On nodes of small degrees and degree profile in preferential dynamic attachment circuits
Abstract
We investigate the joint distribution of nodes of small degrees and the degree profile in preferential dynamic attachment circuits. In particular, we study the joint asymptotic distribution of the number of the nodes of outdegree (terminal nodes) and outdegree in a very large circuit. The expectation and variance of the number of those two types of nodes are both asymptotically linear with respect to the age of the circuit. We show that the numbers of nodes of outdegree and asymptotically follow a two-dimensional Gaussian law via multivariate martingale methods. We also study the exact distribution of the degree of a node, as the circuit ages, via a series of P\'olya-Eggenberger urn models with "hiccups" in between. The exact expectation and variance of the degree of nodes are determined by recurrence methods. Phase transitions of these degrees are discussed briefly. This is an extension of the abstract [20].
Keywords
Cite
@article{arxiv.1610.04675,
title = {On nodes of small degrees and degree profile in preferential dynamic attachment circuits},
author = {Panpan Zhang and Hosam Mahmoud},
journal= {arXiv preprint arXiv:1610.04675},
year = {2018}
}