The phase transition in the multi-type binomial random graph G(n,P)

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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 an ε=ε(n)→0, with probability 1-o(1), the largest component in G(n,P) contains asymptotically 2ε n₁ vertices and all other components are of size o(ε n₁).