Connectivity of Growing Random Networks

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∼k^γ, different behaviors arise for γ<1, γ>1, and γ = 1. For γ<1, the number of sites with k links, N_k, varies as a stretched exponential. For γ>1, a single site connects to nearly all other sites. In the borderline case A_k∼k, the power law N_k∼k^-ν is found, where the exponent ν can be tuned to any value in the range 2<ν<∞.