{"id":186391,"date":"2011-06-15T00:00:00","date_gmt":"2011-06-16T13:08:16","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/phase-transitions-for-modified-erdos-renyi-processes\/"},"modified":"2016-08-22T11:30:50","modified_gmt":"2016-08-22T18:30:50","slug":"phase-transitions-for-modified-erdos-renyi-processes","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/phase-transitions-for-modified-erdos-renyi-processes\/","title":{"rendered":"Phase Transitions for Modified Erdos-Renyi Processes"},"content":{"rendered":"<div class=\"asset-content\">\n<p>A fundamental and very well studied region of the Erdos-Renyi process is the phase transition at m near n\/2 edges in which a giant component suddenly appears.  We review the behavior in the barely subcritical and barely supercritical regimes.  We modify the process, particularly discussing a modification due to Tom Bohman and Alan Frieze in which isolated vertices are given preference.<br \/>\nWhile the position of the phase transition changes and many constants change, we show to a large extent (and conjecture the<br \/>\nrest!) that the critical exponents remain the same and that the two processes belong to the same universality class.  A key role is played by the susceptibility of the graph, a concept taken from theoretical physics with strong application to large finite graphs.  The susceptibility, asymptotically, is shown to satisfy a differential equation from which its barely subcritical behavior may be deduced.  We also discuss other natural processes which appear to have very different behavior.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A fundamental and very well studied region of the Erdos-Renyi process is the phase transition at m near n\/2 edges in which a giant component suddenly appears. We review the behavior in the barely subcritical and barely supercritical regimes. We modify the process, particularly discussing a modification due to Tom Bohman and Alan Frieze in [&hellip;]<\/p>\n","protected":false},"featured_media":196187,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"research-area":[13561],"msr-video-type":[],"msr-locale":[268875],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-186391","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/fcLJXl1R0FQ","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/186391"}],"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\/186391\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/196187"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=186391"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=186391"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=186391"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=186391"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=186391"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=186391"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}