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The phase transition in the multi-type binomial random graph $G(\mathbf{n},P)$

We determine the asymptotic size of the largest component in the $2$-type binomial random graph $G(\mathbf{n},P)$ near criticality using a refined branching process approach. In $G(\mathbf{n},P)$ every vertex has one of two types, the vector $\mathbf{n}$ 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 $\varepsilon=\varepsilon(\mathbf{n})\to0,$ with probability $1-o(1),$ the largest component in $G(\mathbf{n},P)$ contains asymptotically $2\varepsilon \|\mathbf{n}\|_1$ vertices and all other components are of size $o(\varepsilon \|\mathbf{n}\|_1).$