{"id":"https://openalex.org/W7220975003","doi":"https://doi.org/10.48550/arxiv.2610.08711","title":"Asymptotic Null Distributions of Moran's $I$ and Assortativity in Large Networks","display_name":"Asymptotic Null Distributions of Moran's $I$ and Assortativity in Large Networks","publication_year":2026,"publication_date":"2026-10-06","ids":{"openalex":"https://openalex.org/W7220975003","doi":"https://doi.org/10.48550/arxiv.2610.08711"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.08711","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08711","pdf_url":"https://arxiv.org/pdf/2610.08711","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":"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/2610.08711","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5157957602","display_name":"Karin Ait Braham","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I43406934","display_name":"Universit\u00e9 Laval","ror":"https://ror.org/04sjchr03","country_code":"CA","type":"education","lineage":["https://openalex.org/I43406934"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Braham, Karin Ait","raw_affiliation_strings":["Department of Mathematics and Statistics , Universit\u00e9 Laval"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics , Universit\u00e9 Laval","institution_ids":["https://openalex.org/I43406934"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5030478326","display_name":"Louis\u2010Paul Rivest","orcid":"https://orcid.org/0000-0003-4351-4127","observed_orcids":["https://orcid.org/0000-0003-4351-4127"]},"institutions":[{"id":"https://openalex.org/I43406934","display_name":"Universit\u00e9 Laval","ror":"https://ror.org/04sjchr03","country_code":"CA","type":"education","lineage":["https://openalex.org/I43406934"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Rivest, Louis-Paul","raw_affiliation_strings":["Department of Mathematics and Statistics , Universit\u00e9 Laval"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics , Universit\u00e9 Laval","institution_ids":["https://openalex.org/I43406934"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023514231","display_name":"Thierry Duchesne","orcid":"https://orcid.org/0000-0001-9893-604X","observed_orcids":["https://orcid.org/0000-0001-9893-604X"]},"institutions":[{"id":"https://openalex.org/I43406934","display_name":"Universit\u00e9 Laval","ror":"https://ror.org/04sjchr03","country_code":"CA","type":"education","lineage":["https://openalex.org/I43406934"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Duchesne, Thierry","raw_affiliation_strings":["Department of Mathematics and Statistics , Universit\u00e9 Laval"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics , Universit\u00e9 Laval","institution_ids":["https://openalex.org/I43406934"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I43406934"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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/T10064","display_name":"Complex Network Analysis Techniques","score":0.9961000084877014,"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"}},"topics":[{"id":"https://openalex.org/T10064","display_name":"Complex Network Analysis Techniques","score":0.9961000084877014,"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/T11911","display_name":"Spatial and Panel Data Analysis","score":0.0006000000284984708,"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"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.0006000000284984708,"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/moran-s-i","display_name":"Moran's I","score":0.9620000123977661},{"id":"https://openalex.org/keywords/network-assortativity","display_name":"network assortativity","score":0.7749999761581421},{"id":"https://openalex.org/keywords/asymptotic-null-distribution","display_name":"asymptotic null distribution","score":0.8270000219345093},{"id":"https://openalex.org/keywords/normal-approximation","display_name":"Gaussian approximation","score":0.7279999852180481},{"id":"https://openalex.org/keywords/network-heterogeneity","display_name":"network heterogeneity","score":0.609000027179718},{"id":"https://openalex.org/keywords/degree-heterogeneity","display_name":"degree heterogeneity","score":0.9440000057220459},{"id":"https://openalex.org/keywords/dense-networks","display_name":"dense networks","score":0.6790000200271606}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7745000123977661},{"id":"https://openalex.org/C89694873","wikidata":"https://www.wikidata.org/wiki/Q4810299","display_name":"Assortativity","level":3,"score":0.765500009059906},{"id":"https://openalex.org/C65778772","wikidata":"https://www.wikidata.org/wiki/Q12345341","display_name":"Asymptotic distribution","level":3,"score":0.6826000213623047},{"id":"https://openalex.org/C169857963","wikidata":"https://www.wikidata.org/wiki/Q1461038","display_name":"Test statistic","level":3,"score":0.5509999990463257},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.527999997138977},{"id":"https://openalex.org/C203763787","wikidata":"https://www.wikidata.org/wiki/Q371029","display_name":"Null (SQL)","level":2,"score":0.5209000110626221},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.5065000057220459},{"id":"https://openalex.org/C62611344","wikidata":"https://www.wikidata.org/wiki/Q1062658","display_name":"Node (physics)","level":2,"score":0.48069998621940613},{"id":"https://openalex.org/C191988596","wikidata":"https://www.wikidata.org/wiki/Q628374","display_name":"Null hypothesis","level":2,"score":0.4528000056743622},{"id":"https://openalex.org/C87007009","wikidata":"https://www.wikidata.org/wiki/Q210832","display_name":"Statistical hypothesis testing","level":2,"score":0.44119998812675476},{"id":"https://openalex.org/C36382193","wikidata":"https://www.wikidata.org/wiki/Q7068966","display_name":"Null model","level":2,"score":0.43459999561309814},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.4275999963283539},{"id":"https://openalex.org/C89128539","wikidata":"https://www.wikidata.org/wiki/Q1949963","display_name":"Statistic","level":2,"score":0.4147000014781952},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.41029998660087585},{"id":"https://openalex.org/C120639","wikidata":"https://www.wikidata.org/wiki/Q7068958","display_name":"Null distribution","level":4,"score":0.39259999990463257},{"id":"https://openalex.org/C2776157432","wikidata":"https://www.wikidata.org/wiki/Q1375683","display_name":"Normality","level":2,"score":0.3822999894618988},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.38119998574256897},{"id":"https://openalex.org/C199845137","wikidata":"https://www.wikidata.org/wiki/Q145490","display_name":"Network topology","level":2,"score":0.3806000053882599},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.3476000130176544},{"id":"https://openalex.org/C87414783","wikidata":"https://www.wikidata.org/wiki/Q1002603","display_name":"Degree distribution","level":3,"score":0.34689998626708984},{"id":"https://openalex.org/C57945734","wikidata":"https://www.wikidata.org/wiki/Q7977941","display_name":"Weak convergence","level":3,"score":0.33889999985694885},{"id":"https://openalex.org/C136197465","wikidata":"https://www.wikidata.org/wiki/Q1729295","display_name":"Variety (cybernetics)","level":2,"score":0.3343000113964081},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.32580000162124634},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.31060001254081726},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.30640000104904175},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.29820001125335693},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.28349998593330383}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.08711","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08711","pdf_url":"https://arxiv.org/pdf/2610.08711","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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.08711","doi":"https://doi.org/10.48550/arxiv.2610.08711","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.08711","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:2610.08711","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08711","pdf_url":"https://arxiv.org/pdf/2610.08711","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":"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/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220975003.pdf","grobid_xml":"https://content.openalex.org/works/W7220975003.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W338356565","https://openalex.org/W2963768049","https://openalex.org/W4407890654","https://openalex.org/W2097625489","https://openalex.org/W6967167747","https://openalex.org/W7074211560","https://openalex.org/W7045528518","https://openalex.org/W7074658570","https://openalex.org/W6980071807","https://openalex.org/W2475131717"],"abstract_inverted_index":{"This":[0],"study":[1],"investigates":[2],"the":[3,20,32,38,41,45,54,57,67,75,90,105,112,122,129],"asymptotic":[4],"behavior":[5],"of":[6,31,40,50,56,70,97],"two":[7],"dependence":[8,72],"measures":[9,52,73],"defined":[10],"on":[11],"networks,":[12],"Moran's":[13,158],"$I$":[14,159],"statistic":[15,155],"and":[16,81,99,127],"Newman's":[17],"assortativity,":[18],"under":[19,74],"null":[21,76],"hypothesis":[22,77],"that":[23,37,104,146],"a":[24,95,152,164],"Gaussian":[25],"node":[26,166],"attribute":[27],"$Y$":[28],"is":[29,114,132],"independent":[30],"network":[33,42,58,113],"structure.":[34],"We":[35,60],"demonstrate":[36],"structure":[39],"directly":[43],"affects":[44],"convergence":[46,123],"rate":[47,124],"to":[48,94,125,150],"normality":[49,126],"these":[51,71],"as":[53],"size":[55],"increases.":[59],"further":[61,144],"establish":[62],"that,":[63],"in":[64],"some":[65],"instances,":[66],"mean":[68],"values":[69],"remain":[78],"non-negligible":[79],"asymptotically":[80],"must":[82],"therefore":[83],"be":[84,151],"explicitly":[85],"accounted":[86],"for":[87],"when":[88,111],"calculating":[89],"test":[91,154],"statistics.":[92],"Applications":[93],"variety":[96],"simulated":[98],"real":[100],"networks":[101,136],"also":[102],"reveal":[103],"normal":[106],"approximation":[107],"performs":[108],"well":[109,161],"only":[110],"not":[115,141],"strongly":[116],"heterogeneous.":[117],"Network":[118],"topology":[119],"determines":[120],"both":[121],"whether":[128],"limiting":[130],"distribution":[131],"Gaussian.":[133],"In":[134],"dense":[135],"whose":[137],"degree":[138],"heterogeneity":[139],"does":[140],"vanish,":[142],"we":[143],"show":[145],"assortativity":[147],"can":[148],"fail":[149],"valid":[153],"even":[156],"though":[157],"remains":[160],"behaved,":[162],"whereas":[163],"dominating":[165],"invalidates":[167],"both.":[168]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-10-09T00:00:00"}
