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Large degree vertices in random directed acyclic graphs

Rafael Engel

TL;DR

This work analyzes large-degree vertices in random recursive directed acyclic graphs (RRDAGs) by extending Kingman’s coalescent to a generalized setting. It proves that the degrees of a fixed number of uniform vertices, in the large-degree regime, behave as independent geometric variables, and that counts of vertices with large degrees converge to a Poisson point process; the maximum degree has an explicit limiting distribution. Additionally, the ungreedy depth and labels of a single vertex conditioned on large degree admit Gaussian-type limits, and the same conditioning yields product-form limits for the labels of multiple vertices. Collectively, these results extend known RRT-based findings to the broader RRDAG framework and establish a cohesive coalescent-based toolkit for studying large-degree phenomena in random directed acyclic graphs.

Abstract

This Master's thesis examines the properties of large degree vertices in random recursive directed acyclic graphs (RRDAGs), a generalization of the well-studied random recursive tree (RRT) model. Using a novel adaptation of Kingman's coalescent, we extend results from RRTs to RRDAGs, focusing on different vertex properties. For large degrees, we establish the asymptotic joint distribution of the degree of multiple uniform vertices, proving that they follow a multivariate geometric distribution, and obtain results on maximal and near-maximal degree vertices. In addition, we consider a version of vertex depth that we call ungreedy depth and describe its asymptotic behavior, along with the labels, of single uniform vertices with a given large degree. Finally, we extend this analysis to multiple uniform vertices by deriving the asymptotic behavior of their labels conditional on large degrees.

Large degree vertices in random directed acyclic graphs

TL;DR

This work analyzes large-degree vertices in random recursive directed acyclic graphs (RRDAGs) by extending Kingman’s coalescent to a generalized setting. It proves that the degrees of a fixed number of uniform vertices, in the large-degree regime, behave as independent geometric variables, and that counts of vertices with large degrees converge to a Poisson point process; the maximum degree has an explicit limiting distribution. Additionally, the ungreedy depth and labels of a single vertex conditioned on large degree admit Gaussian-type limits, and the same conditioning yields product-form limits for the labels of multiple vertices. Collectively, these results extend known RRT-based findings to the broader RRDAG framework and establish a cohesive coalescent-based toolkit for studying large-degree phenomena in random directed acyclic graphs.

Abstract

This Master's thesis examines the properties of large degree vertices in random recursive directed acyclic graphs (RRDAGs), a generalization of the well-studied random recursive tree (RRT) model. Using a novel adaptation of Kingman's coalescent, we extend results from RRTs to RRDAGs, focusing on different vertex properties. For large degrees, we establish the asymptotic joint distribution of the degree of multiple uniform vertices, proving that they follow a multivariate geometric distribution, and obtain results on maximal and near-maximal degree vertices. In addition, we consider a version of vertex depth that we call ungreedy depth and describe its asymptotic behavior, along with the labels, of single uniform vertices with a given large degree. Finally, we extend this analysis to multiple uniform vertices by deriving the asymptotic behavior of their labels conditional on large degrees.

Paper Structure

This paper contains 19 sections, 25 theorems, 180 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Definition of the model
  4. Statement of the main results
  5. Basic notions
  6. Random directed acyclic graph model
  7. Kingman's coalescent
  8. Selection sets and connection sets
  9. Properties of vertex degrees
  10. Asymptotic joint degree distribution
  11. Moment estimates on the vertex count of a certain degree
  12. Large degree vertices
  13. Properties of single vertices with given large degrees
  14. Analysis of the ungreedy depth of a vertex
  15. Label and ungreedy depth of a vertex with given large degree
  16. ...and 4 more sections

Key Result

Theorem 2.2

Fix $c\in(0,m+1)$ and $k\in\mathbb{N}$. Let $V_1,\ldots,V_k$ be distinct vertices selected uniformly at random from $[n]$. Then, there exists a constant $\alpha>0$ such that uniformly over natural numbers $d_1,\ldots,d_k<c\log n$, as $n\to\infty$.

Figures (3)

  • Figure 1: An example of roots and trees in the directed acyclic graph $V = [3]$, $E = \{(3,1), (3,2)\}$. Vertices 1 and $2$ are roots and the respectively colored ellipses mark the corresponding trees.
  • Figure 2: Possible realization of Kingman's $(2,5)$-coalescent $(F_5,\ldots,F_1)$. For each step, dotted lines represent the added edges. In this case, $(a_{5,1},a_{5,2},a_{5,3})=(1,2,5),(a_{4,1},a_{4,2},a_{4,3})=(2,3,4),(a_{3,1},a_{3,2},a_{3,3})=(1,2,3),(a_{2,1},a_{2,2})=(1,2)$ and $\xi_5=3,\xi_4=2,\xi_3=2,\xi_2=1$.
  • Figure 3: An example for the relabeling based on the graph $F_1$ from Figure \ref{['fig:coalescent_process']}. The left graph shows $F_1$ with its edges labeled by $L_C^-$ and the right graph $F_1$ relabeled by $L_C$.

Theorems & Definitions (39)

  • Definition 2.1: Random recursive directed acyclic graph, short RRDAG
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Proposition 3.1
  • Definition 3.2: Roots and trees in directed acyclic graphs
  • Definition 3.3
  • ...and 29 more