Applying affine urn models to the global profile of hyperrecursive trees

Abstract

Inside the discipline of graph theory exists an extension known as the hypergraph. This generalization of graphs includes vertices along with hyperedges consisting of collections of two or more vertices. One well-studied application of this structure is that of the recursive tree, and we apply its framework within the context of hypergraphs to form hyperrecursive trees, an area that shows promise in network theory. However, when the global profile of these hyperrecursive trees is studied via recursive equations, its recursive nature develops a combinatorial explosion of sorts when deriving mixed moments for higher containment levels. One route to circumvent these issues is through using a special class of urn model, known as an affine urn model, which samples multiple balls at once while maintaining a structure such that the replacement criteria is based on a linear combination of the balls sampled within a draw. We investigate the hyperrecursive tree through its global containment profile, observing the number of vertices found within a particular containment level, and given a set of $k$ containment levels, relate its structure to that of a similar affine urn model in order to derive the asymptotic evolution of its first two mixed moments. We then establish a multivariate central limit theorem for the number of vertices for the first $k$ containment levels. We produce simulations which support our theoretical findings and suggest a relatively quick rate of convergence.