{"id":"https://openalex.org/W7220675107","doi":"https://doi.org/10.48550/arxiv.2610.06237","title":"Bradley-Terry model under general comparison graphs","display_name":"Bradley-Terry model under general comparison graphs","publication_year":2026,"publication_date":"2026-10-05","ids":{"openalex":"https://openalex.org/W7220675107","doi":"https://doi.org/10.48550/arxiv.2610.06237"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.06237","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06237","pdf_url":"https://arxiv.org/pdf/2610.06237","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":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.06237","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5028504100","display_name":"Ruijian Han","orcid":"https://orcid.org/0000-0002-9225-2218","observed_orcids":["https://orcid.org/0000-0002-9225-2218"]},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Han, Ruijian","raw_affiliation_strings":["Department of Data Science and Artificial Intelligence , The Hong Kong Polytechnic University , Hung Hom , Kowloon , Hong Kong SAR"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Data Science and Artificial Intelligence , The Hong Kong Polytechnic University , Hung Hom , Kowloon , Hong Kong SAR","institution_ids":["https://openalex.org/I14243506"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5157777408","display_name":"Yuanhang Luo","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I14243506","display_name":"Hong Kong Polytechnic University","ror":"https://ror.org/0030zas98","country_code":"HK","type":"education","lineage":["https://openalex.org/I14243506"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Luo, Yuanhang","raw_affiliation_strings":["Department of Data Science and Artificial Intelligence , The Hong Kong Polytechnic University , Hung Hom , Kowloon , Hong Kong SAR"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Data Science and Artificial Intelligence , The Hong Kong Polytechnic University , Hung Hom , Kowloon , Hong Kong SAR","institution_ids":["https://openalex.org/I14243506"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5157746962","display_name":"Yiming Xu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I143302722","display_name":"University of Kentucky","ror":"https://ror.org/02k3smh20","country_code":"US","type":"education","lineage":["https://openalex.org/I143302722"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Xu, Yiming","raw_affiliation_strings":["Department of Mathematics , University of Kentucky , Lexington , KY , 40506 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Kentucky , Lexington , KY , 40506 , USA","institution_ids":["https://openalex.org/I143302722"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":null,"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/T10136","display_name":"Statistical Methods and Inference","score":0.33500000834465027,"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"}},"topics":[{"id":"https://openalex.org/T10136","display_name":"Statistical Methods and Inference","score":0.33500000834465027,"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/T13500","display_name":"Probability and Statistical Research","score":0.10819999873638153,"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/T10991","display_name":"Game Theory and Voting Systems","score":0.10199999809265137,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/bradley-terry-model","display_name":"Bradley-Terry model","score":1.0},{"id":"https://openalex.org/keywords/maximum-likelihood-estimator","display_name":"maximum likelihood estimator","score":0.7960000038146973},{"id":"https://openalex.org/keywords/uniform-consistency","display_name":"uniform consistency","score":0.8080000281333923},{"id":"https://openalex.org/keywords/asymptotic-normality","display_name":"asymptotic normality","score":0.8790000081062317},{"id":"https://openalex.org/keywords/effective-resistance","display_name":"effective resistance","score":0.9129999876022339}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8080999851226807},{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.6873999834060669},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.6341000199317932},{"id":"https://openalex.org/C65778772","wikidata":"https://www.wikidata.org/wiki/Q12345341","display_name":"Asymptotic distribution","level":3,"score":0.588100016117096},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.5224999785423279},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4729999899864197},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4399000108242035},{"id":"https://openalex.org/C117251300","wikidata":"https://www.wikidata.org/wiki/Q1849855","display_name":"Parametric statistics","level":2,"score":0.4377000033855438},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.43689998984336853},{"id":"https://openalex.org/C142259097","wikidata":"https://www.wikidata.org/wiki/Q5891314","display_name":"Homogeneity (statistics)","level":2,"score":0.38920000195503235},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.37400001287460327},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3582000136375427},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.35089999437332153},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.33959999680519104},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.3310999870300293},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.32670000195503235},{"id":"https://openalex.org/C31388003","wikidata":"https://www.wikidata.org/wiki/Q7624548","display_name":"Strong consistency","level":3,"score":0.3075000047683716},{"id":"https://openalex.org/C30609935","wikidata":"https://www.wikidata.org/wiki/Q7291969","display_name":"Random geometric graph","level":5,"score":0.30570000410079956},{"id":"https://openalex.org/C114289077","wikidata":"https://www.wikidata.org/wiki/Q3284399","display_name":"Statistical model","level":2,"score":0.29750001430511475},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.2964000105857849},{"id":"https://openalex.org/C2776157432","wikidata":"https://www.wikidata.org/wiki/Q1375683","display_name":"Normality","level":2,"score":0.2944999933242798},{"id":"https://openalex.org/C5134670","wikidata":"https://www.wikidata.org/wiki/Q1626444","display_name":"Cut","level":4,"score":0.29350000619888306},{"id":"https://openalex.org/C24574437","wikidata":"https://www.wikidata.org/wiki/Q7135228","display_name":"Parametric model","level":3,"score":0.289000004529953},{"id":"https://openalex.org/C78023250","wikidata":"https://www.wikidata.org/wiki/Q657596","display_name":"Unary operation","level":2,"score":0.28839999437332153},{"id":"https://openalex.org/C130594498","wikidata":"https://www.wikidata.org/wiki/Q2028293","display_name":"Iterated logarithm","level":3,"score":0.2777999937534332},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.2759999930858612},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.26489999890327454},{"id":"https://openalex.org/C181789720","wikidata":"https://www.wikidata.org/wiki/Q4812191","display_name":"Asymptotically optimal algorithm","level":2,"score":0.25529998540878296},{"id":"https://openalex.org/C122377713","wikidata":"https://www.wikidata.org/wiki/Q4422799","display_name":"Weak consistency","level":4,"score":0.2540000081062317},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.2531999945640564},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.25029999017715454}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.06237","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06237","pdf_url":"https://arxiv.org/pdf/2610.06237","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":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.06237","doi":"https://doi.org/10.48550/arxiv.2610.06237","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.06237","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:2610.06237","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.06237","pdf_url":"https://arxiv.org/pdf/2610.06237","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":"cc-by-nc-sa","license_id":"https://openalex.org/licenses/cc-by-nc-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220675107.pdf","grobid_xml":"https://content.openalex.org/works/W7220675107.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W121824094","https://openalex.org/W1458927723","https://openalex.org/W2004691942","https://openalex.org/W4293072317","https://openalex.org/W4366196988","https://openalex.org/W4390963066","https://openalex.org/W7104574165","https://openalex.org/W7125374114","https://openalex.org/W7171396762"],"related_works":["https://openalex.org/W3210246763","https://openalex.org/W3207622980","https://openalex.org/W6891898465","https://openalex.org/W3087223531","https://openalex.org/W7220958470","https://openalex.org/W4286899093","https://openalex.org/W1569736542","https://openalex.org/W4318691814","https://openalex.org/W4390963066","https://openalex.org/W4413967399"],"abstract_inverted_index":{"The":[0,94],"Bradley-Terry":[1],"model":[2,6],"is":[3,96],"a":[4,68,81,151],"parametric":[5],"for":[7,15,102,135],"ranking":[8],"from":[9],"pairwise":[10,63],"comparisons.":[11],"Existing":[12],"asymptotic":[13,120,157],"theory":[14],"the":[16,24,55,73,77,86,90,123,141,146],"maximum":[17],"likelihood":[18],"estimator":[19],"(MLE)":[20],"in":[21],"regimes":[22],"where":[23],"number":[25],"of":[26,54,67,89,110,122],"objects":[27,79],"grows":[28],"often":[29],"requires":[30],"homogeneity":[31],"assumptions":[32],"or":[33],"compatibility":[34],"conditions":[35],"on":[36],"comparison":[37,60],"graphs,":[38],"which":[39],"limits":[40],"its":[41],"applicability":[42],"to":[43,99],"many":[44],"practical":[45],"settings.":[46],"In":[47],"this":[48],"work,":[49],"we":[50,117,131],"establish":[51,119],"uniform":[52,133],"consistency":[53,134],"MLE":[56],"under":[57,159],"general":[58],"deterministic":[59],"designs.":[61],"Our":[62],"error":[64],"bound":[65,95],"consists":[66],"pair-specific":[69],"term":[70,83],"governed":[71,84],"by":[72,85,150],"effective":[74],"resistance":[75],"between":[76],"corresponding":[78,124],"and":[80],"global":[82],"spectral":[87],"gap":[88],"unnormalized":[91],"graph":[92,104,138],"Laplacian.":[93],"tight":[97],"up":[98],"logarithmic":[100,152],"factors":[101],"certain":[103],"sequences.":[105],"For":[106],"any":[107],"prespecified":[108],"sequence":[109],"pairs":[111],"satisfying":[112],"an":[113,129],"additional":[114,160],"balancing":[115,161],"condition,":[116],"further":[118],"normality":[121,158],"estimated":[125],"utility":[126],"differences.":[127],"As":[128],"application,":[130],"obtain":[132],"independent-edge":[136],"random":[137],"designs":[139],"whenever":[140],"minimum":[142],"edge":[143],"probability":[144],"exceeds":[145],"Erd\u0151s-R\u00e9nyi":[147],"connectivity":[148],"threshold":[149],"factor,":[153],"as":[154,156],"well":[155],"conditions.":[162]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}
