Authors: Kiran R. Bhutani ¹; Ravi Kalpathy ¹; Hosam Mahmoud ²

• 1 Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, U.S.A.
• 2 Department of Statistics, The George Washington University, Washington, D.C. 20052, U.S.A.

The theme in this paper is a composition of random graphs and Pólya urns. The random graphs are generated through a small structure called the seed. Via Pólya 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 Pólya urn is obtained by a new method that originated in this paper and is reported in [25].