Phase Transitions in an Aging Network

We consider a growing network in which an incoming node gets attached to the $i^{th}$ existing node with the probability $\Pi_i \propto {k_i}^{\beta}\tau_i^{\alpha}$, where $k_{i}$ is the degree of the $i^{th}$ node and $\tau_i$ its present age. The phase diagram in the ${{\alpha}-{\beta}}$ plane is obtained. The network shows scale-free behaviour, i.e., the degree distribution $P(k) \sim k^{-\gamma}$ with $\gamma =3$ only along a line in this plane. Small world property, on the other hand, exists over a large region in the phase diagram.