[Paper Review] Exceptional times of the critical dynamical Erd\H{o}s-R\'enyi graph

Matthew I. Roberts, Batı Şengül|arXiv (Cornell University)|Oct 19, 2016

Stochastic processes and statistical mechanics17 references3 citations

TL;DR

This paper studies a time-evolving Erdřs-Rényi random graph where edges are resampled at rate 1, focusing on the largest connected component. It proves that, with high probability, the size of the largest component over time reaches order $ n^{2/3} \log^{1/3} n $, significantly exceeding the static critical graph's $ n^{2/3} $, due to rare 'exceptional times' when the component grows transiently larger.

ABSTRACT

In this paper we introduce a network model which evolves in time, and study its largest connected component. We consider a process of graphs $(G_t:t\in [0,1])$, where initially we start with a critical Erd\H{o}s-R\'enyi graph ER(n, 1/n), and then evolve forwards in time by resampling each edge independently at rate 1. We show that the size of the largest connected component that appears during the time interval $[0, 1]$ is of order $n^{2/3} log^{1/3} n$ with high probability. This is in contrast to the largest component in the static critical Erd\H{o}s-R\'enyi graph, which is of order $n^{2/3}$.

Motivation & Objective

• To analyze the evolution of the largest connected component in a dynamically resampled Erdřs-Rényi graph on $ n $ vertices.
• To understand how the component size fluctuates over time, especially in the critical regime $ p = 1/n $, where the static graph has component size $ n^{2/3} $.
• To identify and characterize rare 'exceptional times' when the component size exceeds the typical $ n^{2/3} $ scale.
• To establish the existence of a logarithmic enhancement in component size due to temporal fluctuations in edge configurations.

Proposed method

• Define a continuous-time Markov process $ (G_t)_{t \in [0,1]} $ where each edge is resampled independently at rate 1, starting from a critical Erdřs-Rényi graph $ \mathrm{ER}(n, 1/n) $.
• Use noise sensitivity techniques and Fourier analysis on the hypercube to study component size fluctuations, particularly focusing on the indicator function of large components.
• Apply pivotality estimates and revealment bounds for breadth-first search algorithms to control the influence of edge configurations on component size.
• Leverage results from the multiplicative coalescent and Brownian excursion approximations to understand the structure of large components at exceptional times.
• Use moment methods and tail estimates to bound the probability that the largest component exceeds $ \beta n^{2/3} \log^{1/3} n $ at any time in $[0,1]$.
• Establish a sharp threshold for the appearance of such exceptional times via a two-phase analysis: upper and lower bounds on the probability of large component sizes.

Experimental results

Research questions

• RQ1 Does the supremum of the largest component size over time converge in probability as $ n \to \infty $, and if so, to what limit?
• RQ2 What is the fractal or geometric structure of the set of exceptional times when the largest component exceeds $ \beta n^{2/3} \log^{1/3} n $?
• RQ3 How does the largest component at exceptional times compare to a static component conditioned to be large?
• RQ4 How does the minimum component size behave over time, and is it significantly smaller than the typical $ n^{2/3} $ scale?
• RQ5 What happens if edges are resampled at a different rate $ n^\gamma $, particularly for $ \gamma = -1/3 $?

Key findings

• With high probability, the supremum over time of the largest connected component size in the dynamical critical Erdřs-Rényi graph is of order $ n^{2/3} \log^{1/3} n $.
• The threshold for the appearance of such exceptional times is sharp: for $ \beta < 2/3^{2/3} $, the probability that the component exceeds $ \beta n^{2/3} \log^{1/3} n $ tends to 1, while for $ \beta \geq 2/3^{1/3} $, it tends to 0.
• The Lebesgue measure of the set of exceptional times converges in probability to zero, indicating that such large components are rare in time.
• The set of exceptional times is conjectured to have Hausdorff dimension $ (1 - 3\beta^{3/8}) \vee 0 $, based on box-counting arguments.
• Noise sensitivity analysis shows that the indicator of large components is noise sensitive at the threshold $ \varepsilon_n \sim n^{-1/6} $, implying that component size is highly sensitive to small edge perturbations.
• The component size at exceptional times is conjectured to resemble a static component conditioned to be large, though structural details remain elusive without further tools like Brownian excursion approximations.