Results 1  10
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10
Random intersection graphs with tunable degree distribution and clustering
, 2008
"... A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this paper a model is developed in which each vertex is given a random weight, and vertices with ..."
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Cited by 12 (2 self)
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A random intersection graph is constructed by assigning independently to each vertex a subset of a given set and drawing an edge between two vertices if and only if their respective subsets intersect. In this paper a model is developed in which each vertex is given a random weight, and vertices with larger weights are more likely to be assigned large subsets. The distribution of the degree of a given vertex is characterized and is shown to depend on the weight of the vertex. In particular, if the weight distribution is a power law, the degree distribution will be so as well. Furthermore, an asymptotic expression for the clustering in the graph is derived. By tuning the parameters of the model, it is possible to generate a graph with arbitrary clustering, expected degree and – in the power law case – tail exponent.
First passage percolation on random graphs with finite mean degrees
, 2009
"... We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the numb ..."
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Cited by 7 (2 self)
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We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the socalled hopcount. We analyze the configuration model with degree powerlaw exponent τ> 2, in which the degrees are assumed to be i.i.d. with a tail distribution which is either of powerlaw form with exponent τ − 1> 1, or has even thinner tails (τ = ∞). In this model, the degrees have a finite first moment, while the variance is finite for τ> 3, but infinite for τ ∈ (2, 3). We prove a central limit theorem for the hopcount, with asymptotically equal means and variances equal to α log n, where α ∈ (0, 1) for τ ∈ (2, 3), while α> 1 for τ> 3. Here n denotes the size of the graph. For τ ∈ (2, 3), it is known that the graph distance between two randomly chosen connected vertices is proportional to log log n [25], i.e., distances are ultra small. Thus, the addition of edge weights causes a marked change in the geometry of the network. We further study the weight of the least weight path, and prove convergence in distribution of an appropriately centered version. This study continues the program initiated in [5] of showing that log n is the correct scaling for the hopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen vertices is of much smaller order. The case of infinite mean degrees (τ ∈ [1, 2)) is studied in [6], where it is proved that the hopcount remains uniformly bounded and converges in distribution.
Extreme value theory, PoissonDirichlet distributions and FPP on random networks
, 2009
"... We study first passage percolation on the configuration model (CM) having powerlaw degrees with exponent τ ∈ [1, 2). To this end, we equip the edges with exponential weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of ..."
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Cited by 5 (2 self)
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We study first passage percolation on the configuration model (CM) having powerlaw degrees with exponent τ ∈ [1, 2). To this end, we equip the edges with exponential weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of edges on the minimal weight path, which can be computed in terms of the PoissonDirichlet distribution. We explicitly describe these limits via the construction of an infinite limiting object describing the FPP problem in the densely connected core of the network. We consider two separate cases, namely, the original CM, in which each edge, regardless of its multiplicity, receives an independent exponential weight, as well as the erased CM, for which there is an independent exponential weight between any pair of direct neighbors. While the results are qualitatively similar, surprisingly the limiting random variables are quite different. Our results imply that the flow carrying properties of the network are markedly different from either the meanfield setting or the locally treelike setting, which occurs as τ> 2, and for which the hopcount between typical vertices scales as log n. In our setting the hopcount is tight and has an explicit limiting distribution, showing that one can transfer information remarkably quickly between different vertices in the network. This efficiency has a down side in that such networks are remarkably fragile to directed attacks. These results continue a general program by the authors to obtain a complete picture of how random disorder changes the inherent geometry of various random network models, see [3, 5, 6].
Diameters in preferential attachment models
, 2009
"... In this paper, we investigate the diameter in preferential attachment (PA) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PAmo ..."
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Cited by 4 (0 self)
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In this paper, we investigate the diameter in preferential attachment (PA) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PAmodels. There is a substantial amount of literature proving that, quite generally, PAgraphs possess powerlaw degree sequences with a powerlaw exponent τ> 2. We prove that the diameter of the PAmodel is bounded above by a constant times log t, where t is the size of the graph. When the powerlaw exponent τ exceeds 3, then we prove that log t is the right order for the diameter, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for τ> 3, distances are of the order log t. For τ ∈ (2, 3), we improve the upper bound to a constant times log log t, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order log log t. These bounds partially prove predictions by physicists that the typical distance in PAgraphs are similar to the ones in other scalefree random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order log log t when τ ∈ (2, 3), and of order log t when τ > 3.
A phase transition for the diameter of the configuration model
 INTERNET MATH
, 2008
"... In this paper, we study the configuration model (CM) with i.i.d. degrees. We establish a phase transition for the diameter when the powerlaw exponent τ of the degrees satisfies τ ∈ (2, 3). Indeed, we show that for τ> 2 and when vertices with degree 1 or 2 are present with positive probability, the ..."
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Cited by 2 (2 self)
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In this paper, we study the configuration model (CM) with i.i.d. degrees. We establish a phase transition for the diameter when the powerlaw exponent τ of the degrees satisfies τ ∈ (2, 3). Indeed, we show that for τ> 2 and when vertices with degree 1 or 2 are present with positive probability, the diameter of the random graph is, with high probability, bounded from below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are 3 or more, we show that, for τ ∈ (2, 3), the diameter of the graph is, with high probability, bounded from above by a constant times the log log of the size of the graph.
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 1 (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. 1
Universal techniques to analyze preferential attachment trees: Global and Local analysis
, 2007
"... We use embeddings in continuous time Branching processes to derive asymptotics for various statistics associated with different models of preferential attachment. This powerful method allows us to deduce, with very little effort, under a common framework, not only local characteristics for a wide cl ..."
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Cited by 1 (0 self)
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We use embeddings in continuous time Branching processes to derive asymptotics for various statistics associated with different models of preferential attachment. This powerful method allows us to deduce, with very little effort, under a common framework, not only local characteristics for a wide class of scale free trees, but also global characteristics such as the height of the tree, maximal degree, and the size and structure of the percolation component attached to the root. We exhibit our computations for a number of different graph models of preferential attachment. Enroute we get exact results for a large number of preferential attachment models including not only the usual preferential attachment but also the preferential attachment with fitness as introduced by Barabasi et al ([6]) and the Competition Induced Preferential attachment of Berger et al ([5]) to name just two. While most of the techniques currently in vogue gain access to the asymptotic degree distribution of the tree, we show how the embedding techniques reveal significantly more information both on local and global characteristics of these trees. Again very soft arguments give us the asymptotic degree distribution and size of the maximal degree in some Preferential attachment network models (not just trees) formulated by Cooper and Frieze [11] and van der Hofstad et al [12]. In the process we find surprising connections between the degree distributions, Yule processes and αstable subordinators. We end with a number of conjectures for the asymptotics for various statistics of these models including size of the maximal component in percolation on these trees.
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 1 (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.