On the block size spectrum of a class of exchangeable dynamic random graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866913833119907840 |
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| author | Alberti, Frederic Boenkost, Florin Cordero, Fernando |
| author_facet | Alberti, Frederic Boenkost, Florin Cordero, Fernando |
| contents | In this work we introduce the dynamic $Θ$-random graph and the associated $Θ$-coalescent with momentum. Dynamic $Θ$-random graphs are a subclass of exchangeable and consistent random graph processes, parametrised by a measure $Θ$ on $[0,1]\times (0,1]$, inspired by the classic $Λ$-coalescent from mathematical population genetics. The $Θ$-coalescent with momentum accounts for the small connected components of this graph; in contrast to the underlying random graph it is exchangeable but not consistent. Our main results specialise on the case where $Θ$ is the product of a beta measure and a Dirac mass at $1$. We prove a dynamic law of large numbers for the block size spectrum, which tracks the numbers of blocks containing $1,...,d$ elements. On top of that, we provide a functional limit theorem for the fluctuations. The limit process satisfies a stochastic differential equation of Ornstein-Uhlenbeck type. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_08972 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the block size spectrum of a class of exchangeable dynamic random graphs Alberti, Frederic Boenkost, Florin Cordero, Fernando Probability In this work we introduce the dynamic $Θ$-random graph and the associated $Θ$-coalescent with momentum. Dynamic $Θ$-random graphs are a subclass of exchangeable and consistent random graph processes, parametrised by a measure $Θ$ on $[0,1]\times (0,1]$, inspired by the classic $Λ$-coalescent from mathematical population genetics. The $Θ$-coalescent with momentum accounts for the small connected components of this graph; in contrast to the underlying random graph it is exchangeable but not consistent. Our main results specialise on the case where $Θ$ is the product of a beta measure and a Dirac mass at $1$. We prove a dynamic law of large numbers for the block size spectrum, which tracks the numbers of blocks containing $1,...,d$ elements. On top of that, we provide a functional limit theorem for the fluctuations. The limit process satisfies a stochastic differential equation of Ornstein-Uhlenbeck type. |
| title | On the block size spectrum of a class of exchangeable dynamic random graphs |
| topic | Probability |
| url | https://arxiv.org/abs/2406.08972 |