Random Recursive Constructions: Asymptotic Geometric and Topological Properties

Asymptotic Properties of Protected Nodes in Random Recursive Trees

We investigate protected nodes in random recursive trees. The exact mean of the number of such nodes is obtained by recurrence, and a linear asymptotic equivalent follows. A nonlinear recurrence for the variance shows that the variance grows linearly, too. It follows that the number of protected nodes in a random recursive tree, upon proper scaling, converges in probability to a constant.

Topological properties of the one dimensional exponential random geometric graph

Topological Properties of an Exponential Random Geometric Graph Process

In this paper we consider a one-dimensional random geometric graph process with the inter-nodal gaps evolving according to an exponential AR(1) process. The transition probability matrix and stationary distribution are derived for the Markov chains concerning connectivity and the number of components. We analyze the algorithm for hitting time regarding disconnectivity. In addition to dynamical ...

Geometric Constructions with Discretized Random Variables

Algorithms computing with discretized random variables have been presented in a number of publications Williamson and Downs [1990], Berleant and Goodman-Strauss [1998], Berleant et al. [2003], Berleant and Zhang [2004b,a], Regan et al. [2004]. However, the algebra of imprecise numbers in its various senses often is a poor model for geometry with imprecisely defined objects. A vector of imprecis...

Asymptotic degree distribution in random recursive trees

The distributions of vertex degrees in random recursive trees and random plane recursive trees are shown to be asymptotically normal. Formulas are given for the asymptotic variances and covariances of the number of vertices with given outdegrees. We also give functional limit theorems for the evolution as vertices are added. The proofs use some old and new results about generalized Pólya urn mo...