Anomalous percolating properties of growing networks

We describe the anomalous phase transition of the emergence of the giant connected component in scale-free networks growing under mechanism of preferential linking. We obtain exact results for the size of a giant connected component and the distribution of vertices among connected components. We show that all the derivatives of the giant connected component size $S$ over the rate $b$ of the emergence of new edges are zero at the percolation threshold $b_c$, and $S \propto \exp\{-d(\gamma)(b-b_c)^{-1/2}\}$, where the coefficient $d$ is a function of the degree distribution exponent $\gamma$. In the entire phase without the giant component, these networks are in a ``critical state'': the probability ${\cal P}(k)$ that a vertex belongs to a connected component of a size $k$ is of a power-law form. At the phase transition point, ${\cal P}(k) \sim 1/(k\ln k)^2$. In the phase with giant component, ${\cal P}(k)$ has an exponential cutoff at $k_c \propto 1/S$. In the simplest particular case, we present exact results for growing exponential networks.