Scale-free percolation

Maria Deijfen; Remco van der Hofstad; Gerard Hooghiemstra

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 3, page 817-838
  • ISSN: 0246-0203

Abstract

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We formulate and study a model for inhomogeneous long-range percolation on ℤ d . Each vertex x ∈ ℤ d is assigned a non-negative weight W x , where ( W x ) x ∈ ℤ d are i.i.d. random variables. Conditionally on the weights, and given two parameters α , λ g t ; 0 , the edges are independent and the probability that there is an edge between x and y is given by p x y = 1 - exp { - λ W x W y / | x - y | α } . The parameter λ is the percolation parameter, while α describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices. First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of W x is regularly varying with exponent τ - 1 , then the tail of the degree distribution is regularly varying with exponent γ = α ( τ - 1 ) / d . The parameter γ turns out to be crucial for the behavior of the model. Conditions on the weight distribution and γ are formulated for the existence of a critical value λ c ∈ ( 0 , ∞ ) such that the graph contains an infinite component when λ g t ; λ c and no infinite component when λ l t ; λ c . Furthermore, a phase transition is established for the graph distances between vertices in the infinite component at the point γ = 2 , that is, at the point where the degrees switch from having finite to infinite second moment. The model can be viewed as an interpolation between long-range percolation and models for inhomogeneous random graphs, and we show that the behavior shares the interesting features of both these models.

How to cite

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Deijfen, Maria, van der Hofstad, Remco, and Hooghiemstra, Gerard. "Scale-free percolation." Annales de l'I.H.P. Probabilités et statistiques 49.3 (2013): 817-838. <http://eudml.org/doc/271988>.

@article{Deijfen2013,
abstract = {We formulate and study a model for inhomogeneous long-range percolation on $\mathbb \{Z\}^\{d\}$. Each vertex $x\in \mathbb \{Z\}^\{d\}$ is assigned a non-negative weight $W_\{x\}$, where $(W_\{x\})_\{x\in \mathbb \{Z\}\}^\{d\}$ are i.i.d. random variables. Conditionally on the weights, and given two parameters $\alpha ,\lambda &gt;0$, the edges are independent and the probability that there is an edge between $x$ and $y$ is given by $p_\{xy\}=1-\exp \lbrace -\lambda W_\{x\}W_\{y\}/|x-y|^\{\alpha \}\rbrace $. The parameter $\lambda $ is the percolation parameter, while $\alpha $ describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices. First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of $W_\{x\}$ is regularly varying with exponent $\tau -1$, then the tail of the degree distribution is regularly varying with exponent $\gamma =\alpha (\tau -1)/d$. The parameter $\gamma $ turns out to be crucial for the behavior of the model. Conditions on the weight distribution and $\gamma $ are formulated for the existence of a critical value $\lambda _\{\mathrm \{c\}\}\in (0,\infty )$ such that the graph contains an infinite component when $\lambda &gt;\lambda _\{\mathrm \{c\}\}$ and no infinite component when $\lambda &lt;\lambda _\{\mathrm \{c\}\}$. Furthermore, a phase transition is established for the graph distances between vertices in the infinite component at the point $\gamma =2$, that is, at the point where the degrees switch from having finite to infinite second moment. The model can be viewed as an interpolation between long-range percolation and models for inhomogeneous random graphs, and we show that the behavior shares the interesting features of both these models.},
author = {Deijfen, Maria, van der Hofstad, Remco, Hooghiemstra, Gerard},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random graphs; Long-range percolation; percolation in random environment; degree distribution; phase transition; chemical distance; graph distance; long-range percolation},
language = {eng},
number = {3},
pages = {817-838},
publisher = {Gauthier-Villars},
title = {Scale-free percolation},
url = {http://eudml.org/doc/271988},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Deijfen, Maria
AU - van der Hofstad, Remco
AU - Hooghiemstra, Gerard
TI - Scale-free percolation
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 3
SP - 817
EP - 838
AB - We formulate and study a model for inhomogeneous long-range percolation on $\mathbb {Z}^{d}$. Each vertex $x\in \mathbb {Z}^{d}$ is assigned a non-negative weight $W_{x}$, where $(W_{x})_{x\in \mathbb {Z}}^{d}$ are i.i.d. random variables. Conditionally on the weights, and given two parameters $\alpha ,\lambda &gt;0$, the edges are independent and the probability that there is an edge between $x$ and $y$ is given by $p_{xy}=1-\exp \lbrace -\lambda W_{x}W_{y}/|x-y|^{\alpha }\rbrace $. The parameter $\lambda $ is the percolation parameter, while $\alpha $ describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices. First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of $W_{x}$ is regularly varying with exponent $\tau -1$, then the tail of the degree distribution is regularly varying with exponent $\gamma =\alpha (\tau -1)/d$. The parameter $\gamma $ turns out to be crucial for the behavior of the model. Conditions on the weight distribution and $\gamma $ are formulated for the existence of a critical value $\lambda _{\mathrm {c}}\in (0,\infty )$ such that the graph contains an infinite component when $\lambda &gt;\lambda _{\mathrm {c}}$ and no infinite component when $\lambda &lt;\lambda _{\mathrm {c}}$. Furthermore, a phase transition is established for the graph distances between vertices in the infinite component at the point $\gamma =2$, that is, at the point where the degrees switch from having finite to infinite second moment. The model can be viewed as an interpolation between long-range percolation and models for inhomogeneous random graphs, and we show that the behavior shares the interesting features of both these models.
LA - eng
KW - random graphs; Long-range percolation; percolation in random environment; degree distribution; phase transition; chemical distance; graph distance; long-range percolation
UR - http://eudml.org/doc/271988
ER -

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