A solution for the time- and age-dependent connectivity distribution of a g
rowing random network is presented. The network is built by adding sites th
at link to earlier sites with a probability A(k) which depends on the numbe
r of preexisting links k to that site. For homogeneous connection kernels,
A(k) similar to k(gamma), different behaviors arise for gamma < 1, <gamma>
> 1, and gamma = 1. For gamma < 1, the number of shes with k links, Nk, var
ies as a stretched exponential. For <gamma> > 1, a single site connects to
nearly all other sites. In the borderline case A(k) similar to k, the power
law N-k similar to k(-nu) is found, where the exponent nu can be tuned to
any value in the range 2 < <nu> < <infinity>.