{"id":188221,"date":"2012-08-09T00:00:00","date_gmt":"2012-08-10T18:52:21","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/the-phase-transition-in-random-graphs-a-simple-proof\/"},"modified":"2016-08-02T06:11:20","modified_gmt":"2016-08-02T13:11:20","slug":"the-phase-transition-in-random-graphs-a-simple-proof","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/the-phase-transition-in-random-graphs-a-simple-proof\/","title":{"rendered":"The Phase Transition in Random Graphs: A Simple Proof"},"content":{"rendered":"<div class=\"asset-content\">\n<p>The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1\/n \u2013 for any &epsilon;0 and p=(1-&epsilon;)\/n, all connected components of G(n,p) are typically of size O(log  n), while for p=(1+&epsilon;)\/n, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime p=(1+&epsilon;)\/n, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games.<\/p>\n<p>Joint work with M. Krivelelvich<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1\/n \u2013 for any &epsilon;0 and p=(1-&epsilon;)\/n, all connected components of G(n,p) are typically of size O(log n), while for p=(1+&epsilon;)\/n, with high probability there exists a connected component of size linear in n. We provide a [&hellip;]<\/p>\n","protected":false},"featured_media":197065,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[206954],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-188221","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-video-type-microsoft-research-talks","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/HtBWmw4d8tg","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/188221","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/188221\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/197065"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=188221"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=188221"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=188221"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=188221"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=188221"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=188221"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=188221"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=188221"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=188221"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=188221"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}