On nodes of small degrees and degree profile in preferential dynamic attachment circuits

TL;DR

This paper analyzes the distribution of small-degree nodes and degree profiles in large preferential dynamic attachment circuits, revealing Gaussian asymptotics and exact degree distributions through advanced probabilistic models.

Contribution

It introduces a detailed analysis of small-degree node distributions and degree profiles in preferential attachment circuits, extending previous work with new probabilistic methods.

Findings

Number of terminal and degree-1 nodes grows linearly with circuit age.

Joint distribution of small-degree nodes converges to a Gaussian law.

Exact degree distribution derived using Pólya-Eggenberger urn models.

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 $0$ (terminal nodes) and outdegree $1$ 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 $0$ and $1$ 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].