  The theme in this paper is a composition of random graphs and P\'olya urns.
The random graphs are generated through a small structure called the seed. Via
P\'olya urns, we study the asymptotic degree structure in a random $m$-ary
hooking network and identify strong laws. We further upgrade the result to
second-order asymptotics in the form of multivariate Gaussian limit laws. We
give a few concrete examples and explore some properties with a full
representation of the Gaussian limit in each case. The asymptotic covariance
matrix associated with the P\'olya urn is obtained by a new method that
originated in this paper and is reported in [25].
