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180
Girth of Sparse Graphs
 2002), 194  200. ILWOO CHO
"... Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the nonnegative reals, although the d ..."
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Cited by 78 (6 self)
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Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the nonnegative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for largescale realworld networks. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ(n 2) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons, which are essentially bounded kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasirandom graphs are in a
Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges
, 2010
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Graph limits and exchangeable random graphs
, 2007
"... We develop a clear connection between deFinetti’s theorem for exchangeable arrays (work of Aldous–Hoover–Kallenberg) and the emerging area of graph limits (work of Lovász and many coauthors). Along the way, we translate the graph theory into more classical probability. ..."
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Cited by 50 (9 self)
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We develop a clear connection between deFinetti’s theorem for exchangeable arrays (work of Aldous–Hoover–Kallenberg) and the emerging area of graph limits (work of Lovász and many coauthors). Along the way, we translate the graph theory into more classical probability.
Distances in random graphs with finite mean and infinite variance degrees.
 Electron. J. Probab.,
, 2007
"... Abstract In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent τ ∈ (2, 3). The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph wi ..."
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Cited by 38 (13 self)
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Abstract In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent τ ∈ (2, 3). The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph with N nodes is investigated when N → ∞. When τ ∈ (2, 3), this graph distance grows like 2 log log N  log(τ −2) . In different papers, the cases τ > 3 and τ ∈ (1, 2) have been studied. We also study the fluctuations around these asymptotic means, and describe their distributions. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.
The method of moments and degree distributions for network models
 Ann. Statist
, 2011
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Critical random graphs: diameter and mixing time
"... Abstract. Let C1 denote the largest connected component of the critical ErdősRényi random graph G(n, 1). We show that, typically, the diameter of C1 is of n order n 1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Worm ..."
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Cited by 21 (7 self)
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Abstract. Let C1 denote the largest connected component of the critical ErdősRényi random graph G(n, 1). We show that, typically, the diameter of C1 is of n order n 1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald [5]. These results extend to clusters of size n 2/3 of pbond percolation on any dregular nvertex graph where such clusters exist, provided that p(d − 1) ≤ 1 + O(n −1/3). 1.
Asymptotic equivalence and contiguity of some random graphs
, 2008
"... Abstract. We show that asymptotic equivalence, in a strong form, holds between two random graph models with slightly differing edge probabilities under substantially weaker conditions than what might naively be expected. One application is a simple proof of a recent result by van den Esker, van der ..."
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Cited by 19 (5 self)
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Abstract. We show that asymptotic equivalence, in a strong form, holds between two random graph models with slightly differing edge probabilities under substantially weaker conditions than what might naively be expected. One application is a simple proof of a recent result by van den Esker, van der Hofstad and Hooghiemstra on the equivalence between graph distances for some random graph models. 1.