{"id":182719,"date":"2008-01-04T00:00:00","date_gmt":"2009-10-31T09:56:54","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/candidate-talk-critical-percolation-on-finite-graphs\/"},"modified":"2016-09-09T09:45:40","modified_gmt":"2016-09-09T16:45:40","slug":"candidate-talk-critical-percolation-on-finite-graphs","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/candidate-talk-critical-percolation-on-finite-graphs\/","title":{"rendered":"Candidate Talk: Critical percolation on finite graphs"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Bond percolation on a graph G with parameter p in [0,1] is the random subgraph G<sub>p<\/sub> of G obtained by independently deleting each edge with probability 1-p and retaining it with probability p. For many graphs, the size of the largest component of G<sub>p<\/sub> exhibits a phase transition: it changes sharply from logarithmic to linear as p increases. When G is the complete graph, this model is known as the Erdos-Renyi random graph: at the phase transition, i.e. p=1\/n, the largest component satisfies a power-law of order 2\/3.<\/p>\n<p>For which d-regular graphs does percolation with p=1\/(d-1) exhibit similar &#8220;mean-field&#8221; behavior? We show that this occurs for graphs where the probability of a non-backtracking random walk to return to its initial location behaves as it does on the complete graph. In particular, the celebrated Lubotzky-Phillips-Sarnak expander graphs and Cartesian products of 2 or 3 complete graphs exhibit mean-field behavior at p=1\/(d-1); surprisingly, a product of 4 complete graphs does not.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Bond percolation on a graph G with parameter p in [0,1] is the random subgraph Gp of G obtained by independently deleting each edge with probability 1-p and retaining it with probability p. For many graphs, the size of the largest component of Gp exhibits a phase transition: it changes sharply from logarithmic to linear [&hellip;]<\/p>\n","protected":false},"featured_media":194740,"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":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-182719","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/AdfaZrZ-ogk","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/182719","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\/182719\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/194740"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=182719"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=182719"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=182719"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=182719"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=182719"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=182719"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=182719"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=182719"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=182719"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=182719"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}