{"id":"https://openalex.org/W7214727061","doi":"https://doi.org/10.48550/arxiv.2609.33968","title":"Two-Sample Testing for Inhomogeneous Random Graphs in Non-Integral $L_r$ Norms","display_name":"Two-Sample Testing for Inhomogeneous Random Graphs in Non-Integral $L_r$ Norms","publication_year":2026,"publication_date":"2026-09-27","ids":{"openalex":"https://openalex.org/W7214727061","doi":"https://doi.org/10.48550/arxiv.2609.33968"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.33968","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33968","pdf_url":"https://arxiv.org/pdf/2609.33968","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/2609.33968","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153446974","display_name":"Soham Dan","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Dan, Soham","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5153446974"],"corresponding_institution_ids":[],"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":0.6772000193595886,"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":0.6772000193595886,"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/T10136","display_name":"Statistical Methods and Inference","score":0.15719999372959137,"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":0.052000001072883606,"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/two-sample-test","display_name":"two-sample test","score":0.8999999761581421},{"id":"https://openalex.org/keywords/random-graphs","display_name":"random graphs","score":0.6959999799728394},{"id":"https://openalex.org/keywords/sample-complexity","display_name":"sample complexity","score":0.7239999771118164},{"id":"https://openalex.org/keywords/minimax-lower-bounds","display_name":"minimax lower bounds","score":0.7549999952316284},{"id":"https://openalex.org/keywords/network-inference","display_name":"network inference","score":0.777999997138977},{"id":"https://openalex.org/keywords/hypothesis-testing","display_name":"hypothesis testing","score":0.36500000953674316}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7922999858856201},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.574400007724762},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.538100004196167},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.5259000062942505},{"id":"https://openalex.org/C149728462","wikidata":"https://www.wikidata.org/wiki/Q751319","display_name":"Minimax","level":2,"score":0.5127000212669373},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4991999864578247},{"id":"https://openalex.org/C89128539","wikidata":"https://www.wikidata.org/wiki/Q1949963","display_name":"Statistic","level":2,"score":0.4634999930858612},{"id":"https://openalex.org/C2778445095","wikidata":"https://www.wikidata.org/wiki/Q18354077","display_name":"Sample complexity","level":2,"score":0.43549999594688416},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.42829999327659607},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.4147999882698059},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.4142000079154968},{"id":"https://openalex.org/C44082924","wikidata":"https://www.wikidata.org/wiki/Q1767128","display_name":"Order statistic","level":2,"score":0.4092000126838684},{"id":"https://openalex.org/C169857963","wikidata":"https://www.wikidata.org/wiki/Q1461038","display_name":"Test statistic","level":3,"score":0.3693999946117401},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.366100013256073},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.33559998869895935},{"id":"https://openalex.org/C87007009","wikidata":"https://www.wikidata.org/wiki/Q210832","display_name":"Statistical hypothesis testing","level":2,"score":0.3149000108242035},{"id":"https://openalex.org/C129848803","wikidata":"https://www.wikidata.org/wiki/Q2564360","display_name":"Sample size determination","level":2,"score":0.31290000677108765},{"id":"https://openalex.org/C141042865","wikidata":"https://www.wikidata.org/wiki/Q200125","display_name":"Expected value","level":2,"score":0.2800999879837036},{"id":"https://openalex.org/C198531522","wikidata":"https://www.wikidata.org/wiki/Q485146","display_name":"Sample (material)","level":2,"score":0.2784999907016754},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.2727999985218048},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.26570001244544983},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.26440000534057617},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.26019999384880066},{"id":"https://openalex.org/C164680029","wikidata":"https://www.wikidata.org/wiki/Q1055293","display_name":"Multiple","level":2,"score":0.257099986076355},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.2531999945640564}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.33968","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33968","pdf_url":"https://arxiv.org/pdf/2609.33968","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.2609.33968","doi":"https://doi.org/10.48550/arxiv.2609.33968","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.33968","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:2609.33968","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.33968","pdf_url":"https://arxiv.org/pdf/2609.33968","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":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214727061.pdf","grobid_xml":"https://content.openalex.org/works/W7214727061.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W1977486191","https://openalex.org/W2728580124","https://openalex.org/W2748846107","https://openalex.org/W4307701377","https://openalex.org/W4414690762","https://openalex.org/W4416252586"],"related_works":["https://openalex.org/W2748846107","https://openalex.org/W2730935900","https://openalex.org/W2614531606","https://openalex.org/W7144224320","https://openalex.org/W2969285614","https://openalex.org/W4366460283","https://openalex.org/W4392342787","https://openalex.org/W4362598589","https://openalex.org/W4402427354","https://openalex.org/W4404291506"],"abstract_inverted_index":{"Testing":[0],"whether":[1],"two":[2,90,204],"populations":[3],"of":[4,93,149,169],"networks":[5],"share":[6],"the":[7,24,29,32,37,56,66,100,120,130,144,154,167,186,192,198,203,207,210,214,227],"same":[8,101,121],"edge":[9],"probabilities":[10],"is":[11,21,41,148,195,217],"a":[12,86,182,222],"basic":[13],"problem":[14],"in":[15],"network":[16],"inference.":[17],"How":[18],"hard":[19],"it":[20],"depends":[22],"on":[23,81,99],"norm":[25,47],"used":[26],"to":[27,71],"measure":[28],"difference.":[30],"For":[31,52,188],"inhomogeneous":[33],"Erd\u0151s--R\u00e9nyi":[34],"(IER)":[35],"model,":[36],"optimal":[38],"sample":[39,122,146],"complexity":[40,147],"known":[42,57],"for":[43,49,78,140,172,190],"every":[44,141],"integer":[45],"$L_r$":[46],"and":[48,59,65,96,103,165,226],"$1\\le":[50],"r<2$.":[51],"non-integral":[53],"$r>2$,":[54],"however,":[55],"upper":[58],"lower":[60,67],"bounds":[61],"do":[62],"not":[63],"match,":[64],"bound":[68],"was":[69],"conjectured":[70,131],"be":[72],"tight.":[73],"We":[74,84,124],"study":[75],"this":[76,127,137],"gap":[77],"two-sample":[79],"testing":[80],"aligned":[82],"vertices.":[83],"propose":[85],"test":[87,128],"that":[88,116,126,139],"runs":[89],"published":[91],"statistics,":[92],"orders":[94],"$2$":[95],"$\\lceil":[97],"r\\rceil$,":[98],"data":[102],"rejects":[104],"if":[105],"either":[106],"one":[107],"rejects.":[108],"Its":[109],"thresholds":[110],"come":[111],"from":[112],"H\u00f6lder":[113],"interpolation,":[114],"so":[115],"both":[117],"statistics":[118,205],"have":[119],"cost.":[123],"prove":[125],"attains":[129],"rate.":[132],"Combined":[133],"with":[134,157,161,179,185],"earlier":[135],"results,":[136],"shows":[138],"fixed":[142],"$r\\ge1$":[143],"minimax":[145],"order":[150],"$n^{\\max\\{4/r-1,\\,2/r\\}}/\u03b5^2$,":[151],"even":[152],"when":[153,220,230],"separation":[155],"changes":[156],"$n$.":[158],"In":[159],"simulations":[160],"$n$":[162,180],"between":[163],"32":[164],"256,":[166],"number":[168],"graphs":[170],"needed":[171],"80\\%":[173],"power":[174],"at":[175,181],"level":[176],"$0.05$":[177],"grows":[178],"rate":[183],"consistent":[184],"theory.":[187],"$r=2.5$,":[189],"example,":[191],"fitted":[193],"exponent":[194],"$0.78$,":[196],"against":[197],"theoretical":[199],"value":[200],"$0.8$.":[201],"Interestingly,":[202],"split":[206],"work":[208],"as":[209],"interpolation":[211],"argument":[212],"suggests:":[213],"higher-order":[215],"statistic":[216,229],"more":[218],"powerful":[219],"only":[221],"few":[223],"edges":[224,232],"change,":[225],"$L_2$":[228],"many":[231],"change.":[233]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}
