Nous définissons et étudions un modèle de percolation inhomogènes à longue portée sur . A chaque site est assigné un poids positif , où les sont des variables aléatoires indépendantes et identiquement distribuées. Conditionnellement aux poids et étant donnés deux paramètres , les arêtes sont indépendantes et la probabilité qu’il existe un lien entre et est . Le paramètre est le paramètre de percolation tandis que caractérise la portée des interactions. Nous étudierons la distribution des degrés dans le graphe résultant et l’existence éventuelle d’une composante infinie ainsi que la distance de graphe entre deux sites éloignés. Nous montrons d’abord que la queue de la distribution des degrés est liée à la queue de la distribution des poids. Quand la queue de la distribution de est à variation régulière d’indice , alors la queue de la distribution des degrés est à variation régulière d’indice . Le paramètre s’avère crucial pour décrire le modèle. Des conditions sur la distribution des poids et de sont formulées pour l’existence d’une valeur critique telle que le graphe contienne une composante infinie quand et aucune composante infinie quand . De plus, une transition de phase est établie pour la distance dans le graphe de la composante infinie au point , c’est à dire au point où les degrés n’ont plus de second moment fini. Notre modèle peut être vu comme une interpolation entre la percolation à longue portée et des modèles de graphes aléatoires inhomogènes. Nous montrons qu’il possède les caractéristiques des deux modèles précédents.
We formulate and study a model for inhomogeneous long-range percolation on . Each vertex is assigned a non-negative weight , where are i.i.d. random variables. Conditionally on the weights, and given two parameters , the edges are independent and the probability that there is an edge between and is given by . The parameter is the percolation parameter, while 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 is regularly varying with exponent , then the tail of the degree distribution is regularly varying with exponent . 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 such that the graph contains an infinite component when and no infinite component when . Furthermore, a phase transition is established for the graph distances between vertices in the infinite component at the point , 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.
Mots-clés : random graphs, Long-range percolation, percolation in random environment, degree distribution, phase transition, chemical distance, graph distance
@article{AIHPB_2013__49_3_817_0, author = {Deijfen, Maria and van der Hofstad, Remco and Hooghiemstra, Gerard}, title = {Scale-free percolation}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {817--838}, publisher = {Gauthier-Villars}, volume = {49}, number = {3}, year = {2013}, doi = {10.1214/12-AIHP480}, mrnumber = {3112435}, zbl = {1274.60290}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/12-AIHP480/} }
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 SP - 817 EP - 838 VL - 49 IS - 3 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/12-AIHP480/ DO - 10.1214/12-AIHP480 LA - en ID - AIHPB_2013__49_3_817_0 ER -
%0 Journal Article %A Deijfen, Maria %A van der Hofstad, Remco %A Hooghiemstra, Gerard %T Scale-free percolation %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 817-838 %V 49 %N 3 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/12-AIHP480/ %R 10.1214/12-AIHP480 %G en %F AIHPB_2013__49_3_817_0
Deijfen, Maria; van der Hofstad, Remco; Hooghiemstra, Gerard. Scale-free percolation. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 817-838. doi : 10.1214/12-AIHP480. http://archive.numdam.org/articles/10.1214/12-AIHP480/
[1] Discontinuity of the percolation density in one dimensional percolation models. Comm. Math. Phys. 107 (1986) 611-647. | MR | Zbl
and .[2] Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 (1997) 812-854. | MR | Zbl
.[3] Geometry of the uniform spanning forest: Transitions in diameters 4, 8, 12, … . Ann. of Math. 160 (2004) 465-491. | MR | Zbl
, , and .[4] Transience, recurrence and critical behavior for long-range percolation. Comm. Math. Phys. 226 (3) (2002) 531-558. | MR | Zbl
.[5] A lower bound for chemical distances in sparse long-range percolation models. Preprint, 2004. Available at arXiv:math/0409021v1.
.[6] Novel scaling limits for critical inhomogeneous random graphs. Preprint, 2009. | MR | Zbl
, and .[7] Scaling limits for critical inhomogeneous random graphs with finite third moments. Electron. J. Probab. 15 (2010) 1682-1702. | MR | Zbl
, and .[8] Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge, 1987. | MR | Zbl
, and .[9] On the scaling of the chemical distance in long range percolation models. Ann. Probab. 32 (2004) 2933-2977. | MR | Zbl
.[10] Percolation. Cambridge Univ. Press, New York, 2006. | MR | Zbl
and .[11] The phase transition in inhomogeneous random graphs. Rand. Struct. Alg. 31 (2007) 3-122. | MR | Zbl
, and .[12] Generating simple random graphs with prescribed degree distribution. J. Stat. Phys. 124 (2006) 1377-1397. | MR | Zbl
, and .[13] The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA 99 (2002) 15879-15882. | MR | Zbl
and .[14] Connected components in random graphs with given expected degree sequences. Ann. Comb. 6 (2002) 125-145. | MR | Zbl
and .[15] The diameter of a long-range percolation graph. Rand. Struct. Alg. 21 (2002) 1-13. | MR | Zbl
, and .[16] Diameters in preferential attachment graphs. J. Stat. Phys. 139 (2010) 72-107. | MR | Zbl
, and .[17] An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edition. Wiley, New York, 1971. | MR | Zbl
.[18] Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields 92 (1992) 511-527. | MR | Zbl
, and .[19] Percolation, 2nd edition. Springer, Berlin, 1999. | MR
.[20] The scaling window for a random graph with a given degree sequence. Preprint, 2009. | MR | Zbl
and .[21] Mean-field behavior for long- and finite range ising model, percolation and self-avoiding walk. J. Stat. Phys. 132 (5) (2008) 1001-1049. | MR | Zbl
, and .[22] Asymptotic equivalence and contiguity of some random graphs. Rand. Struct. Alg. 36 (1) 2010 26-45. | MR | Zbl
.[23] Domination by product measure. Ann. Probab. 25 (1997) 71-95. | MR | Zbl
, and .[24] Continuum Percolation. Cambridge Univ. Press, Cambridge, 1996. | MR | Zbl
and .[25] One-dimensional percolation models: The existence of a transition for . Comm. Math. Phys. 104 (1986) 547-571. | MR | Zbl
and .[26] On a conditionally Poissonian graph process. Adv. in Appl. Probab. 38 (2006) 59-75. | MR | Zbl
and .[27] Long range percolation in one dimension. J. Phys. A 16 (1983) L639-L641. | MR
.[28] Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 (3) (2001) 239-244. | MR | Zbl
.[29] Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1. Preprint, 2009. | MR | Zbl
.[30] Ultra-small scale-free geometric networks. J. Appl. Probab. 43 (2006) 665-677. | MR | Zbl
.[31] Critical behavior in inhomogeneous random graphs. Preprint, 2009. | MR | Zbl
.[32] Distances in random graphs with finite variance degrees. Rand. Struct. Alg. 26 (2005) 76-123. | MR | Zbl
, and .[33] Distances in random graphs with finite mean and infinite variance degrees. Electron. J. Probab. 12 (2007) 703-766. | MR | Zbl
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