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 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. The rate of convergence is derived analytically. We also study the exact distribution of the degree of a node, as the circuit ages, via a series of Pólya-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 (Zhang 2016).

Appendix

1.1 A.1 Coefficients in Proposition 2

$$ \begin{array}{@{}rcl@{}} \mathcal{C}_{1} &=& 4m(m - 1)(mn - m + n - 2),\\ \mathcal{C}_{2} &=& 4m(mn - m + n - 3)(mn - m + n - 2),\\ \mathcal{C}_{3} &=& (mn - m + n - 4)(mn - m + n - 3)(mn - m + n - 2),\\ \mathcal{C}_{4} &=& 4m(m^{2}n^{2} - m^{2}n + 2mn^{2} + m^{2} - 7mn + n^{2} + m - 6n + 10),\\ \mathcal{C}_{5} &=& 2m(mn - m + n - 2)(2mn - m + 2n - 7). \end{array} $$

1.2 A.2 Exact Moment of $\mathbb {E}[B^{2}_{n}]$ in Proposition 3

$$ \begin{array}{@{}rcl@{}} \mathbb{E}[B_{n}^{2}] &=& \frac{n - 1}{(mn + n - 3)(mn + n - 2)(mn + n - 1)} \left( \frac{4m^{2}(m + 1)^{5}n^{4}}{(3m + 1)^{2}(2m + 1)^{2}}\right.\\ &&+\frac{(m + 1)^{3}}{(5m + 1)(4m + 1)(3m + 1)^{2}(2m + 1)^{2}}\\ &&\times \left( 4(m + 1)m(116m^{4} + 47m^{3} + 39m^{2} + 13m + 1)n^{3}\right.\\ &&\left. + (304m^{6} - 1772m^{5} - 1836m^{4} - 1043m^{3} - 173m^{2} + 7m + 1)n^{2}\right)\\ &&-\frac{(m + 1)}{(5m + 1)(4m + 1)(3m + 1)^{2}(3m - 1)(2m + 1)^{2}}\\ &&\times\left( (m + 1)(528m^{8} + 6100m^{7} + 156m^{6} + 261m^{5} \right.\\ &&+ 1731m^{4} + 1594m^{3} + 86m^{2} - 83m - 5)n\\ &&+ 2(144m^{9} - 300m^{8} - 5182m^{7} - 7839m^{6}\\ && \left.- 6918m^{5} - 3483m^{4} - 828m^{3} + 99m^{2} + 58m + 3)\right). \end{array} $$