  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 $o(\varepsilon
\|\mathbf{n}\|_1).$
