Degrees in random $m$-ary hooking networks

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].