{"id":"https://openalex.org/W7167885481","doi":"https://doi.org/10.48550/arxiv.2607.08696","title":"Spectrum of Directed Inhomogeneous Random Graphs","display_name":"Spectrum of Directed Inhomogeneous Random Graphs","publication_year":2026,"publication_date":"2026-07-09","ids":{"openalex":"https://openalex.org/W7167885481","doi":"https://doi.org/10.48550/arxiv.2607.08696"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2607.08696","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.08696","pdf_url":"https://arxiv.org/pdf/2607.08696","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","license_id":"https://openalex.org/licenses/cc-by","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/2607.08696","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5086330568","display_name":"Rajat Subhra Hazra","orcid":"https://orcid.org/0000-0002-9254-4763","observed_orcids":["https://orcid.org/0000-0002-9254-4763"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hazra, Rajat Subhra","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5094008696","display_name":"Giacomo Passuello","orcid":"https://orcid.org/0009-0000-4178-7055","observed_orcids":["https://orcid.org/0009-0000-4178-7055"]},"institutions":[{"id":"https://openalex.org/I121797337","display_name":"Leiden University","ror":"https://ror.org/027bh9e22","country_code":"NL","type":"education","lineage":["https://openalex.org/I121797337"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Passuello, Giacomo","raw_affiliation_strings":["MATHEMATICAL INSTITUTE , LEIDEN UNIVERSITY , GORLAEUS GEBOUW , BW-VLEUGEL , EINSTEINWEG 55 , 2333 CC LEI- DEN , THE NETHERLANDS"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"MATHEMATICAL INSTITUTE , LEIDEN UNIVERSITY , GORLAEUS GEBOUW , BW-VLEUGEL , EINSTEINWEG 55 , 2333 CC LEI- DEN , THE NETHERLANDS","institution_ids":["https://openalex.org/I121797337"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"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/T11716","display_name":"Random Matrices and Applications","score":0.9980000257492065,"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/T11716","display_name":"Random Matrices and Applications","score":0.9980000257492065,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.0005000000237487257,"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":0.0003000000142492354,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/adjacency-spectrum","display_name":"adjacency spectrum","score":0.9070000052452087},{"id":"https://openalex.org/keywords/circular-law","display_name":"circular law","score":0.847000002861023},{"id":"https://openalex.org/keywords/chung-lu-model","display_name":"Chung-Lu model","score":0.8849999904632568},{"id":"https://openalex.org/keywords/singular-value-distribution","display_name":"singular value distribution","score":0.859000027179718},{"id":"https://openalex.org/keywords/free-probability","display_name":"free probability","score":0.9259999990463257},{"id":"https://openalex.org/keywords/gaussian-fluctuations","display_name":"Gaussian fluctuations","score":0.6150000095367432},{"id":"https://openalex.org/keywords/rank-one-perturbations","display_name":"rank-one perturbation","score":0.5979999899864197}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7993999719619751},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.6237999796867371},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.5895000100135803},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5720999836921692},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5582000017166138},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.5536999702453613},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.49880000948905945},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.44040000438690186},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.41350001096725464},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4124999940395355},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.4068000018596649},{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.39649999141693115},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.36959999799728394},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36899998784065247},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.3612000048160553},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.34599998593330383},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.3294000029563904},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.32359999418258667},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.32339999079704285},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.31119999289512634},{"id":"https://openalex.org/C168110828","wikidata":"https://www.wikidata.org/wiki/Q1331626","display_name":"Spectral density","level":2,"score":0.305400013923645},{"id":"https://openalex.org/C18653775","wikidata":"https://www.wikidata.org/wiki/Q1333358","display_name":"Joint probability distribution","level":2,"score":0.2996000051498413},{"id":"https://openalex.org/C95707001","wikidata":"https://www.wikidata.org/wiki/Q5121666","display_name":"Circular law","level":5,"score":0.29910001158714294},{"id":"https://openalex.org/C74003402","wikidata":"https://www.wikidata.org/wiki/Q3180727","display_name":"Spectral graph theory","level":5,"score":0.2948000133037567},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.28999999165534973},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.2824999988079071},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.2815000116825104},{"id":"https://openalex.org/C78913703","wikidata":"https://www.wikidata.org/wiki/Q5597087","display_name":"Graph energy","level":5,"score":0.27959999442100525},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.26820001006126404},{"id":"https://openalex.org/C130402806","wikidata":"https://www.wikidata.org/wiki/Q5361768","display_name":"Random field","level":2,"score":0.2653000056743622},{"id":"https://openalex.org/C98306654","wikidata":"https://www.wikidata.org/wiki/Q2409122","display_name":"Spectral theory","level":3,"score":0.26330000162124634},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.26190000772476196},{"id":"https://openalex.org/C13929819","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Random function","level":3,"score":0.25609999895095825},{"id":"https://openalex.org/C2779982251","wikidata":"https://www.wikidata.org/wiki/Q25053762","display_name":"Stochastic block model","level":3,"score":0.2554999887943268},{"id":"https://openalex.org/C53622274","wikidata":"https://www.wikidata.org/wiki/Q2496679","display_name":"Correlation function (quantum field theory)","level":3,"score":0.25369998812675476},{"id":"https://openalex.org/C33962027","wikidata":"https://www.wikidata.org/wiki/Q1930697","display_name":"Wishart distribution","level":3,"score":0.2515000104904175}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2607.08696","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.08696","pdf_url":"https://arxiv.org/pdf/2607.08696","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2607.08696","doi":"https://doi.org/10.48550/arxiv.2607.08696","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.08696","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2607.08696","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.08696","pdf_url":"https://arxiv.org/pdf/2607.08696","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7167885481.pdf","grobid_xml":"https://content.openalex.org/works/W7167885481.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W2735601436","https://openalex.org/W2958819415","https://openalex.org/W2969415100","https://openalex.org/W4387994855","https://openalex.org/W4405193659","https://openalex.org/W4417098830","https://openalex.org/W7124458492","https://openalex.org/W7165791210"],"related_works":["https://openalex.org/W2991293040","https://openalex.org/W4405070858","https://openalex.org/W3208294655","https://openalex.org/W3088059916","https://openalex.org/W2619587099","https://openalex.org/W2264907590","https://openalex.org/W4404569647","https://openalex.org/W4319067399","https://openalex.org/W2783326523","https://openalex.org/W2980519147"],"abstract_inverted_index":{"We":[0,17],"study":[1],"the":[2,5,36,49,62,71,89,109,116,127],"spectrum":[3],"of":[4,9,48,70,129],"adjacency":[6],"matrix":[7],"$A_n$":[8,20],"directed":[10,37,42,102,136],"inhomogeneous":[11,137],"random":[12,39,138],"graphs":[13],"on":[14],"$n$":[15],"vertices.":[16],"assume":[18],"that":[19,53,61,106],"has":[21],"independent":[22],"entries":[23],"and":[24,41,52,132],"diverging":[25],"average":[26],"degree":[27],"scale":[28,117],"$s_n$.":[29],"This":[30],"framework":[31],"includes,":[32],"as":[33],"special":[34],"cases,":[35],"Chung--Lu":[38,64],"graph":[40],"stochastic":[43],"block":[44],"models.":[45],"Assuming":[46],"boundedness":[47],"variance":[50],"profile":[51],"$s_n$":[54],"diverges":[55],"faster":[56],"than":[57],"$\\log(n)^2$,":[58],"we":[59,87,104],"show":[60],"rank-one":[63],"model":[65],"satisfies":[66],"a":[67],"non-homogeneous":[68],"version":[69],"circular":[72],"law,":[73],"which":[74],"in":[75],"some":[76],"situations":[77],"allows":[78],"for":[79,100],"an":[80,120],"explicit":[81,121],"expression.":[82],"Moreover,":[83],"under":[84],"mild":[85],"conditions,":[86],"identify":[88],"asymptotic":[90],"singular":[91],"value":[92],"distribution":[93],"using":[94],"tools":[95],"from":[96],"free":[97],"probability.":[98],"Finally,":[99],"finite-rank":[101],"models,":[103],"establish":[105],"eigenvalues":[107],"outside":[108],"bulk":[110],"have":[111],"joint":[112],"Gaussian":[113],"fluctuations":[114,134],"at":[115],"$\\sqrt{s_n/n}$,":[118],"with":[119],"covariance":[122],"matrix.":[123],"These":[124],"results":[125],"extend":[126],"theory":[128],"spectral":[130],"outliers":[131],"their":[133],"to":[135],"graphs.":[139]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-07-11T00:00:00"}
