Scale-free percolation
By Maria Deijfen, Remco van der Hofstad, Gerard Hooghiemstra. 2011
111 citations recorded by the source corpus.
Abstract We formulate and study a model for inhomogeneous long-range percolation on Zd. Each vertex x?Zd is assigned a non-negative weight Wx, where (Wx)x?Zd are i.i.d. random variables. Conditionally on the weights, and given two parameters a,?>0, the edges are independent and the probability that there is an edge between x and y is given by pxy=1-exp{-?WxWy/|x-y|a}. The parameter ? is the percolation parameter, while a 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 Wx is regularly varying with exponent t-1, then the tail of the degree distribution is regularly varying with exponent ?=a(t-1)/d. The parameter ? turns out to be crucial for the behavior of the model. Conditions on the weight distribution and ? are formulated for the existence of a critical value ?c?(0,8) such that the graph contains an infinite component when ?>?c and no infinite component when ?0, les aretes sont independantes et la probabilite qu’il existe un lien entre x et y est pxy=1-exp{-?WxWy/|x-y|a}. Le parametre ? est le parametre de percolation tandis que a caracterise la portee des interactions. Nous etudierons la distribution des degres dans le graphe resultant et l’existence eventuelle d’une composante infinie ainsi que la distance de graphe entre deux sites eloignes. Nous montrons d’abord que la queue de la distribution des degres est liee a la queue de la distribution des poids. Quand la queue de la distribution de Wx est a variation reguliere d’indice t-1, alors la queue de la distribution des degres est a variation reguliere d’indice ?=a(t-1)/d. Le parametre ? s’avere crucial pour decrire le modele. Des conditions sur la distribution des poids et de ? sont formulees pour l’existence d’une valeur critique ?c?(0,8) telle que le graphe contienne une composante infinie quand ?>?c et aucune composante infinie quand ?
Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/521ee5c9cc420bd7a226dc73572334ba0f5efcfb
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In Condensed Matter Physics
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Random Graphs and Complex Networks (2016, methodology): “7 (Power-law degrees for power-law weights Deijfen et al. (2013)) Fix d ≥ 1.”