13 research outputs found
Scaling limits for critical inhomogeneous random graphs with finite third moments
We identify the scaling limits for the sizes of the largest components at
criticality for inhomogeneous random graphs when the degree exponent
satisfies . We see that the sizes of the (rescaled) components converge
to the excursion lengths of an inhomogeneous Brownian motion, extending results
of \cite{Aldo97}. We rely heavily on martingale convergence techniques, and
concentration properties of (super)martingales. This paper is part of a
programme to study the critical behavior in inhomogeneous random graphs of
so-called rank-1 initiated in \cite{Hofs09a}.Comment: Final versio
Critical behavior in inhomogeneous random graphs
We study the critical behavior of inhomogeneous random graphs where edges are
present independently but with unequal edge occupation probabilities. The edge
probabilities are moderated by vertex weights, and are such that the degree of
vertex i is close in distribution to a Poisson random variable with parameter
w_i, where w_i denotes the weight of vertex i. We choose the weights such that
the weight of a uniformly chosen vertex converges in distribution to a limiting
random variable W, in which case the proportion of vertices with degree k is
close to the probability that a Poisson random variable with random parameter W
takes the value k. We pay special attention to the power-law case, in which
P(W\geq k) is proportional to k^{-(\tau-1)} for some power-law exponent \tau>3,
a property which is then inherited by the asymptotic degree distribution.
We show that the critical behavior depends sensitively on the properties of
the asymptotic degree distribution moderated by the asymptotic weight
distribution W. Indeed, when P(W\geq k) \leq ck^{-(\tau-1)} for all k\geq 1 and
some \tau>4 and c>0, the largest critical connected component in a graph of
size n is of order n^{2/3}, as on the Erd\H{o}s-R\'enyi random graph. When,
instead, P(W\geq k)=ck^{-(\tau-1)}(1+o(1)) for k large and some \tau\in (3,4)
and c>0, the largest critical connected component is of the much smaller order
n^{(\tau-2)/(\tau-1)}.Comment: 26 page
Scale-free percolation
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 arêtes sont indépendantes et la probabilité qu’il existe un lien entre x et y est pxy=1-exp{-¿WxWy/|x-y|a}. Le paramètre ¿ est le paramètre de percolation tandis que a 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 Wx est à variation régulière d’indice t-1, alors la queue de la distribution des degrés est à variation régulière d’indice ¿=a(t-1)/d. 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 ¿c¿(0,8) telle que le graphe contienne une composante infinie quand ¿>¿c et aucune composante infinie quand
Cluster tails for critical power-law inhomogeneous random graphs
Recently, the scaling limit of cluster sizes for critical inhomogeneous
random graphs of rank-1 type having finite variance but infinite third moment
degrees was obtained (see previous work by Bhamidi, van der Hofstad and van
Leeuwaarden). It was proved that when the degrees obey a power law with
exponent in the interval (3,4), the sequence of clusters ordered in decreasing
size and scaled appropriately converges as n goes to infinity to a sequence of
decreasing non-degenerate random variables.
Here, we study the tails of the limit of the rescaled largest cluster, i.e.,
the probability that the scaling limit of the largest cluster takes a large
value u, as a function of u. This extends a related result of Pittel for the
Erd\H{o}s-R\'enyi random graph to the setting of rank-1 inhomogeneous random
graphs with infinite third moment degrees. We make use of delicate large
deviations and weak convergence arguments.Comment: corrected and updated first referenc
Critical random forests
Let denote a random forest on a set of vertices, chosen
uniformly from all forests with edges. Let denote the forest
obtained by conditioning the Erdos-Renyi graph to be acyclic. We
describe scaling limits for the largest components of and , in
the critical window or . Aldous
described a scaling limit for the largest components of within the
critical window in terms of the excursion lengths of a reflected Brownian
motion with time-dependent drift. Our scaling limit for critical random forests
is of a similar nature, but now based on a reflected diffusion whose drift
depends on space as well as on time
Parameterised branching processes:A functional version of Kesten & Stigum theorem
Let (Zn,n≥0) be a supercritical Galton–Watson process whose offspring distribution μ has mean λ>1 and is such that ∫xlog+(x)dμ(x)<+∞. According to the famous Kesten & Stigum theorem, (Zn/λn) converges almost surely, as n→+∞. The limiting random variable has mean 1, and its distribution is characterised as the solution of a fixed point equation. In this paper, we consider a family of Galton–Watson processes (Zn(λ),n≥0) defined for λ ranging in an interval I⊂(1,∞), and where we interpret λ as the time (when n is the generation). The number of children of an individual at time λ is given by X(λ), where (X(λ))λ∈I is a cà dlà g integer-valued process which is assumed to be almost surely non-decreasing and such that E(X(λ))=λ>1 for all λ∈I. This allows us to define Zn(λ) the number of elements in the nth generation at time λ. Set Wn(λ)=Zn(λ)/λn for all n≥0 and λ∈I. We prove that, under some moment conditions on the process X, the sequence of processes (Wn(λ),λ∈I)n≥0 converges in probability as n tends to infinity in the space of cà dlà g processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.</p