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20
Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges
, 2010
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An approximate version of Sidorenko’s conjecture
 Geom. Funct. Anal
"... A beautiful conjecture of ErdősSimonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent ana ..."
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Cited by 23 (7 self)
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A beautiful conjecture of ErdősSimonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs. 1
Extremal results in sparse pseudorandom graphs
 ADV. MATH. 256 (2014), 206–290
, 2014
"... Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extendin ..."
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Cited by 13 (8 self)
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Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi’s regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a wellknown open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rödl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several wellknown combinatorial theorems, including the removal lemmas for graphs and groups, the ErdősStoneSimonovits theorem and Ramsey’s
QuasiRandomness and Algorithmic Regularity for Graphs with General Degree Distributions
"... We deal with two intimately related subjects: quasirandomness and regular partitions. The purpose of the concept of quasirandomness is to measure how much a given graph “resembles ” a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasirandom graphs. ..."
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Cited by 12 (3 self)
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We deal with two intimately related subjects: quasirandomness and regular partitions. The purpose of the concept of quasirandomness is to measure how much a given graph “resembles ” a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasirandom graphs. Regarding quasirandomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we present a novel concept of regularity that takes into account the graph’s degree distribution, and show that if G = (V, E) satisfies a certain boundedness condition, then G admits a regular partition. In addition, building on the work of Alon and Naor [4], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph G in polynomial time.
Tuning topology generators using spectral distributions
 In Lecture Notes in Computer Science, Volume 5119, SPEC International Performance Evaluation Workshop
, 2008
"... Abstract. An increasing number of synthetic topology generators are available, each claiming to produce representative Internet topologies. Every generator has its own parameters, allowing the user to generate topologies with different characteristics. However, there exist no clear guidelines on tun ..."
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Cited by 9 (6 self)
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Abstract. An increasing number of synthetic topology generators are available, each claiming to produce representative Internet topologies. Every generator has its own parameters, allowing the user to generate topologies with different characteristics. However, there exist no clear guidelines on tuning the value of these parameters in order to obtain a topology with specific characteristics. In this paper we optimize the parameters of several topology generators to match a given Internet topology. The optimization is performed either with respect to the link density, or to the spectrum of the normalized Laplacian matrix. Contrary to approaches in the literature that rely only on the largest eigenvalues, we take into account the set of all eigenvalues. However, we show that on their own the eigenvalues cannot be used to construct a metric for optimizing parameters. Instead we present a weighted spectral method which simultaneously takes into account all the properties of the graph. Keywords: Internet Topology, Graph Spectrum. 1
Reconstruction of complete interval tournaments. II.
, 2010
"... Let a, b (b ≥ a) and n (n ≥ 2) be nonnegative integers and let T (a, b, n) be the set of such generalised tournaments, in which every pair of distinct players is connected at most with b, and at least with a arcs. In [40] we gave a necessary and sufficient condition to decide whether a given sequen ..."
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Cited by 7 (3 self)
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Let a, b (b ≥ a) and n (n ≥ 2) be nonnegative integers and let T (a, b, n) be the set of such generalised tournaments, in which every pair of distinct players is connected at most with b, and at least with a arcs. In [40] we gave a necessary and sufficient condition to decide whether a given sequence of nonnegative integers D = (d1, d2,..., dn) can be realized as the outdegree sequence of a T ∈ T (a, b, n). Extending the results of [40] we show that for any sequence of nonnegative integers D there exist f and g such that some element T ∈ T (g, f, n) has D as its outdegree sequence, and for any (a, b, n)tournament T ′ with the same outdegree sequence D hold a ≤ g and b ≥ f. We propose a Θ(n) algorithm to determine f and g and an O(dnn 2) algorithm to construct a corresponding tournament T.
Relating singular values and discrepancy of weighted directed graphs
"... Various parameters have been discovered which give a measurement of the “randomness” of a graph. We consider two such parameters for directed graphs: the singular values of the (normalized) adjacency matrix and discrepancy (a measurement of how randomly edges have been placed). We will show that th ..."
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Cited by 5 (3 self)
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Various parameters have been discovered which give a measurement of the “randomness” of a graph. We consider two such parameters for directed graphs: the singular values of the (normalized) adjacency matrix and discrepancy (a measurement of how randomly edges have been placed). We will show that these two are equivalent by bounding one by the other so that if one is small then both are small. We will also give a related result for discrepancy of walks when the indegree and outdegree at each vertex is equal. Both of these results follow from a more general discrepancy property of nonnegative matrices which we will state and prove.
The Poset of Hypergraph Quasirandomness
, 2012
"... Chung and Graham began the systematic study of hypergraph quasirandom properties soon after the foundational results of Thomason and ChungGrahamWilson on quasirandom graphs. One feature that became apparent in the early work on hypergraph quasirandomness is that properties that are equivalent for ..."
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Cited by 4 (2 self)
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Chung and Graham began the systematic study of hypergraph quasirandom properties soon after the foundational results of Thomason and ChungGrahamWilson on quasirandom graphs. One feature that became apparent in the early work on hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and a fundamental open problem that has emerged is to determine the relationship between these quasirandom properties. We completely determine the poset of implications between essentially all hypergraph quasirandom properties that have been studied in the literature. This answers a recent question of Chung, and in some sense completes the project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990’s. 1
Eigenvalues and Quasirandom Hypergraphs
, 2012
"... Let p(k) denote the partition function of k. For each k ≥ 2, we describe a list of p(k) − 1 quasirandom properties that a kuniform hypergraph can have. Our work connects previous notions on hypergraph quasirandomness, beginning with the early work of Chung and Graham and FranklRödl related to str ..."
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Cited by 4 (0 self)
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Let p(k) denote the partition function of k. For each k ≥ 2, we describe a list of p(k) − 1 quasirandom properties that a kuniform hypergraph can have. Our work connects previous notions on hypergraph quasirandomness, beginning with the early work of Chung and Graham and FranklRödl related to strong hypergraph regularity, the spectral approach of FriedmanWigderson, and more recent results of KohayakawaRödlSkokan and ConlonHànPersonSchacht on weak hypergraph regularity and its relation to counting linear hypergraphs. For each of the quasirandom properties that are described, we define a hypergraph eigenvalue analogous to the graph case and a hypergraph extension of a graph cycle of even length whose count determines if a hypergraph satisfies the property. This answers a question of Conlon et al. Our work can be viewed as an extension to hypergraphs of the seminal results of ChungGrahamWilson for graphs. Our results yield the following applications. First, motivated by Sidorenko’s Conjecture on the minimum homomorphism density of bipartite graphs in arbitrary graphs, we show that an analog of the conjecture for hypergraphs holds for a variety of hypergraph cycles. These are the first infinite families of hypergraphs with minimum degree two where this has been verified. Second, we give an efficient certification algorithm for hypergraph quasirandomnes which leads to an efficient strong refutation algorithm for random kSAT. For nvertex, kuniform hypergraphs with k ≥ 4 and at least n k/2+ √ k edges, we provide an algorithm with running time O(n kω polylog n) that certifies quasirandomness for almost all hypergraphs. This improves the previous best running time for such certification due to CojaOghlanCooperFrieze and HánPersonSchacht, in addition to also certifying a stronger quasirandom property than these previous results.