Upper tail bounds for irregular graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913785596346368 |
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| author | Basak, Anirban Karmakar, Shaibal |
| author_facet | Basak, Anirban Karmakar, Shaibal |
| contents | We consider the upper tail large deviations of subgraph counts for irregular graphs $\mathrm{H}$ in $\mathbb{G}(n,p)$, the sparse Erdős-Rényi graph on $n$ vertices with edge connectivity probability $p \in (0,1)$. For $n^{-1/Δ} \ll p \ll 1$, where $Δ$ is the maximum degree of $\mathrm{H}$, we derive the upper tail large deviations for any irregular graph $\mathrm{H}$. On the other hand, we show that for $p$ such that $1 \ll n^{v_{\mathrm{H}}} p^{e_{\mathrm{H}}} \ll (\log n)^{α^{*}_{\mathrm{H}}/\left(α^{*}_{\mathrm{H}}-1\right)}$, where $v_{\mathrm{H}}$ and $e_{\mathrm{H}}$ denote the number of vertices and edges of $\mathrm{H}$, and $α^*_{\mathrm{H}}$ denotes the fractional independence number, the upper tail large deviations of the number of unlabelled copies of $\mathrm{H}$ in $\mathbb{G}(n,p)$ is given by that of a sequence of Poisson random variables with diverging mean, for any strictly balanced graph $\mathrm{H}$. Restricting to the $r$-armed star graph we further prove a localized behavior in the intermediate range of $p$ (left open by the above two results) and show that the mean-field approximation is asymptotically tight for the logarithm of the upper tail probability. This work further identifies the typical structures of $\mathbb{G}(n,p)$ conditioned on upper tail rare events in the localized regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05311 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper tail bounds for irregular graphs Basak, Anirban Karmakar, Shaibal Probability Combinatorics 60C05, 60F10, 05C80 We consider the upper tail large deviations of subgraph counts for irregular graphs $\mathrm{H}$ in $\mathbb{G}(n,p)$, the sparse Erdős-Rényi graph on $n$ vertices with edge connectivity probability $p \in (0,1)$. For $n^{-1/Δ} \ll p \ll 1$, where $Δ$ is the maximum degree of $\mathrm{H}$, we derive the upper tail large deviations for any irregular graph $\mathrm{H}$. On the other hand, we show that for $p$ such that $1 \ll n^{v_{\mathrm{H}}} p^{e_{\mathrm{H}}} \ll (\log n)^{α^{*}_{\mathrm{H}}/\left(α^{*}_{\mathrm{H}}-1\right)}$, where $v_{\mathrm{H}}$ and $e_{\mathrm{H}}$ denote the number of vertices and edges of $\mathrm{H}$, and $α^*_{\mathrm{H}}$ denotes the fractional independence number, the upper tail large deviations of the number of unlabelled copies of $\mathrm{H}$ in $\mathbb{G}(n,p)$ is given by that of a sequence of Poisson random variables with diverging mean, for any strictly balanced graph $\mathrm{H}$. Restricting to the $r$-armed star graph we further prove a localized behavior in the intermediate range of $p$ (left open by the above two results) and show that the mean-field approximation is asymptotically tight for the logarithm of the upper tail probability. This work further identifies the typical structures of $\mathbb{G}(n,p)$ conditioned on upper tail rare events in the localized regime. |
| title | Upper tail bounds for irregular graphs |
| topic | Probability Combinatorics 60C05, 60F10, 05C80 |
| url | https://arxiv.org/abs/2503.05311 |