Minimum stationary values of sparse random directed graphs

We consider the stationary distribution of the simple random walk on the directed configuration model with bounded degrees. Provided that the minimum out-degree is at least 2, with high probability (whp) there is a unique stationary distribution (uniqueness regime). We show that the minimum positive...

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Autores: Cai, Xing Shi, Perarnau Llobet, Guillem|||0000-0002-1953-9511
Formato: artículo
Fecha de publicación:2026
País:España
Recursos:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/459039
Acesso em linha:https://hdl.handle.net/2117/459039
https://dx.doi.org/10.1214/25-AAP2247
Access Level:acceso abierto
Palavra-chave:Graph theory
Directed configuration model
Random walks
Stationary distribution
Grafs, Teoria de
Classificació AMS::05 Combinatorics::05C Graph theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
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repository_id_str
spelling Minimum stationary values of sparse random directed graphsCai, Xing ShiPerarnau Llobet, Guillem|||0000-0002-1953-9511Graph theoryDirected configuration modelRandom walksStationary distributionGrafs, Teoria deClassificació AMS::05 Combinatorics::05C Graph theoryÀrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafsWe consider the stationary distribution of the simple random walk on the directed configuration model with bounded degrees. Provided that the minimum out-degree is at least 2, with high probability (whp) there is a unique stationary distribution (uniqueness regime). We show that the minimum positive stationary value is whp n-(1+C+o(1)) for some constant C=0 determined by the degree distribution, answering a question raised by Bordenave, Caputo and Salez (Probab. Theory Related Fields 170 (2018) 933–960). In particular, C is the competing combination of two factors: (1) the contribution of atypically “thin” in-neighbourhoods, controlled by subcritical branching processes; and (2) the contribution of atypically “light” trajectories, controlled by large deviation rate functions. Additionally, we give estimates for the expected lower tail of the empirical stationary distribution. As a by-product of our proof, we obtain that the hitting and the cover time are both n1+C+o(1) whp. Our results are in sharp contrast to those of Caputo and Quattropani (Probab. Theory Related Fields 178 (2020) 1011–1066) who showed that under the additional condition of minimum in-degree at least 2 (ergodicity regime), stationary values only have logarithmic fluctuations around n-1.We would like to thanks Pietro Caputo and Matteo Quattropani for insightful discussions on the topic. We are also grateful to the three anonymous reviewers, whose insightful comments have greatly contributed to improving this paper. G.P. was supported by the grants PID2020-113082GB-I00, PID2023-147202NB-I00, PCI2024-155080-2 and the Programme Severo Ochoa y María de Maeztu por Centros y Unidades de Excelencia en I&D (CEX2020-001084-M), all of them funded by MICIU/AEI/10.13039/501100011033.Peer Reviewed20262026-02-0120262026-03-19journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/459039https://dx.doi.org/10.1214/25-AAP2247reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengAgencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PID2020-113082GB-I00 COMBINATORIA: NUEVAS TENDENCIAS Y APLICACIONESAgencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2023-147202NB-I00 COMBINATORIA CONTEMPORANEA Y SUS APLICACIONESAgencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2024-2027 PCI2024-155080-2 STRUCTURE, RANDOMNESS AND COMPUTATIONAL METHODS IN EXTREMAL COMBINATORICSopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/4590392026-05-27T15:37:01Z
dc.title.none.fl_str_mv Minimum stationary values of sparse random directed graphs
title Minimum stationary values of sparse random directed graphs
spellingShingle Minimum stationary values of sparse random directed graphs
Cai, Xing Shi
Graph theory
Directed configuration model
Random walks
Stationary distribution
Grafs, Teoria de
Classificació AMS::05 Combinatorics::05C Graph theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
title_short Minimum stationary values of sparse random directed graphs
title_full Minimum stationary values of sparse random directed graphs
title_fullStr Minimum stationary values of sparse random directed graphs
title_full_unstemmed Minimum stationary values of sparse random directed graphs
title_sort Minimum stationary values of sparse random directed graphs
dc.creator.none.fl_str_mv Cai, Xing Shi
Perarnau Llobet, Guillem|||0000-0002-1953-9511
author Cai, Xing Shi
author_facet Cai, Xing Shi
Perarnau Llobet, Guillem|||0000-0002-1953-9511
author_role author
author2 Perarnau Llobet, Guillem|||0000-0002-1953-9511
author2_role author
dc.subject.none.fl_str_mv Graph theory
Directed configuration model
Random walks
Stationary distribution
Grafs, Teoria de
Classificació AMS::05 Combinatorics::05C Graph theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
topic Graph theory
Directed configuration model
Random walks
Stationary distribution
Grafs, Teoria de
Classificació AMS::05 Combinatorics::05C Graph theory
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs
description We consider the stationary distribution of the simple random walk on the directed configuration model with bounded degrees. Provided that the minimum out-degree is at least 2, with high probability (whp) there is a unique stationary distribution (uniqueness regime). We show that the minimum positive stationary value is whp n-(1+C+o(1)) for some constant C=0 determined by the degree distribution, answering a question raised by Bordenave, Caputo and Salez (Probab. Theory Related Fields 170 (2018) 933–960). In particular, C is the competing combination of two factors: (1) the contribution of atypically “thin” in-neighbourhoods, controlled by subcritical branching processes; and (2) the contribution of atypically “light” trajectories, controlled by large deviation rate functions. Additionally, we give estimates for the expected lower tail of the empirical stationary distribution. As a by-product of our proof, we obtain that the hitting and the cover time are both n1+C+o(1) whp. Our results are in sharp contrast to those of Caputo and Quattropani (Probab. Theory Related Fields 178 (2020) 1011–1066) who showed that under the additional condition of minimum in-degree at least 2 (ergodicity regime), stationary values only have logarithmic fluctuations around n-1.
publishDate 2026
dc.date.none.fl_str_mv 2026
2026-02-01
2026
2026-03-19
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/459039
https://dx.doi.org/10.1214/25-AAP2247
url https://hdl.handle.net/2117/459039
https://dx.doi.org/10.1214/25-AAP2247
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PID2020-113082GB-I00 COMBINATORIA: NUEVAS TENDENCIAS Y APLICACIONES
Agencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2023-147202NB-I00 COMBINATORIA CONTEMPORANEA Y SUS APLICACIONES
Agencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2024-2027 PCI2024-155080-2 STRUCTURE, RANDOMNESS AND COMPUTATIONAL METHODS IN EXTREMAL COMBINATORICS
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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