Slow emergence of the giant component in the growing m-out graph

Abstract

Let H m(n) be a random graph on n vertices, grown by adding vertices one at a time, joining each new vertex to a uniformly chosen set of m earlier vertices. If edges of H m(n) are deleted independently, each being retained with probability p, then there is a “phase transition”. There is a certain critical value pc of p such that, with high probability, a component of order T(n) remains as n → ∞ if and only if p > pc. Among other results, we obtain the exact value of pc, which depends on m in a nontrivial way, and show that the phase transition has “infinite order”; in fact, for p = pc + e, the largest component has order exp(-T(1/$\sqrt{\varepsilon}$))n with high probability. Analogous results were proved recently in by Bollobas, Janson, and Riordan [Random Structures Algorithms 26 (2005), 1–36] for a related model in which edges are present independently. The model we study is considerably more difficult to analyze, since the dependence between the edges is very important, affecting the value of pc, so many new complications arise. In overcoming these complications we make use of the techniques developed by the authors [Internet Math 1 (2003), 1–35] to analyze a very different model. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005