Cluster-size decay in supercritical kernel-based spatial random graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914974730813440 |
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| author | Jorritsma, Joost Komjáthy, Júlia Mitsche, Dieter |
| author_facet | Jorritsma, Joost Komjáthy, Júlia Mitsche, Dieter |
| contents | We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent $ζ\in(0,1)$ of the decay of the cluster-size distribution. That is, with $|\mathcal{C}(0)|$ denoting the number of vertices in the component of the vertex at $0\in \mathbb{R}^d$, we prove \[
\mathbb{P}(k< |\mathcal{C}(0)|<\infty)=\exp\big(-Θ(k^ζ)\big), \qquad \text{as }k\to\infty.
\] The value of $ζ$ undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension $d$, the power-law tail exponent $τ$ of the degree distribution and a long-range parameter $α$ governing the presence of long edges in Euclidean space. In this paper we present the proof for the region in the phase diagram where the model is a generalization of continuum scale-free percolation and/or hyperbolic random graphs: $ζ$ in this regime depends both on $τ,α$. We also prove that the second-largest component in a box of volume $n$ is of size $Θ((\log n)^{1/ζ})$ with high probability. We develop a deterministic algorithm, the cover expansion, as new methodology. This algorithm enables us to prevent too large components that may be de-localized or locally dense in space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_00724 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cluster-size decay in supercritical kernel-based spatial random graphs Jorritsma, Joost Komjáthy, Júlia Mitsche, Dieter Probability 05C80, 60K35 We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent $ζ\in(0,1)$ of the decay of the cluster-size distribution. That is, with $|\mathcal{C}(0)|$ denoting the number of vertices in the component of the vertex at $0\in \mathbb{R}^d$, we prove \[ \mathbb{P}(k< |\mathcal{C}(0)|<\infty)=\exp\big(-Θ(k^ζ)\big), \qquad \text{as }k\to\infty. \] The value of $ζ$ undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension $d$, the power-law tail exponent $τ$ of the degree distribution and a long-range parameter $α$ governing the presence of long edges in Euclidean space. In this paper we present the proof for the region in the phase diagram where the model is a generalization of continuum scale-free percolation and/or hyperbolic random graphs: $ζ$ in this regime depends both on $τ,α$. We also prove that the second-largest component in a box of volume $n$ is of size $Θ((\log n)^{1/ζ})$ with high probability. We develop a deterministic algorithm, the cover expansion, as new methodology. This algorithm enables us to prevent too large components that may be de-localized or locally dense in space. |
| title | Cluster-size decay in supercritical kernel-based spatial random graphs |
| topic | Probability 05C80, 60K35 |
| url | https://arxiv.org/abs/2303.00724 |