{"id":4128,"date":"2014-02-17T08:49:03","date_gmt":"2014-02-17T07:49:03","guid":{"rendered":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/?p=4128"},"modified":"2014-02-17T08:52:20","modified_gmt":"2014-02-17T07:52:20","slug":"properties-of-random-graphs-subgraph-containment","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/properties-of-random-graphs-subgraph-containment\/","title":{"rendered":"Properties of random graphs &#8211; subgraph containment"},"content":{"rendered":"<p>Se ha publicado un art\u00edculo de razonamiento formalizado en <a href=\"http:\/\/www.cl.cam.ac.uk\/research\/hvg\/Isabelle\/index.html\">Isabelle\/HOL<\/a> sobre grafos titulado <a href=\"http:\/\/afp.sourceforge.net\/browser_info\/current\/AFP\/Random_Graph_Subgraph_Threshold\/document.pdf\">Properties of random graphs &#8211; subgraph containment<\/a>.<\/p>\n<p>Sus autor es <a href=\"http:\/\/www21.in.tum.de\/~hupel\">Lars Hupel<\/a> (de la Univ. Polit\u00e9cnica de Munich).<\/p>\n<p>Su resumen es<\/p>\n<blockquote><p>\n<a href=\"http:\/\/en.wikipedia.org\/wiki\/Random_graph\">Random graphs<\/a> have been introduced by Erd\u00f6s and R\u00e9nyi in [<a href=\"http:\/\/www.renyi.hu\/~p_erdos\/1960-10.pdf\">1<\/a>]. They describe a probability space where, for a fixed number of vertices, each possible edge is present with a certain probability independent from other edges, but with the same probability for each edge. They study what properties emerge when increasing the number of vertices, or as they call it, &#8220;the evolution of such a random graph&#8221;. The theorem which we will prove here is a slightly different version from that in the first section of that paper. <\/p>\n<p>Here, we are interested in the probability that a random graph contains a certain pattern, for example a cycle or a clique. A very high edge probability gives rise to perhaps too many edges, which is usually undesired since it degrades the performance of many algorithms, whereas a low edge probability might result in a disconnected graph. The central theorem determines a threshold probability such that a higher edge probability will asymptotically almost surely produce a random graph with the desired subgraph. <\/p>\n<p>The proof is outlined in [<a href=\"http:\/\/www.esi2.us.es\/~mbilbao\/pdffiles\/DiestelGT.pdf\">2<\/a>, \u00a711.4] and [<a href=\"http:\/\/bit.ly\/1eHcpMZ\">3<\/a>, \u00a73]. The work is based on the comprehensive formalization of probability theory in Isabelle\/HOL and on a previous definition of graphs in a work by Noschinski [<a href=\"http:\/\/afp.sourceforge.net\/browser_info\/current\/AFP\/Girth_Chromatic\/document.pdf\">4<\/a>]. There, Noschinski formalized the proof that graphs with arbitrarily large girth and chromatic number exist. While the proof in this paper uses a different approach, the definition of a probability space on edges turned out to be quite useful.\n<\/p><\/blockquote>\n<p>El trabajo se ha publicado en <a href=\"http:\/\/afp.sourceforge.net\/entries\/Random_Graph_Subgraph_Threshold.shtml\">Archive of Formal Proofs<\/a><\/p>\n<p>El c\u00f3digo de las correspondientes teor\u00edas en Isabelle\/HOL se encuentra <a href=\"http:\/\/afp.sourceforge.net\/release\/afp-Random_Graph_Subgraph_Threshold-current.tar.gz\">aqu\u00ed<\/a>. <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Se ha publicado un art\u00edculo de razonamiento formalizado en Isabelle\/HOL sobre grafos titulado Properties of random graphs &#8211; subgraph containment. Sus autor es Lars Hupel (de la Univ. Polit\u00e9cnica de Munich). Su resumen es Random graphs have been introduced by Erd\u00f6s and R\u00e9nyi in [1]. They describe a probability space where, for a fixed number&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[100],"tags":[144,285],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":false,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/4128"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/comments?post=4128"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/4128\/revisions"}],"predecessor-version":[{"id":4131,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/posts\/4128\/revisions\/4131"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/media?parent=4128"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/categories?post=4128"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/vestigium\/wp-json\/wp\/v2\/tags?post=4128"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}