Phase Transitions in an Aging Network

Abstract

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