{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T01:25:53Z","timestamp":1773537953980,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We consider a model for complex networks that was introduced by Krioukov et al.\u00a0 In this model, $N$ points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an\u00a0 edge if they are within a certain hyperbolic distance.\u00a0 The $N$ points are distributed according to a quasi-uniform distribution, which is a distorted version of\u00a0 the uniform distribution. The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges. Namely, it exhibits a power-law degree sequence and small distances but, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering.\r\nThe model is controlled by two parameters $\\alpha$ and $\\nu$ where, roughly speaking, $\\alpha$ controls the exponent of the power-law and $\\nu$ controls the average degree. The present paper focuses on the evolution of the component structure of the random graph.\u00a0 We show that (a) for $\\alpha &gt; 1$ and $\\nu$ arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order; (b) for $\\alpha &lt; 1$ and $\\nu$ arbitrary with high probability there is a \"giant\" component\u00a0 of linear order,\u00a0 and (c) when $\\alpha=1$ then there is a non-trivial phase transition for the existence of a linear-sized component in terms of $\\nu$.\r\nA corrigendum was added to this paper 29 Dec 2018.<\/jats:p>","DOI":"10.37236\/4958","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:26:38Z","timestamp":1578669998000},"source":"Crossref","is-referenced-by-count":25,"title":["On the Largest Component of a Hyperbolic Model of Complex Networks"],"prefix":"10.37236","volume":"22","author":[{"given":"Michel","family":"Bode","sequence":"first","affiliation":[]},{"given":"Nikolaos","family":"Fountoulakis","sequence":"additional","affiliation":[]},{"given":"Tobias","family":"M\u00fcller","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,8,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i3p24\/7758","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i3p24\/7758","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:13:32Z","timestamp":1579256012000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i3p24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,8,14]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2015,7,1]]}},"URL":"https:\/\/doi.org\/10.37236\/4958","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,8,14]]},"article-number":"P3.24"}}