We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1\u00a1p. Let V be the number of vertices in G, and let \u00ad be their degree. We define the critical threshold pc = pc(G; \u00b8) to be the value of p for which the expected cluster size of a fixed vertex attains the value \u00b8V 1=3, where \u00b8 is fixed and positive. We show that for any such model, there is a phase transition at pc analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold pc. In particular, we show that the largest cluster inside a scaling window of size jp\u00a1pcj = \u00a3(\u00ad\u00a11V \u00a11=3) is of size \u00a3(V 2=3), while below this scaling window, it is much smaller, of order O(\u00b2\u00a12 log(V \u00b23)), with \u00b2 = \u00ad(pc\u00a1p). We also obtain an upper bound O(\u00ad(p \u00a1 pc)V ) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order \u00a3(\u00ad(p\u00a1pc)). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the n-cube and certain Hamming cubes, as well as the spread-out n-dimensional torus for n > 6.<\/p>\n","protected":false},"excerpt":{"rendered":"
We study random subgraphs of an arbitrary finite connected transitive graph G obtained by independently deleting edges with probability 1\u00a1p. Let V be the number of vertices in G, and let \u00ad be their degree. We define the critical threshold pc = pc(G; \u00b8) to be the value of p for which the expected cluster […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561,13546],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-317936","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"","msr_edition":"Random Structures and Algorithms 27","msr_affiliation":"","msr_published_date":"2004-09-14","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"137-184","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"459579","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"Random Subgraphs of Finite Graphs: I. 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