Evolution of networks with aging of sites

Abstract

We study the growth of a network with aging of sites. Each new site of the network is connected to some old site with probability proportional (i) to the connectivity of the old site as in the Barab\'asi-Albert's model and (ii) to ${\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\alpha}},$ where $\ensuremath{\tau}$ is the age of the old site. We find both from simulation and analytically that the network shows scaling behavior only in the region $\ensuremath{\alpha}<1.$ When $\ensuremath{\alpha}$ increases from $\ensuremath{-}\ensuremath{\infty}$ to $0,$ the exponent $\ensuremath{\gamma}$ of the distribution of connectivities $[P(k)\ensuremath{\propto}{k}^{\ensuremath{-}\ensuremath{\gamma}}$ for large $k]$ grows from $2$ to the value for the network without aging. The ensuing increase of $\ensuremath{\alpha}$ to $1$ causes $\ensuremath{\gamma}$ to grow to $\ensuremath{\infty}.$ For $\ensuremath{\alpha}>1,$ the distribution $P(k)$ is exponentional.