A novel approach to the giant component fluctuations
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arXiv
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| Format: | Preprint |
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2024
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| author | Corujo, Josué Lemaire, Sophie Limic, Vlada |
| author_facet | Corujo, Josué Lemaire, Sophie Limic, Vlada |
| contents | We present a novel approach to study the evolution of the size (i.e. the number of vertices) of the giant component of a random graph process. It is based on the exploration algorithm called simultaneous breadth-first walk, introduced by Limic in 2019, that encodes the dynamic of the evolution of the sizes of the connected components of a large class of random graph processes. We limit our study to the variant of the Erdős-Rényi graph process $(G_n(s))_{s\geq 0}$ with $n$ vertices where an edge connecting a pair of vertices appears at an exponential rate 1 waiting time, independently over pairs. We first use the properties of the simultaneous breadth-first walk to obtain an alternative and self-contained proof of the functional central limit theorem recently established by Enriquez, Faraud and Lemaire in the super-critical regime ($s=\frac{c}{n}$ and $c>1$). Next, to show the versatility of our approach, we prove a functional central limit theorem in the barely super-critical regime ($s=\frac{1+tε_n}{n}$ where $t>0$ and $(ε_n)_n$ is a sequence of positive reals that converges to 0 such that $(nε_n^3)_n$ tends to $+\infty$). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_06995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A novel approach to the giant component fluctuations Corujo, Josué Lemaire, Sophie Limic, Vlada Probability Combinatorics Primary 05C80, Secondary 60F17, 60C05 We present a novel approach to study the evolution of the size (i.e. the number of vertices) of the giant component of a random graph process. It is based on the exploration algorithm called simultaneous breadth-first walk, introduced by Limic in 2019, that encodes the dynamic of the evolution of the sizes of the connected components of a large class of random graph processes. We limit our study to the variant of the Erdős-Rényi graph process $(G_n(s))_{s\geq 0}$ with $n$ vertices where an edge connecting a pair of vertices appears at an exponential rate 1 waiting time, independently over pairs. We first use the properties of the simultaneous breadth-first walk to obtain an alternative and self-contained proof of the functional central limit theorem recently established by Enriquez, Faraud and Lemaire in the super-critical regime ($s=\frac{c}{n}$ and $c>1$). Next, to show the versatility of our approach, we prove a functional central limit theorem in the barely super-critical regime ($s=\frac{1+tε_n}{n}$ where $t>0$ and $(ε_n)_n$ is a sequence of positive reals that converges to 0 such that $(nε_n^3)_n$ tends to $+\infty$). |
| title | A novel approach to the giant component fluctuations |
| topic | Probability Combinatorics Primary 05C80, Secondary 60F17, 60C05 |
| url | https://arxiv.org/abs/2412.06995 |