  We examine the global organization of growing networks in which a new vertex
is attached to already existing ones with a probability depending on their age.
We find that the network is infinite- or finite-dimensional depending on
whether the attachment probability decays slower or faster than $(age)^{-1}$.
The network becomes one-dimensional when the attachment probability decays
faster than $(age)^{-2}$. We describe structural characteristics of these
phases and transitions between them.
