  The organizational development of growing random networks is investigated.
These growing networks are built by adding nodes successively and linking each
to an earlier node of degree k with attachment probability A_k. When A_k grows
slower than linearly with k, the number of nodes with k links, N_k(t), decays
faster than a power-law in k, while for A_k growing faster than linearly in k,
a single node emerges which connects to nearly all other nodes. When A_k is
asymptotically linear, N_k(t) tk^{-nu}, with nu dependent on details of the
attachment probability, but in the range 2<nu<infty. The combined age and
degree distribution of nodes shows that old nodes typically have a large
degree. There is also a significant correlation in the degrees of neighboring
nodes, so that nodes of similar degree are more likely to be connected. The
size distributions of the in-components and out-components of the network with
respect to a given node -- namely, its "descendants" and "ancestors" -- are
also determined. The in-component exhibits a robust s^{-2} power-law tail,
where s is the component size. The out component has a typical size of order ln
t and it provides basic insights about the genealogy of the network.
