{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T14:31:05Z","timestamp":1649082665959},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2012,10,3]],"date-time":"2012-10-03T00:00:00Z","timestamp":1349222400000},"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":[[2013,1]]},"abstract":"<jats:p>The study of the phase transition of random graph processes, and recently in particular Achlioptas processes, has attracted much attention. Achlioptas, D'Souza and Spencer (<jats:italic>Science<\/jats:italic>, 2009) gave strong numerical evidence that a variety of edge-selection rules in Achlioptas processes exhibit a discontinuous phase transition. However, Riordan and Warnke (<jats:italic>Science<\/jats:italic>, 2011) recently showed that all these processes have a continuous phase transition.<\/jats:p><jats:p>In this work we prove discontinuous phase transitions for three random graph processes: all three start with the empty graph on <jats:italic>n<\/jats:italic> vertices and, depending on the process, we connect in every step (i) one vertex chosen randomly from all vertices and one chosen randomly from a restricted set of vertices, (ii) two components chosen randomly from the set of all components, or (iii) a randomly chosen vertex and a randomly chosen component.<\/jats:p>","DOI":"10.1017\/s0963548312000442","type":"journal-article","created":{"date-parts":[[2012,10,3]],"date-time":"2012-10-03T12:05:23Z","timestamp":1349265923000},"page":"133-145","source":"Crossref","is-referenced-by-count":5,"title":["Explosive Percolation in Erd\u0151s\u2013R\u00e9nyi-Like Random Graph Processes"],"prefix":"10.1017","volume":"22","author":[{"given":"KONSTANTINOS","family":"PANAGIOTOU","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"RETO","family":"SP\u00d6HEL","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ANGELIKA","family":"STEGER","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"HENNING","family":"THOMAS","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2012,10,3]]},"reference":[{"key":"S0963548312000442_ref2","doi-asserted-by":"publisher","DOI":"10.2307\/3318611"},{"key":"S0963548312000442_ref3","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.1019"},{"key":"S0963548312000442_ref6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.105.255701"},{"key":"S0963548312000442_ref9","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581274"},{"key":"S0963548312000442_ref17","doi-asserted-by":"publisher","DOI":"10.1007\/s00493-007-2163-2"},{"key":"S0963548312000442_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02186868"},{"key":"S0963548312000442_ref11","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.103.255701"},{"key":"S0963548312000442_ref10","first-page":"17","article-title":"On the evolution of random graphs","volume":"5","author":"Erd\u0151s","year":"1960","journal-title":"Magyar Tud. 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