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