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72
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
Flag algebras
 JOURNAL OF SYMBOLIC LOGIC
"... Asymptotic extremal combinatorics deals with questions that in the language of model theory can be restated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a randomly chosen ..."
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Cited by 70 (6 self)
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Asymptotic extremal combinatorics deals with questions that in the language of model theory can be restated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a randomly chosen submodel of N with M  elements is isomorphic to M. Which asymptotic relations exist between the quantities p(M1, N),..., p(Mh, N), where M1,..., Mh are fixed “template ” models and N  grows to infinity? In this paper we develop a formal calculus that captures many standard arguments in the area, both previously known and apparently new. We give the first application of this formalism by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density.
Szemerédi’s regularity lemma for sparse graphs
 Foundations of Computational Mathematics
, 1997
"... A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This farreaching result has proved to play a central r^ole in many areas of combinatorics, both `pure ' and `algorithmic. ' The ..."
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Cited by 68 (18 self)
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A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This farreaching result has proved to play a central r^ole in many areas of combinatorics, both `pure ' and `algorithmic. ' The quest for an equally powerful variant of this lemma for sparse graphs has not yet been successful, but some progress has been achieved recently. The aim of this note is to report on the successes so far.
PseudoRandom Graphs
 IN: MORE SETS, GRAPHS AND NUMBERS, BOLYAI SOCIETY MATHEMATICAL STUDIES 15
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Graph partitioning via adaptive spectral techniques
 Comb. Probab. Comput
"... Abstract. In this paper we study the use of spectral techniques for graph partitioning. Let G = (V,E) be a graph whose vertex set has a “latent ” partition V1,..., Vk. Moreover, consider a “density matrix” E = (Evw)v,w∈V such that for v ∈ Vi and w ∈ Vj the entry Evw is the fraction of all possible V ..."
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Cited by 38 (0 self)
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Abstract. In this paper we study the use of spectral techniques for graph partitioning. Let G = (V,E) be a graph whose vertex set has a “latent ” partition V1,..., Vk. Moreover, consider a “density matrix” E = (Evw)v,w∈V such that for v ∈ Vi and w ∈ Vj the entry Evw is the fraction of all possible ViVjedges that are actually present in G. We show that on input (G, k) the partition V1,..., Vk can (almost) be recovered in polynomial time via spectral methods, provided that the following holds: E approximates the adjacency matrix of G in the operator norm, for vertices v ∈ Vi, w ∈ Vj 6 = Vi the corresponding column vectors Ev, Ew are separated, and G is sufficiently “regular ” w.r.t. the matrix E. This result in particular applies to sparse graphs with bounded average degree as n = #V →∞, and it yields interesting consequences on partitioning random graphs.
Mixing time of exponential random graphs
, 2008
"... A plethora of random graph models have been developed in recent years to study a range of problems on networks, driven by the wide availability of data from many social, telecommunication, biochemical and other networks. A key model, extensively used in the sociology literature, is the exponential r ..."
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Cited by 24 (1 self)
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A plethora of random graph models have been developed in recent years to study a range of problems on networks, driven by the wide availability of data from many social, telecommunication, biochemical and other networks. A key model, extensively used in the sociology literature, is the exponential random graph model. This model seeks to incorporate in random graphs the notion of reciprocity, that is, the larger than expected number of triangles and other small subgraphs. Sampling from these distributions is crucial for answering almost any problem of parameter estimation hypothesis testing or to understand the inherent network model itself. In practice this sampling is typically carried out using either the Glauber dynamics or the MetropolisHasting Markov chain Monte Carlo procedure. In this paper we characterize the high and low temperature regimes. We establish that in the high temperature regime the mixing time of the Glauber dynamics is Θ(n 2 log n), where n is the number of vertices in the graph; in contrast, we show that in the low temperature regime the mixing is exponentially slow for any local Markov chain. Our results, moreover, give a rigorous basis for criticisms made of such models. In the high temperature regime, where sampling with MCMC is possible, we show that any finite collection of edges are asymptotically independent; thus, the model does not possess the desired reciprocity property, and is not appreciably different from the ErdősRényi random graph. 1
Regular pairs in sparse random graphs I
 RANDOM STRUCTURES ALGORITHMS
, 2003
"... We consider bipartite subgraphs of sparse random graphs that are regular in the sense of Szemeredi and, among other things, show that they must satisfy a certain local pseudorandom property. This property and its consequences turn out to be useful when considering embedding problems in subgraphs of ..."
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Cited by 21 (9 self)
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We consider bipartite subgraphs of sparse random graphs that are regular in the sense of Szemeredi and, among other things, show that they must satisfy a certain local pseudorandom property. This property and its consequences turn out to be useful when considering embedding problems in subgraphs of sparse random graphs.
An optimal algorithm for checking regularity
 SIAM J. ON COMPUTING
"... We present a deterministic algorithm A that, in O(m 2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemerédi [E. Szemerédi, Regular partitions of graphs, Problèmes Combinatoires et Théorie des Graphes (Colloq. Internat. CNRS, Univ. Orsay, ..."
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Cited by 21 (6 self)
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We present a deterministic algorithm A that, in O(m 2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemerédi [E. Szemerédi, Regular partitions of graphs, Problèmes Combinatoires et Théorie des Graphes (Colloq. Internat. CNRS, Univ. Orsay,
Sparse graphs: metrics and random models
"... 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 13 (1 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 kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasirandom graphs are in a sense completely