{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,17]],"date-time":"2024-07-17T21:13:51Z","timestamp":1721250831065},"reference-count":34,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2012,8,28]],"date-time":"2012-08-28T00:00:00Z","timestamp":1346112000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Random Struct Algorithms"],"published-print":{"date-parts":[[2013,7]]},"abstract":"Abstract<\/jats:title>We study the critical behavior of inhomogeneous random graphs in the so\u2010called rank\u20101 case, where edges are present independently but with unequal edge occupation probabilities. The edge occupation probabilities are moderated by vertex weights<\/jats:italic>, and are such that the degree of vertex i<\/jats:italic> is close in distribution to a Poisson random variable with parameter w<\/jats:italic>i<\/jats:italic><\/jats:sub>, where w<\/jats:italic>i<\/jats:italic><\/jats:sub> denotes the weight of vertex i<\/jats:italic>. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W<\/jats:italic>. In this case, the proportion of vertices with degree k<\/jats:italic> is close to the probability that a Poisson random variable with random<\/jats:italic> parameter W<\/jats:italic> takes the value k<\/jats:italic>. We pay special attention to the power\u2010law case<\/jats:italic>, i.e., the case where\n\\documentclass{article}\\usepackage{mathrsfs}\\usepackage{amsmath, amssymb}\\pagestyle{empty}\\begin{document}\\begin{align*}{\\mathbb{P}}(W\\geq k)\\end{align*} \\end{document}<\/jats:styled-content>\nis proportional to k<\/jats:italic>\u2010(\u03c4\u20101)<\/jats:sup> for some power\u2010law exponent \u03c4 > 3, a property which is then inherited by the asymptotic degree distribution.<\/jats:p>We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W<\/jats:italic>. Indeed, when\n\\documentclass{article}\\usepackage{mathrsfs}\\usepackage{amsmath, amssymb}\\pagestyle{empty}\\begin{document}\\begin{align*}{\\mathbb{P}}(W > k) \\leq ck^{-(\\tau-1)}\\end{align*} \\end{document}<\/jats:styled-content>\nfor all k<\/jats:italic> \u2265 1 and some \u03c4 > 4 and c<\/jats:italic> > 0, the largest critical connected component in a graph of size n<\/jats:italic> is of order n<\/jats:italic>2\/3<\/jats:sup>, as it is for the critical Erd\u0151s\u2010R\u00e9nyi random graph. When, instead,\n\\documentclass{article}\\usepackage{mathrsfs}\\usepackage{amsmath, amssymb}\\pagestyle{empty}\\begin{document}\\begin{align*}{\\mathbb{P}}(W > k)=ck^{-(\\tau-1)}(1+o(1))\\end{align*} \\end{document}<\/jats:styled-content>\nfor k<\/jats:italic> large and some \u03c4\u2208(3,4) and c<\/jats:italic> > 0, the largest critical connected component is of the much smaller order n<\/jats:italic>(\u03c4\u20102)\/(\u03c4\u20101)<\/jats:sup>. \u00a9 2012 Wiley Periodicals, Inc. Random Struct. Alg., 42, 480\u2013508, 2013<\/jats:p>","DOI":"10.1002\/rsa.20450","type":"journal-article","created":{"date-parts":[[2012,8,28]],"date-time":"2012-08-28T13:03:20Z","timestamp":1346159000000},"page":"480-508","source":"Crossref","is-referenced-by-count":28,"title":["Critical behavior in inhomogeneous random graphs"],"prefix":"10.1002","volume":"42","author":[{"given":"Remco","family":"van der Hofstad","sequence":"first","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2012,8,28]]},"reference":[{"key":"e_1_2_8_2_2","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1024404421"},{"key":"e_1_2_8_3_2","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v15-817"},{"key":"e_1_2_8_4_2","article-title":"Novel scaling limits for critical inhomogeneous random graphs","volume":"1472","author":"Bhamidi S.","year":"2009","journal-title":"To appear in Annals of Applied Probability"},{"key":"e_1_2_8_5_2","doi-asserted-by":"publisher","DOI":"10.2307\/1999405"},{"key":"e_1_2_8_6_2","volume-title":"Volume 73 of Cambridge studies in advanced mathematics","author":"Bollob\u00e1s B.","year":"2001"},{"key":"e_1_2_8_7_2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20168"},{"key":"e_1_2_8_8_2","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20051"},{"key":"e_1_2_8_9_2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199910\/12)15:3\/4<368::AID-RSA9>3.0.CO;2-B"},{"key":"e_1_2_8_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-006-9168-x"},{"key":"e_1_2_8_11_2","doi-asserted-by":"publisher","DOI":"10.1007\/PL00012580"},{"key":"e_1_2_8_12_2","doi-asserted-by":"publisher","DOI":"10.1080\/15427951.2004.10129081"},{"key":"e_1_2_8_13_2","volume-title":"Complex graphs and networks, Volume 107 of CBMS regional conference series in mathematics. 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