1.0Microsoft Researchhttps://www.microsoft.com/en-us/researchChristian Borgshttps://www.microsoft.com/en-us/research/people/borgs/rich600338<blockquote class="wp-embedded-content" data-secret="V5va6z9Gsn"><a href="https://www.microsoft.com/en-us/research/publication/percolation-dense-graph-sequences/">Percolation on Dense Graph Sequences</a></blockquote><iframe sandbox="allow-scripts" security="restricted" src="https://www.microsoft.com/en-us/research/publication/percolation-dense-graph-sequences/embed/#?secret=V5va6z9Gsn" width="600" height="338" title="“Percolation on Dense Graph Sequences” — Microsoft Research" data-secret="V5va6z9Gsn" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" class="wp-embedded-content"></iframe><script type="text/javascript"> /* <![CDATA[ */ /*! This file is auto-generated */ !function(d,l){"use strict";l.querySelector&&d.addEventListener&&"undefined"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!/[^a-zA-Z0-9]/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret="'+t.secret+'"]'),o=l.querySelectorAll('blockquote[data-secret="'+t.secret+'"]'),c=new RegExp("^https?:$","i"),i=0;i<o.length;i++)o[i].style.display="none";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute("style"),"height"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):"link"===t.message&&(r=new URL(s.getAttribute("src")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener("message",d.wp.receiveEmbedMessage,!1),l.addEventListener("DOMContentLoaded",function(){for(var e,t,s=l.querySelectorAll("iframe.wp-embedded-content"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute("data-secret"))||(t=Math.random().toString(36).substring(2,12),e.src+="#?secret="+t,e.setAttribute("data-secret",t)),e.contentWindow.postMessage({message:"ready",secret:t},"*")},!1)))}(window,document); //# sourceURL=https://www.microsoft.com/en-us/research/wp-includes/js/wp-embed.min.js /* ]]> */ </script> In this paper, we determine the percolation threshold for an arbitrary sequence of dense graphs (Gn). Let ¸n be the largest eigenvalue of the adjacency matrix of Gn, and let Gn(pn) be the random subgraph of Gn that is obtained by keeping each edge independently with probability pn. We show that the appearance of a […]