{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T15:18:38Z","timestamp":1777562318363,"version":"3.51.4"},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2011,1,27]],"date-time":"2011-01-27T00:00:00Z","timestamp":1296086400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2011,5]]},"abstract":"<jats:p>Let <jats:italic><jats:bold>d<\/jats:bold><\/jats:italic> = (<jats:italic>d<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>d<\/jats:italic><jats:sub>2<\/jats:sub>, .\u00a0.\u00a0., <jats:italic>d<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence <jats:italic><jats:bold>d<\/jats:bold><\/jats:italic>, including the probability of a given subgraph or induced subgraph.<\/jats:p><jats:p>Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of <jats:italic>n<\/jats:italic>.<\/jats:p><jats:p>Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory developed for bipartite graphs by Greenhill and McKay (2009).<\/jats:p>","DOI":"10.1017\/s0963548311000034","type":"journal-article","created":{"date-parts":[[2011,1,27]],"date-time":"2011-01-27T13:08:13Z","timestamp":1296133693000},"page":"413-433","source":"Crossref","is-referenced-by-count":21,"title":["Subgraphs of Dense Random Graphs with Specified Degrees"],"prefix":"10.1017","volume":"20","author":[{"given":"BRENDAN D.","family":"McKAY","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2011,1,27]]},"reference":[{"key":"S0963548311000034_ref14","first-page":"213","article-title":"Subgraphs of random graphs with specified degrees","volume":"33","author":"McKay","year":"1981","journal-title":"Congr. 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Alg., published online, doi: 10.1002\/rsa.20324.","DOI":"10.1002\/rsa.20324"},{"key":"S0963548311000034_ref12","first-page":"346","article-title":"Random regular graphs of high degree","volume":"18","author":"Krivelevich","year":"2001","journal-title":"Europ. J. Combin."},{"key":"S0963548311000034_ref5","unstructured":"[5] Chatterjee S. , Diaconis P. and Sly A. (2010) Random graphs with a given degree sequence. Preprint, available at arxiv.org\/abs\/1005.1136."},{"key":"S0963548311000034_ref19","doi-asserted-by":"publisher","DOI":"10.1016\/S0195-6698(13)80042-X"},{"key":"S0963548311000034_ref15","first-page":"15","article-title":"Asymptotics for symmetric 0\u20131 matrices with prescribed row sums","volume":"19A","author":"McKay","year":"1985","journal-title":"Ars Combin."},{"key":"S0963548311000034_ref17","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548398003642"},{"key":"S0963548311000034_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2005.03.005"},{"key":"S0963548311000034_ref1","unstructured":"[1] Barvinok A. and Hartigan J. A. (2010) The number of graphs and a random graph with a given degree sequence. Preprint, available at arxiv.org\/abs\/1003.0356."},{"key":"S0963548311000034_ref13","first-page":"139","volume-title":"Third Caribbean Conference on Combinatorics and Computing","author":"McKay","year":"1981"},{"key":"S0963548311000034_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/BF02128671"},{"key":"S0963548311000034_ref18","doi-asserted-by":"publisher","DOI":"10.1006\/jcta.1996.0003"},{"key":"S0963548311000034_ref7","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1017\/S0963548302005254","article-title":"Random regular graphs of non-constant degree: Independence and chromatic number","volume":"11","author":"Cooper","year":"2002","journal-title":"Combin. Probab. Comput."},{"key":"S0963548311000034_ref21","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511721335.010"},{"key":"S0963548311000034_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2007.03.009"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548311000034","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T08:46:58Z","timestamp":1556354818000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548311000034\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1,27]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2011,5]]}},"alternative-id":["S0963548311000034"],"URL":"https:\/\/doi.org\/10.1017\/s0963548311000034","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1,27]]}}}