Large deviation analysis for layered percolation problems on the complete graph

Abstract

AbstractWe analyze the large deviation properties for the (multitype) version of percolation on the complete graph – the simplest substitutive generalization of the Erd&0151;s‐Rènyi random graph that was treated in article by Bollobás et al. (Random Structures Algorithms 31 (2007), 3–122). Here the vertices of the graph are divided into a fixed finite number of sets (called layers) the probability of {u,v} being in our edge set depends on the respective layers of u and v. We determine the exponential rate function for the probability that a giant component occupies a fixed fraction of the graph, while all other components are small. We also determine the exponential rate function for the probability that a particular exploration process on the random graph will discover a certain fraction of vertices in each layer, without encountering a giant component.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 460–492, 2012