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The process of fluctuations of the giant component of an Erdős-Rényi graph

Nathanaël Enriquez, Gabriel Faraud, Sophie Lemaire

TL;DR

A detailed study of the evolution of the giant component of the Erd\H{o}s-R\'enyi graph process as the mean degree increases from 1 to infinity leads to the identification of the limiting process of the rescaled fluctuations of its order around its deterministic asymptotic.

Abstract

We present a detailed study of the evolution of the giant component of the Erdős-Rényi graph process as the mean degree increases from 1 to infinity. It leads to the identification of the limiting process of the rescaled fluctuations of its order around its deterministic asymptotic. This process is Gaussian with an explicit covariance.

The process of fluctuations of the giant component of an Erdős-Rényi graph

TL;DR

A detailed study of the evolution of the giant component of the Erd\H{o}s-R\'enyi graph process as the mean degree increases from 1 to infinity leads to the identification of the limiting process of the rescaled fluctuations of its order around its deterministic asymptotic.

Abstract

We present a detailed study of the evolution of the giant component of the Erdős-Rényi graph process as the mean degree increases from 1 to infinity. It leads to the identification of the limiting process of the rescaled fluctuations of its order around its deterministic asymptotic. This process is Gaussian with an explicit covariance.
Paper Structure (9 sections, 13 theorems, 124 equations, 5 figures)

This paper contains 9 sections, 13 theorems, 124 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Heuristic
  3. A representation of the infinitesimal evolution of the giant component
  4. Convergence of the finite dimensional distributions
  5. The first three moments of Xn(t+h)-Xn(t)
  6. Assumptions of Proposition \ref{['normanth1']} for the sequence Zn
  7. Tightness
  8. Rare events
  9. Proof of Proposition \ref{['normanth1']}

Key Result

Theorem 1.1

Let $\{B(t),\,t\geq 0\}$ be a standard Brownian motion on $\mathbb{R}$. Set for all $t>1$, Let $t_0,t_1$ be two real numbers such that $1<t_0<t_1$. The process $\{X_n(t),\; t\in[t_0,t_1]\}$ converges weakly, with respect to the Skorohod topology, to the Gaussian process $\{X(t), \,t\in[t_0,t_1]\}$ defined by:

Figures (5)

  • Figure 1: A realization of graph $\mathscr{G}_{n}(t)$ for which $L_n(t)=20$.
  • Figure 2: A realization of graph $\tilde{\mathscr{G}}_{n}(t)$ obtained from $\mathscr{G}_{n}(t)$ by resampling independently with probability $\frac{t}{n}$, all edges 'external' to $\mathscr{C}^{max}_{n}(t)$.
  • Figure 3: Graph $\tilde{\mathscr{G}}_{n}(t,h)$ obtained from $\tilde{\mathscr{G}}_{n}(t)$ by adding edges (drawn as dashed lines) independently with probability $\frac{h}{n-t}$. In this realization, there are ${Y_n=5}$ bridges drawn as green dashed lines and nine external edges drawn in blue. Its largest component is the union of $\mathscr{C}^{max}_{n}(t)$ and $\cup_{i=1}^{5}C_{x_i}$.
  • Figure 4: Subgraph $\tilde{\mathscr{G}}_{n}^{e,1}(t,h)$. New edges with respect to $\tilde{\mathscr{G}}_{n}^{e}(t,h)$ are drawn in red and obtained from an independent resampling of edges having an endpoint in $\tilde{C}_{x_1}$, with probability $\frac{t+h}{n}$. In this example, vertices $x_3$ and $x_4$ are in $\tilde{C}_{x_2}$. Therefore, $\tilde{C}_{x_3}=\tilde{C}_{x_4}=\emptyset$ and $\tilde{\mathscr{G}}_{n}^{e,3}(t,h)=\tilde{\mathscr{G}}_{n}^{e,2}(t,h)=\tilde{\mathscr{G}}_{n}^{e,1}(t,h)$.
  • Figure 5: Subgraph $\tilde{\mathscr{G}}_{n}^{e,4}(t,h)$. New edges with respect to $\tilde{\mathscr{G}}_{n}^{e,3}(t,h)$ are drawn in orange and obtained from an independent resampling of edges having an endpoint in $\cup_{i=1}^4\tilde{C}_{x_i}$, with probability $\frac{t+h}{n}$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Lemma 3.1: Lemma 10 of Behrisch
  • Lemma 3.2
  • Remark 3.3
  • Theorem : Theorems 4.2 and 4.3 in vdh
  • proof : Proof of Lemma \ref{['encadrement']}
  • Proposition 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 9 more