"https://openalex.org/keywords/transitivity","display_name":"transitivity","score":0.5809999704360962},{"id":"https://openalex.org/keywords/triadic-closure","display_name":"triadic closure","score":0.7409999966621399},{"id":"https://openalex.org/keywords/network-connectivity","display_name":"network connectivity","score":0.46799999475479126},{"id":"https://openalex.org/keywords/sharp-phase-transition","display_name":"sharp phase transition","score":0.6309999823570251},{"id":"https://openalex.org/keywords/bounded-degree-graphs","display_name":"bounded degree graphs","score":0.6629999876022339},{"id":"https://openalex.org/keywords/component-size","display_name":"component size","score":0.5550000071525574},{"id":"https://openalex.org/keywords/network-fragmentation","display_name":"network fragmentation","score":0.6340000033378601},{"id":"https://openalex.org/keywords/real-world-networks","display_name":"real-world networks","score":0.6389999985694885}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7570000290870667},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.510699987411499},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5006999969482422},{"id":"https://openalex.org/C149288129","wikidata":"https://www.wikidata.org/wiki/Q185357","display_name":"Phase transition","level":2,"score":0.42640000581741333},{"id":"https://openalex.org/C203519979","wikidata":"https://www.wikidata.org/wiki/Q865360","display_name":"Jaccard index","level":3,"score":0.41749998927116394},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38580000400543213},{"id":"https://openalex.org/C38976095","wikidata":"https://www.wikidata.org/wiki/Q752641","display_name":"Asymmetry","level":2,"score":0.38019999861717224},{"id":"https://openalex.org/C193435613","wikidata":"https://www.wikidata.org/wiki/Q2997928","display_name":"Connected component","level":2,"score":0.37290000915527344},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.36800000071525574},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.35249999165534973},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.34549999237060547},{"id":"https://openalex.org/C89128539","wikidata":"https://www.wikidata.org/wiki/Q1949963","display_name":"Statistic","level":2,"score":0.3321000039577484},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.32659998536109924},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.32249999046325684},{"id":"https://openalex.org/C168167062","wikidata":"https://www.wikidata.org/wiki/Q1117970","display_name":"Component (thermodynamics)","level":2,"score":0.31459999084472656},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.296099990606308},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.28690001368522644},{"id":"https://openalex.org/C191399111","wikidata":"https://www.wikidata.org/wiki/Q64861","display_name":"Transitive relation","level":2,"score":0.27619999647140503},{"id":"https://openalex.org/C14103023","wikidata":"https://www.wikidata.org/wiki/Q11681459","display_name":"Pairing","level":3,"score":0.2734000086784363},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.2703999876976013},{"id":"https://openalex.org/C122377713","wikidata":"https://www.wikidata.org/wiki/Q4422799","display_name":"Weak consistency","level":4,"score":0.26260000467300415},{"id":"https://openalex.org/C57945734","wikidata":"https://www.wikidata.org/wiki/Q7977941","display_name":"Weak convergence","level":3,"score":0.2605000138282776},{"id":"https://openalex.org/C166785042","wikidata":"https://www.wikidata.org/wiki/Q190391","display_name":"Central limit theorem","level":2,"score":0.2533999979496002},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.25099998712539673}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1093/pnasnexus/pgag364","doi":"https://doi.org/10.1093/pnasnexus/pgag364","is_oa":true,"landing_page_url":"https://doi.org/10.1093/pnasnexus/pgag364","pdf_url":"https://academic.oup.com/pnasnexus/advance-article-pdf/doi/10.1093/pnasnexus/pgag364/72371457/pgag364.pdf","source":{"id":"https://openalex.org/S4220651275","display_name":"PNAS Nexus","issn_l":"2752-6542","issn":["2752-6542"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-c","jufo-1","ki-jl-1","norway-1","vabb-shw"],"host_organization":"https://openalex.org/P4310311648","host_organization_name":"Oxford University Press","host_organization_lineage":["https://openalex.org/P4310311648","https://openalex.org/P4310311647"],"host_organization_lineage_names":["Oxford University Press","University of Oxford"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"PNAS Nexus","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1093/pnasnexus/pgag364","doi":"https://doi.org/10.1093/pnasnexus/pgag364","is_oa":true,"landing_page_url":"https://doi.org/10.1093/pnasnexus/pgag364","pdf_url":"https://academic.oup.com/pnasnexus/advance-article-pdf/doi/10.1093/pnasnexus/pgag364/72371457/pgag364.pdf","source":{"id":"https://openalex.org/S4220651275","display_name":"PNAS Nexus","issn_l":"2752-6542","issn":["2752-6542"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["cwts-core","doaj","dongbi-c","jufo-1","ki-jl-1","norway-1","vabb-shw"],"host_organization":"https://openalex.org/P4310311648","host_organization_name":"Oxford University Press","host_organization_lineage":["https://openalex.org/P4310311648","https://openalex.org/P4310311647"],"host_organization_lineage_names":["Oxford University Press","University of Oxford"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"PNAS Nexus","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G3503781647","display_name":"Matrix Completion with Non-uniform Missing Patterns, a New Measure of Conditional Dependence, and Applications to Feature Selection","funder_award_id":"2113242","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6455832783","display_name":"Mathematical Foundations for Yang-Mills Theory, Randomly Growing Surfaces, and Related Systems","funder_award_id":"2153654","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":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7223209936.pdf","grobid_xml":"https://content.openalex.org/works/W7223209936.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7096228057","https://openalex.org/W7095398692","https://openalex.org/W2136852793","https://openalex.org/W4383647225","https://openalex.org/W2069668029","https://openalex.org/W2739123860","https://openalex.org/W2349307166","https://openalex.org/W7098940043","https://openalex.org/W3102050987","https://openalex.org/W1982075554"],"abstract_inverted_index":{"Abstract":[0],"Granovetter's":[1],"``strength":[2],"of":[3,15,66,70,79,110,117,204],"weak":[4,16],"ties''":[5],"argument":[6],"links":[7],"local":[8],"triadic":[9],"closure":[10,46],"to":[11],"the":[12,36,67,113,118,179,192,205],"global":[13,154],"importance":[14],"relationships,":[17],"but":[18,172],"it":[19],"does":[20],"not":[21,178],"by":[22,39,106,143],"itself":[23],"quantify":[24],"when":[25],"strong":[26,40,68,77],"ties":[27,41],"can":[28,129],"or":[29],"cannot":[30],"sustain":[31],"large-scale":[32],"connectivity.":[33],"We":[34,136],"study":[35],"subgraph":[37],"formed":[38],"under":[42],"a":[43,63,82,88,145,187,195,201],"uniform,":[44],"edgewise":[45,181],"condition.":[47],"A":[48,74],"strong-tie":[49,101,134],"graph":[50],"is":[51,185],"called":[52],"p-transitive":[53,127],"when,":[54],"for":[55,149],"every":[56,100],"ordered":[57],"adjacent":[58],"pair":[59],"(A,B),":[60],"at":[61,92],"least":[62],"fraction":[64],"p":[65,93,97,111,123],"neighbors":[69,78],"B":[71],"other":[72],"than":[73,200],"are":[75,170,177],"also":[76,137],"A.":[80],"Under":[81],"fixed":[83],"maximum-degree":[84],"bound,":[85],"we":[86],"prove":[87],"sharp":[89],"structural":[90,189],"transition":[91],"=":[94],"2/3.":[95,207],"If":[96,122],">":[98],"2/3,":[99,125],"component":[102,160],"has":[103],"size":[104],"bounded":[105],"an":[107],"explicit":[108],"function":[109],"and":[112,152,175],"degree":[114],"bounds,":[115],"independently":[116],"total":[119],"network":[120],"size.":[121,161],"<":[124],"bounded-degree":[126],"graphs":[128],"have":[130],"arbitrarily":[131],"large":[132],"connected":[133],"components.":[135],"revisit":[138],"three":[139],"unweighted":[140],"real-world":[141],"networks":[142],"thresholding":[144],"Jaccard":[146],"neighborhood-overlap":[147],"proxy":[148],"tie":[150],"strength":[151],"plotting":[153],"transitivity":[155],"against":[156],"normalized":[157],"size-biased":[158],"mean":[159],"These":[162],"empirical":[163],"curves":[164],"show":[165],"fragmentation":[166],"as":[167],"low-overlap":[168],"edges":[169],"removed,":[171],"their":[173],"statistic":[174],"thresholds":[176],"theorem's":[180],"parameter.":[182],"The":[183],"theorem":[184],"therefore":[186],"conditional":[188],"result,":[190],"while":[191],"data":[193],"illustrate":[194],"related":[196],"clustering--connectivity":[197],"tradeoff":[198],"rather":[199],"direct":[202],"validation":[203],"constant":[206]},"counts_by_year":[],"updated_date":"2026-10-26T11:51:55.731654","created_date":"2026-10-23T00:00:00"}