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Dynamic random graphs with vertex removal

Josep Díaz, Lyuben Lichev, Bas Lodewijks

TL;DR

It is proved that the DRGVR converges to a local limit and determine this limit and several concentration and stability results complete the study.

Abstract

We introduce and analyse a Dynamic Random Graph with Vertex Removal (DRGVR) defined as follows. At every step, with probability $p > 1/2$ a new vertex is introduced, and with probability $1-p$ a vertex, chosen uniformly at random among the present ones (if any), is removed from the graph together with all edges adjacent to it. In the former case, the new vertex connects by an edge to every other vertex with probability inversely proportional to the number of vertices already present. We prove that the DRGVR converges to a local limit and determine this limit. Moreover, we analyse its component structure and distinguish a subcritical and a supercritical regime with respect to the existence of a giant component. As a byproduct of this analysis, we obtain upper and lower bounds for the critical parameter. Furthermore, we provide precise expression of the maximum degree (as well as in- and out-degree for a natural orientation of the DRGVR). Several concentration and stability results complete the study.

Dynamic random graphs with vertex removal

TL;DR

It is proved that the DRGVR converges to a local limit and determine this limit and several concentration and stability results complete the study.

Abstract

We introduce and analyse a Dynamic Random Graph with Vertex Removal (DRGVR) defined as follows. At every step, with probability a new vertex is introduced, and with probability a vertex, chosen uniformly at random among the present ones (if any), is removed from the graph together with all edges adjacent to it. In the former case, the new vertex connects by an edge to every other vertex with probability inversely proportional to the number of vertices already present. We prove that the DRGVR converges to a local limit and determine this limit. Moreover, we analyse its component structure and distinguish a subcritical and a supercritical regime with respect to the existence of a giant component. As a byproduct of this analysis, we obtain upper and lower bounds for the critical parameter. Furthermore, we provide precise expression of the maximum degree (as well as in- and out-degree for a natural orientation of the DRGVR). Several concentration and stability results complete the study.
Paper Structure (37 sections, 32 theorems, 319 equations, 1 figure)

This paper contains 37 sections, 32 theorems, 319 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation.
  3. A brief introduction to local convergence
  4. Local convergence in distribution of marked rooted graphs.
  5. Results
  6. Main ideas of the proofs.
  7. Organisation of the paper.
  8. Preliminaries
  9. General probabilistic preliminaries
  10. Preliminaries for the DRGVR
  11. An alternative viewpoint on the vertex process $(V_n)_{n\ge 0}$.
  12. Further preliminary results.
  13. Theorem \ref{['thrm:LWC intro']}: setting up the framework
  14. The exploration process.
  15. Step $1$:
  16. ...and 22 more sections

Key Result

Theorem 1.3

Fix $\beta>0$ and $\varepsilon\in(0,1/2]$, and consider the DRGVR model and the Binomial birth-death tree given in Definitions def:drgvr and def:wll, respectively. Let $k_0$ be a vertex selected uniformly at random from $V_n$. Then, for any $r\in\mathbb N$, In particular, $(G_n,k_0,M_{n})_{n\ge 1}$ converges locally in distribution as a marked rooted graph to the random marked rooted tree $(\math

Figures (1)

  • Figure 1: A plot of the lower bound and the two upper bounds for $\beta_c$ from Proposition \ref{['prop beta_c intro']}. The sharper upper bound given by $f(p) := \inf_{t\in(-1/2,\infty)} (\frac{(1+2t)(2t^2+7t+4+1/p)}{(1+t)^2(t+1/p)(2t+2/p-1)})^{-1/2}$ is approximated numerically.

Theorems & Definitions (68)

  • Definition 1.1: Dynamic Random Graph with Vertex Removal (DRGVR)
  • Definition 1.2: Binomial birth-death tree
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Proposition 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 58 more