{"id":"https://openalex.org/W2146999606","doi":"https://doi.org/10.37236/9485","title":"Diameter and Stationary Distribution of Random $r$-Out Digraphs","display_name":"Diameter and Stationary Distribution of Random $r$-Out Digraphs","publication_year":2020,"publication_date":"2020-08-07","ids":{"openalex":"https://openalex.org/W2146999606","doi":"https://doi.org/10.37236/9485","mag":"2146999606"},"language":"en","primary_location":{"id":"doi:10.37236/9485","is_oa":true,"landing_page_url":"https://doi.org/10.37236/9485","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i3p28/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i3p28/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5085128089","display_name":"Louigi Addario\u2010Berry","orcid":"https://orcid.org/0000-0001-7125-5746"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Louigi Addario-Berry","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5085679401","display_name":"Borja Balle","orcid":"https://orcid.org/0009-0003-8726-2803"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Borja Balle","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5086363230","display_name":"Guillem Perarnau","orcid":"https://orcid.org/0000-0002-1953-9511"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Guillem Perarnau","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5085128089"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.566,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.7788691,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"27","issue":"3","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9991999864578247,"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":0.9991999864578247,"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":0.998199999332428,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9980000257492065,"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/combinatorics","display_name":"Combinatorics","score":0.8369019031524658},{"id":"https://openalex.org/keywords/multigraph","display_name":"Multigraph","score":0.6827818155288696},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6095965504646301},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.5665930509567261},{"id":"https://openalex.org/keywords/pi","display_name":"Pi","score":0.4969065487384796},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4881342351436615},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.35395121574401855},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32519882917404175},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.13554084300994873},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.12747550010681152},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.09783846139907837}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8369019031524658},{"id":"https://openalex.org/C17758045","wikidata":"https://www.wikidata.org/wiki/Q2642629","display_name":"Multigraph","level":3,"score":0.6827818155288696},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6095965504646301},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.5665930509567261},{"id":"https://openalex.org/C53009064","wikidata":"https://www.wikidata.org/wiki/Q167","display_name":"Pi","level":2,"score":0.4969065487384796},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4881342351436615},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.35395121574401855},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32519882917404175},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.13554084300994873},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.12747550010681152},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.09783846139907837}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.37236/9485","is_oa":true,"landing_page_url":"https://doi.org/10.37236/9485","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i3p28/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1504.06840","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1504.06840","pdf_url":"https://arxiv.org/pdf/1504.06840","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"},{"id":"mag:2146999606","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1504.06840","pdf_url":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:upcommons.upc.edu:2117/342134","is_oa":true,"landing_page_url":"http://hdl.handle.net/2117/342134","pdf_url":null,"source":{"id":"https://openalex.org/S4210207057","display_name":"QRU Quaderns de Recerca en Urbanisme","issn_l":"2014-9689","issn":["2014-9689","2385-6777"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310322448","host_organization_name":"Q71272178","host_organization_lineage":["https://openalex.org/P4310322448"],"host_organization_lineage_names":["Q71272178"],"type":"journal"},"license":"cc-by-nd","license_id":"https://openalex.org/licenses/cc-by-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.1504.06840","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1504.06840","pdf_url":null,"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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.37236/9485","is_oa":true,"landing_page_url":"https://doi.org/10.37236/9485","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v27i3p28/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Sustainable cities and communities","score":0.44999998807907104,"id":"https://metadata.un.org/sdg/11"}],"awards":[{"id":"https://openalex.org/G2862252062","display_name":null,"funder_award_id":"Grant","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"}],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2146999606.pdf","grobid_xml":"https://content.openalex.org/works/W2146999606.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W35268004","https://openalex.org/W138607541","https://openalex.org/W142090792","https://openalex.org/W1487835737","https://openalex.org/W1495867784","https://openalex.org/W1520252399","https://openalex.org/W1522735212","https://openalex.org/W1538240903","https://openalex.org/W1577466578","https://openalex.org/W1704463895","https://openalex.org/W1752963096","https://openalex.org/W1896273530","https://openalex.org/W2019363670","https://openalex.org/W2031541804","https://openalex.org/W2044456918","https://openalex.org/W2046054273","https://openalex.org/W2071896207","https://openalex.org/W2074010768","https://openalex.org/W2079495652","https://openalex.org/W2088285637","https://openalex.org/W2106435631","https://openalex.org/W2142399242","https://openalex.org/W2154599653","https://openalex.org/W2157754126","https://openalex.org/W2158978530","https://openalex.org/W2217322393","https://openalex.org/W2260314326","https://openalex.org/W2295392808","https://openalex.org/W2949995719","https://openalex.org/W2963870251","https://openalex.org/W3103572255","https://openalex.org/W3104831136","https://openalex.org/W6603756267","https://openalex.org/W6636021564"],"related_works":["https://openalex.org/W1987070596","https://openalex.org/W2271879067","https://openalex.org/W2983515507","https://openalex.org/W1980985200","https://openalex.org/W2564242734","https://openalex.org/W2901838223","https://openalex.org/W2163461030","https://openalex.org/W2083459869","https://openalex.org/W2056389572","https://openalex.org/W2045904790","https://openalex.org/W1600293573","https://openalex.org/W3210111603","https://openalex.org/W2953224632","https://openalex.org/W2255223375","https://openalex.org/W1918058354","https://openalex.org/W2310381493","https://openalex.org/W2149119567","https://openalex.org/W2949155560","https://openalex.org/W1548765768","https://openalex.org/W900403624"],"abstract_inverted_index":{"Let":[0],"$D(n,r)$":[1],"be":[2],"a":[3,66],"random":[4,47,68],"$r$-out":[5],"regular":[6],"directed":[7],"multigraph":[8],"on":[9,70],"the":[10,62,76,83,86,90,108,117,130],"set":[11],"of":[12,46,65,79,89,119],"vertices":[13,109,124],"$\\{1,\\ldots,n\\}$.":[14],"In":[15,72],"this":[16],"work,":[17],"we":[18,74],"establish":[19],"that":[20,30,95,107],"for":[21,128],"every":[22],"$r":[23],"\\ge":[24],"2$,":[25],"there":[26],"exists":[27],"$\\eta_r>0$":[28],"such":[29,123],"$\\mathrm{diam}(D(n,r))=(1+\\eta_r+o(1))\\log_r{n}$.":[31],"The":[32],"constant":[33],"$\\eta_r$":[34],"is":[35],"related":[36,60],"to":[37,55,61,113,135],"branching":[38],"processes":[39],"and":[40,81,85,102],"also":[41,52,126],"appears":[42],"in":[43],"other":[44],"models":[45],"undirected":[48],"graphs.":[49],"Our":[50,104],"techniques":[51],"allow":[53],"us":[54],"bound":[56],"some":[57],"extremal":[58],"quantities":[59],"stationary":[63,91],"distribution":[64],"simple":[67],"walk":[69],"$D(n,r)$.":[71],"particular,":[73],"determine":[75],"asymptotic":[77],"behaviour":[78],"$\\pi_{\\max}$":[80],"$\\pi_{\\min}$,":[82],"maximum":[84],"minimum":[87],"values":[88],"distribution.":[92],"We":[93],"show":[94],"with":[96,110],"high":[97],"probability":[98],"$\\pi_{\\max}":[99],"=":[100],"n^{-1+o(1)}$":[101],"$\\pi_{\\min}=n^{-(1+\\eta_r)+o(1)}$.":[103],"proof":[105],"shows":[106],"$\\pi(v)$":[111],"near":[112],"$\\pi_{\\min}$":[114],"lie":[115],"at":[116],"top":[118],"\"narrow,":[120],"slippery":[121],"tower\";":[122],"are":[125],"responsible":[127],"increasing":[129],"diameter":[131],"from":[132],"$(1+o(1))\\log_r":[133],"n$":[134],"$(1+\\eta_r+o(1))\\log_r{n}$.":[136]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-04-16T08:26:57.006410","created_date":"2025-10-10T00:00:00"}