Results 1  10
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19
Scaling limits for critical inhomogeneous random graphs with finite third moments
, 2009
"... We find scaling limits for the sizes of the largest components at criticality for the rank1 inhomogeneous random graphs with powerlaw degrees with exponent τ. We investigate the case where τ ∈ (3, 4), so that the degrees have finite variance but infinite third moment. The sizes of the largest clus ..."
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Cited by 17 (7 self)
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We find scaling limits for the sizes of the largest components at criticality for the rank1 inhomogeneous random graphs with powerlaw degrees with exponent τ. We investigate the case where τ ∈ (3, 4), so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by n −(τ −2)/(τ −1) , converge to hitting times of a ‘thinned ’ Lévy process. This process is intimately connected to the general multiplicative coalescents studied in [1] and [3]. In particular, we use the results in [3] to show that, when interpreting the location λ inside the critical window as time, the limiting process is a multiplicative process with diffusion constant 0 and the entrance boundary describing the size of relative components in the λ → − ∞ regime proportional to ( −1/(τ
Universality for the distance in finite variance random graphs
, 2008
"... We generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs. Among others, this class contains the Poissonian random graph, the expected degree random graph and the generalized random graph (including the ..."
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Cited by 13 (7 self)
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We generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs. Among others, this class contains the Poissonian random graph, the expected degree random graph and the generalized random graph (including the classical ErdősRényi graph). In the paper we assign to each node a deterministic capacity and the probability that there exists an edge between a pair of nodes is equal to a function of the product of the capacities of the pair divided by the total capacity of all the nodes. We consider capacities which are such that the degrees of a node has uniformly bounded moments of order strictly larger than two, so that, in particular, the degrees have finite variance. We prove that the graph distance grows like log ν N, where the ν depends on the capacities. In addition, the random fluctuations around this asymptotic mean log ν N are shown to be tight. We also consider the case where the capacities are independent copies of a positive random Λ with P (Λ> x) ≤ cx 1−τ, for some constant c and τ> 3, again resulting in graphs where the degrees have finite variance. The method of proof of these results is to couple each member of the class to the Poissonian random graph, for which we then give the complete proof by adapting the arguments of [13].
Sparse random graphs with clustering
, 2008
"... In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random ..."
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Cited by 9 (6 self)
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In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (nonPoisson) multitype branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with powerlaw degree sequences with a wide range of degree exponents and clustering coefficients.
Critical behavior in inhomogeneous random graphs
, 2009
"... We study the critical behavior of inhomogeneous random graphs where edges are present independently but with unequal edge occupation probabilities. We show that the critical behavior depends sensitively on the properties of the asymptotic degrees. Indeed, when the proportion of vertices with degree ..."
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Cited by 9 (2 self)
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We study the critical behavior of inhomogeneous random graphs where edges are present independently but with unequal edge occupation probabilities. We show that the critical behavior depends sensitively on the properties of the asymptotic degrees. Indeed, when the proportion of vertices with degree at least k is bounded above by k −τ+1 for some τ> 4, the largest critical connected component is of order n 2/3, where n denotes the size of the graph, as on the ErdősRényi random graph. The restriction τ> 4 corresponds to finite third moment of the degrees. When, the proportion of vertices with degree at least k is asymptotically equal to ck −τ+1 for some τ ∈ (3,4), the largest critical connected component is of order n (τ−2)/(τ−1) , instead. Our results show that, for inhomogeneous random graphs with a powerlaw degree sequence, the critical behavior admits a transition when the third moment of the degrees turns from finite to infinite. Similar phase transitions have been shown to occur for typical distances in such random graphs when the variance of the degrees turns from finite to infinite. We present further results related to the size of the critical or scaling window, and state conjectures for this and related random graph models.
THE LARGEST COMPONENT IN A SUBCRITICAL RANDOM Graph with a Power Law Degree Distribution
, 2008
"... It is shown that in a subcritical random graph with given vertex degrees satisfying a power law degree distribution with exponent γ>3, the largest component is of order n 1/(γ −1). More precisely, the order of the largest component is approximatively given by a simple constant times the largest v ..."
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Cited by 8 (0 self)
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It is shown that in a subcritical random graph with given vertex degrees satisfying a power law degree distribution with exponent γ>3, the largest component is of order n 1/(γ −1). More precisely, the order of the largest component is approximatively given by a simple constant times the largest vertex degree. These results are extended to several other random graph models with power law degree distributions. This proves a conjecture by Durrett.
The cut metric, random graphs, and branching processes
, 2009
"... In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an app ..."
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Cited by 8 (5 self)
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In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in [4], as well as related results of Bollobás, Borgs, Chayes and Riordan [3], all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering [5]. 1 Introduction and
Scalefree percolation
, 2011
"... We formulate and study a model for inhomogeneous longrange percolation on Zd. Each vertex x ∈ Zd is assigned a nonnegative weight Wx, where (Wx) x∈Zd are i.i.d. random variables. Conditionally on the weights, and given two parameters α, λ> 0, the edges are independent and the probability that t ..."
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Cited by 7 (2 self)
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We formulate and study a model for inhomogeneous longrange percolation on Zd. Each vertex x ∈ Zd is assigned a nonnegative weight Wx, where (Wx) x∈Zd are i.i.d. random variables. Conditionally on the weights, and given two parameters α, λ> 0, the edges are independent and the probability that there is an edge between x and y is given by pxy = 1 − exp{−λWxWy/x − y  α}. The parameter λ is the percolation parameter, while α describes the longrange 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 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 λc ∈ (0, ∞) such that the graph contains an infinite component when λ> λc and no infinite component when λ < λc. 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 longrange percolation and models for inhomogeneous random graphs, and we show that the behavior shares the interesting features of both these models.
Universality for distances in powerlaw random graphs
, 2008
"... We survey the recent work on phase transition and distances in various random graph models with general degree sequences. We focus on inhomogeneous random graphs, the configuration model and affine preferential attachment models, and pay special attention to the setting where these random graphs hav ..."
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Cited by 4 (0 self)
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We survey the recent work on phase transition and distances in various random graph models with general degree sequences. We focus on inhomogeneous random graphs, the configuration model and affine preferential attachment models, and pay special attention to the setting where these random graphs have a powerlaw degree sequence. This means that the proportion of vertices with degree k in large graphs is approximately proportional to k −τ, for some τ> 1. Since many real networks have been empirically shown to have powerlaw degree sequences, these random graphs can be seen as more realistic models for real complex networks. It is often suggested that the behavior of random graphs should have a large amount of universality, meaning, in this case, that random graphs with similar degree sequences share similar behavior. We survey the available results on graph distances in powerlaw random graphs that are consistent with this prediction.