Connectivity of Growing Random Networks

Abstract

A solution for the time- and age-dependent connectivity distribution of a growing random network is presented. The network is built by adding sites that link to earlier sites with a probability $A_k$ which depends on the number of preexisting links $k$ to that site. For homogeneous connection kernels, $A_k\sim k^{\gamma }$, different behaviors arise for $\gamma <1\mathrm{}$, $\gamma >1\mathrm{}$, and $\gamma =1\mathrm{}$. For $\gamma <1\mathrm{}$, the number of sites with $k$ links, $N_k$, varies as a stretched exponential. For $\gamma >1\mathrm{}$, a single site connects to nearly all other sites. In the borderline case $A_k\sim k$, the power law $N_k\sim k^{-\nu }$ is found, where the exponent $\nu$ can be tuned to any value in the range $2<\mathrm{}\nu <\infty$.