  Preferential attachment networks with power law exponent $\tau>3$ are known
to exhibit a phase transition. There is a value $\rho_{\rm c}>0$ such that, for
small edge densities $\rho\leq \rho_c$ every component of the graph comprises
an asymptotically vanishing proportion of vertices, while for large edge
densities $\rho>\rho_c$ there is a unique giant component comprising an
asymptotically positive proportion of vertices. In this paper we study the
decay in the size of the giant component as the critical edge density is
approached from above. We show that the size decays very rapidly, like
$\exp(-c/ \sqrt{\rho-\rho_c})$ for an explicit constant $c>0$ depending on the
model implementation. This result is in contrast to the behaviour of the class
of rank-one models of scale-free networks, including the configuration model,
where the decay is polynomial. Our proofs rely on the local neighbourhood
approximations of [Dereich, Morters, 2013] and recent progress in the theory of
branching random walks [Gantert, Hu, Shi, 2011].
