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34
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 191 (33 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 27 (7 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.
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 19 (11 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].
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 18 (1 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.
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 15 (4 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 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 15 (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
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 13 (4 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.
Universality for first passage percolation on sparse random graphs. arXiv:1210.6839 [math.PR
, 2012
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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 8 (3 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.
Flooding in Weighted Random Graphs
"... In this paper, we study the impact of edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen n ..."
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Cited by 8 (1 self)
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In this paper, we study the impact of edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node, and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of n, when the size of the graph n tends to infinity. We also derive the exact value for the prefactors. These allow us to analyze an asynchronous randomized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average. 1