{"id":"https://openalex.org/W2158059367","doi":"https://doi.org/10.1017/s0963548310000325","title":"The Diameter of Sparse Random Graphs","display_name":"The Diameter of Sparse Random Graphs","publication_year":2010,"publication_date":"2010-10-05","ids":{"openalex":"https://openalex.org/W2158059367","doi":"https://doi.org/10.1017/s0963548310000325","mag":"2158059367"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548310000325","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548310000325","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/0808.4067","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"OLIVER RIORDAN","orcid":null},"institutions":[{"id":"https://openalex.org/I4210120676","display_name":"Mathematical Institute of the Slovak Academy of Sciences","ror":"https://ror.org/02fhy7464","country_code":"SK","type":"facility","lineage":["https://openalex.org/I207624831","https://openalex.org/I4210120676"]}],"countries":["SK"],"is_corresponding":true,"raw_author_name":"OLIVER RIORDAN","raw_affiliation_strings":["Mathematical Institute, University of Oxford, 24\u201329 St Giles', Oxford OX1 3LB, UK (e-mail:"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Oxford, 24\u201329 St Giles', Oxford OX1 3LB, UK (e-mail:","institution_ids":["https://openalex.org/I4210120676"]}]},{"author_position":"last","author":{"id":null,"display_name":"NICHOLAS WORMALD","orcid":null},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"NICHOLAS WORMALD","raw_affiliation_strings":["Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, Canada (e-mail:"],"affiliations":[{"raw_affiliation_string":"Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, Canada (e-mail:","institution_ids":["https://openalex.org/I151746483"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I4210120676"],"apc_list":null,"apc_paid":null,"fwci":1.6389,"has_fulltext":false,"cited_by_count":47,"citation_normalized_percentile":{"value":0.82481267,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"19","issue":"5-6","first_page":"835","last_page":"926"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.4120999872684479,"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":0.4120999872684479,"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/T10064","display_name":"Complex Network Analysis Techniques","score":0.1729000061750412,"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/T10591","display_name":"Theoretical and Computational Physics","score":0.09870000183582306,"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/random-graph","display_name":"Random graph","score":0.5817000269889832},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.44749999046325684},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.4131999909877777},{"id":"https://openalex.org/keywords/range","display_name":"Range (aeronautics)","score":0.3840999901294708},{"id":"https://openalex.org/keywords/limiting","display_name":"Limiting","score":0.3758000135421753},{"id":"https://openalex.org/keywords/branching-process","display_name":"Branching process","score":0.35260000824928284},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.34209999442100525},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.3327000141143799},{"id":"https://openalex.org/keywords/term","display_name":"Term (time)","score":0.3312999904155731}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.796500027179718},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6025999784469604},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.5817000269889832},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4837000072002411},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.44749999046325684},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.4131999909877777},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.3840999901294708},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.3758000135421753},{"id":"https://openalex.org/C177255742","wikidata":"https://www.wikidata.org/wiki/Q4109953","display_name":"Branching process","level":2,"score":0.35260000824928284},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.34209999442100525},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.3327000141143799},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.3312999904155731},{"id":"https://openalex.org/C30609935","wikidata":"https://www.wikidata.org/wiki/Q7291969","display_name":"Random geometric graph","level":5,"score":0.3260999917984009},{"id":"https://openalex.org/C4255713","wikidata":"https://www.wikidata.org/wiki/Q7291988","display_name":"Random regular graph","level":5,"score":0.3231000006198883},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.32280001044273376},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.31189998984336853},{"id":"https://openalex.org/C22679943","wikidata":"https://www.wikidata.org/wiki/Q159375","display_name":"Standard deviation","level":2,"score":0.30250000953674316},{"id":"https://openalex.org/C2778751112","wikidata":"https://www.wikidata.org/wiki/Q835016","display_name":"Window (computing)","level":2,"score":0.2903999984264374},{"id":"https://openalex.org/C13929819","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Random function","level":3,"score":0.27639999985694885},{"id":"https://openalex.org/C200985842","wikidata":"https://www.wikidata.org/wiki/Q3375503","display_name":"Random permutation","level":3,"score":0.27489998936653137},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.273499995470047},{"id":"https://openalex.org/C2776839635","wikidata":"https://www.wikidata.org/wiki/Q14942679","display_name":"Random tree","level":4,"score":0.27090001106262207},{"id":"https://openalex.org/C141547133","wikidata":"https://www.wikidata.org/wiki/Q7291996","display_name":"Random variate","level":3,"score":0.2655999958515167},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.2615000009536743},{"id":"https://openalex.org/C154547637","wikidata":"https://www.wikidata.org/wiki/Q776602","display_name":"Expander graph","level":3,"score":0.2549999952316284}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1017/s0963548310000325","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548310000325","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:0808.4067","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0808.4067","pdf_url":"https://arxiv.org/pdf/0808.4067","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:0808.4067","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0808.4067","pdf_url":"https://arxiv.org/pdf/0808.4067","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":28,"referenced_works":["https://openalex.org/W1592213313","https://openalex.org/W1981896702","https://openalex.org/W2014075779","https://openalex.org/W2027377866","https://openalex.org/W2032049676","https://openalex.org/W2036312115","https://openalex.org/W2039800039","https://openalex.org/W2040232420","https://openalex.org/W2040419749","https://openalex.org/W2049529696","https://openalex.org/W2061118827","https://openalex.org/W2068902429","https://openalex.org/W2117495845","https://openalex.org/W2121526364","https://openalex.org/W2145617053","https://openalex.org/W2149745907","https://openalex.org/W2151221105","https://openalex.org/W2169015768","https://openalex.org/W2169534116","https://openalex.org/W2478722229","https://openalex.org/W2769133055","https://openalex.org/W2950850272","https://openalex.org/W2952242105","https://openalex.org/W4241669766","https://openalex.org/W4242518419","https://openalex.org/W4244638572","https://openalex.org/W4245956052","https://openalex.org/W4250665764"],"related_works":[],"abstract_inverted_index":{"In":[0],"this":[1],"paper":[2],"we":[3,48,70,91,123,170],"study":[4],"the":[5,8,19,75,99,103,107,133,136,139,152,177,185,188],"diameter":[6,137],"of":[7,30,53,106,135,138,151,154,181,184],"random":[9,140],"graph":[10,141],"G":[11,142],"(":[12,35,143,166],"n":[13,36,42,86,144,167],",":[14,18,145],"p":[15,32,34,39,62,82,113,146,163,165],"),":[16],"i.e.":[17],"largest":[20],"finite":[21],"distance":[22],"between":[23],"two":[24],"vertices,":[25],"for":[26,160],"a":[27,50,93],"wide":[28],"range":[29],"functions":[31,162],"=":[33,40,164],").":[37,168],"For":[38,81],"\u03bb/":[41],"with":[43,59,87,110,126],"\u03bb":[44],"&gt;":[45],"1":[46],"constant":[47],"give":[49],"simple":[51],"proof":[52],"an":[54,60,111,149],"essentially":[55],"best":[56],"possible":[57],"result,":[58],"O":[61,112],"(1)":[63],"additive":[64,115],"correction":[65,116],"term.":[66],"Using":[67],"similar":[68],"techniques,":[69],"establish":[71],"two-point":[72],"concentration":[73],"in":[74],"case":[76],"that":[77,96],"np":[78],"\u2192":[79,89],"\u221e.":[80],"=(1":[83],"+":[84],"\u03b5)/":[85],"\u03b5":[88],"0,":[90],"obtain":[92],"corresponding":[94],"result":[95],"applies":[97],"all":[98,161],"way":[100],"down":[101],"to":[102,148,191],"scaling":[104],"window":[105],"phase":[108],"transition,":[109],"(1/\u03b5)":[114],"term":[117],"whose":[118],"(appropriately":[119],"scaled)":[120],"limiting":[121],"distribution":[122],"describe.":[124],"Combined":[125],"earlier":[127],"results,":[128],"our":[129],"new":[130],"results":[131],"complete":[132],"determination":[134],")":[147],"accuracy":[150],"order":[153],"its":[155],"standard":[156],"deviation":[157],"(or":[158],"better),":[159],"Throughout":[169],"use":[171],"branching":[172],"process":[173],"methods,":[174],"rather":[175],"than":[176],"more":[178],"common":[179],"approach":[180],"separate":[182],"analysis":[183],"2-core":[186],"and":[187],"trees":[189],"attached":[190],"it.":[192]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":7},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":7},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-02T15:55:50.835912","created_date":"2025-10-10T00:00:00"}
