Transition from small to large world in growing networks

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.