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21
The phase transition in inhomogeneous random graphs, preprint available from http://www.arxiv.org/abs/math.PR/0504589
"... Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, powerlaw degree distributions. ..."
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Cited by 99 (30 self)
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Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, powerlaw degree distributions. Thus there has been a lot of recent interest in defining and studying ‘inhomogeneous ’ random graph models. One of the most studied properties of these new models is their ‘robustness’, or, equivalently, the ‘phase transition ’ as an edge density parameter is varied. For G(n, p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogenous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this already, and others can be approximated by models with
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.
The diameter of sparse random graphs
, 2010
"... In this paper we study the diameter of the random graph G(n,p), i.e., the largest finite distance between two vertices, for a wide range of functions p = p(n). For p = λ/n with λ> 1 constant we give a simple proof of an essentially best possible result, with an Op(1) additive correction term. Usi ..."
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Cited by 7 (0 self)
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In this paper we study the diameter of the random graph G(n,p), i.e., the largest finite distance between two vertices, for a wide range of functions p = p(n). For p = λ/n with λ> 1 constant we give a simple proof of an essentially best possible result, with an Op(1) additive correction term. Using similar techniques, we establish twopoint concentration in the case that np → ∞. For p = (1 + ε)/n with ε → 0, we obtain a corresponding result that applies all the way down to the scaling window of the phase transition, with an Op(1/ε) additive correction term whose (appropriately scaled) limiting distribution we describe. Combined with earlier results, our new results complete the determination of the diameter of the random graph G(n,p) to an accuracy of the order of its standard deviation (or better), for all functions p = p(n). Throughout we use branching process methods, rather than the more common approach of separate analysis of the 2core and the trees attached to it. 1 Introduction and main results Throughout, we write diam(G) for the diameter of a graph G, meaning the largest graph
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 6 (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.
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 5 (3 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.
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].
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 5 (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.
First Passage percolation on locally tree like networks I: Dense random graphs
, 2007
"... We study various properties of least cost paths under iid disorder for the complete graph and dense ErdosRenyii random graphs in the connected phase, with iid exponential and uniform weights on edges. Using a simple heuristic, we compute explicitly, limiting distributions for (properly recentered) ..."
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Cited by 4 (4 self)
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We study various properties of least cost paths under iid disorder for the complete graph and dense ErdosRenyii random graphs in the connected phase, with iid exponential and uniform weights on edges. Using a simple heuristic, we compute explicitly, limiting distributions for (properly recentered) lengths of shortest paths between typical nodes, as well as multiple source destination pairs; we also derive asymptotics for the number of edges on the shortest path, namely the hopcount and find that the addition of edge weights converts these graphs from ultrasmall world networks to small world networks. Finally we study the VickreyClarkeGrooves measure of overpayment for the complete graph with exponential edge weights and show that the complete graph is far from monopolistic for large n. Key words. VickreyClarkeGrooves measure of overpayment, flow, random graph, random network, first passage percolation, Cox point process, hopcount, Yule process
Random graphs with arbitrary i.i.d. degrees
, 2005
"... In this paper we study distances and connectivity properties of random graphs with an arbitrary i.i.d. degree sequence. When the tail of the degree distribution is regularly varying with exponent 1 − τ there are three distinct cases: (i) τ> 3, where the degrees have finite variance, (ii) τ ∈ (2, ..."
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Cited by 4 (2 self)
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In this paper we study distances and connectivity properties of random graphs with an arbitrary i.i.d. degree sequence. When the tail of the degree distribution is regularly varying with exponent 1 − τ there are three distinct cases: (i) τ> 3, where the degrees have finite variance, (ii) τ ∈ (2, 3), where the degrees have infinite variance, but finite mean, and (iii) τ ∈ (1, 2), where the degrees have infinite mean. These random graphs can serve as models for complex networks where degree power laws are observed. The distances between pairs of nodes in the three cases mentioned above have been studied in three previous publications, and we survey the results obtained there. Apart from the critical cases τ = 1, τ = 2 and τ = 3, this completes the scaling picture. We explain the results heuristically and describe related work and open problems. We also compare the behavior in this model to Internet data, where a degree power law with exponent τ ≈ 2.2 is observed. Furthermore, in this paper we derive results concerning the connected components and the diameter. We give a criterion when there exists a unique largest connected component of size proportional to the size of the graph, and study sizes of the other connected components. Also, we show that for τ ∈ (2, 3), which is most often observed in real networks, the diameter in this model grows much faster than the typical distance between two arbitrary nodes.
RANDOM NETWORKS WITH CONCAVE PREFERENTIAL ATTACHMENT RULE
, 2010
"... Many of the phenomena in the complex world in which we live have a rough description as a large network of interacting components. Random network theory tries to describe the global structure of such networks from basic local principles. One such principle is the preferential attachment paradigm whi ..."
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Cited by 3 (2 self)
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Many of the phenomena in the complex world in which we live have a rough description as a large network of interacting components. Random network theory tries to describe the global structure of such networks from basic local principles. One such principle is the preferential attachment paradigm which suggests that networks are built by adding nodes and links successively, in such a way that new nodes prefer to be connected to existing nodes if they have a high degree. Our research gives the first comprehensive and mathematically rigorous treatment of the case when this preference follows a nonlinear, or more precisely concave, rule. We survey results obtained so far and some ongoing developments.