The phase transition in multitype binomial random graphs

We determine the asymptotic size of the largest component in the 2-type binomial random graph G(n, P) near criticality using a refined branching process approach. In G(n, P) every vertex has one of two types, the vector n describes the number of vertices of each type, and any edge {u, v} is present independently with a probability that is given by an entry of the probability matrix P according to the types of u and v. We prove that in the weakly supercritical regime, i.e., if the "distance" to the critical point of the phase transition is given by ε = ε(n) → 0, with probability 1-o(1), the largest component in G(n, P) contains asymptotically 2ε||n||1 vertices and all other components are of size o(ε||n||1).