The stable graph: the metric space scaling limit of a critical random graph with i.i.d. power-law degrees
We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail be- haviour with exponent α + 1, where α ∈ (1,2). The limiting components are constructed from random R-trees encoded by the excursions above its run- ning inf...
| Main Authors: | , |
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| Formato: | Journal article |
| Idioma: | English |
| Publicado: |
Institute of Mathematical Statistics
2023
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| Summary: | We prove a metric space scaling limit for a critical random graph with
independent and identically distributed degrees having power-law tail be-
haviour with exponent α + 1, where α ∈ (1,2). The limiting components are
constructed from random R-trees encoded by the excursions above its run-
ning infimum of a process whose law is locally absolutely continuous with
respect to that of a spectrally positive α-stable Lévy process. These spanning
R-trees are measure-changed α-stable trees. In each such R-tree, we make
a random number of vertex identifications, whose locations are determined
by an auxiliary Poisson process. This generalises results, which were already
known in the case where the degree distribution has a finite third moment
(a model which lies in the same universality class as the Erd˝os–Rényi ran-
dom graph) and where the role of the α-stable Lévy process is played by a
Brownian motion. |
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