Phase transitions in an aging network

Abstract

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