Some Large Deviation Results for Sparse Random Graphs

Download or read book Some Large Deviation Results for Sparse Random Graphs written by Neil O'Connell and published by . This book was released on 1996 with total page 15 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: "We obtain a large deviation principle (LDP) for the relative size of the largest connected component in a random graph with small edge probability. The rate function, which is not convex in general, is determined explicitly using a new technique. As a corollary we present an asymptotic formula for the probability that the random graph is connected. We also present an LDP and related result for the number of isolated vertices. Here we make use of a simple but apparently unknown characterisation, wheich is obtained by embedding the random graph in a random directed graph. The results demonstrate that, at this scaling, the properties 'connected' and 'contains no isolated vertices' are not asymptotically equivalent. (At the threshold probability they are asymptotically equivalent.)."