Limits of dense graph sequences
László Lovász,Balázs Szegedy +1 more
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It is shown that if a sequence of dense graphs G has the property that for every fixed graph F, the density of copies of F in G"n tends to a limit, then there is a natural ''limit object,'' namely a symmetric measurable function W:[0,1]^2->[0, 1].About:
This article is published in Journal of Combinatorial Theory, Series B.The article was published on 2006-11-01 and is currently open access. It has received 882 citations till now. The article focuses on the topics: Universal graph & Forbidden graph characterization.read more
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Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing
TL;DR: In Part I of this series, we showed that left convergence is equivalent to convergence in metric, both for simple graphs and for graphs with nodeweights and edgeweights as discussed by the authors.
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Community Detection and Stochastic Block Models
TL;DR: The recent developments that establish the fundamental limits for community detection in the stochastic block model are surveyed, both with respect to information-theoretic and computational thresholds, and for various recovery requirements such as exact, partial and weak recovery.
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Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints
TL;DR: It is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that the authors consider.
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Matrix estimation by Universal Singular Value Thresholding
TL;DR: This paper introduces a simple estimation procedure, called Universal Singular Value Thresholding (USVT), that works for any matrix that has "a little bit of structure" and achieves the minimax error rate up to a constant factor.
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Convergent Sequences of Dense Graphs II. Multiway Cuts and Statistical Physics
TL;DR: In this article, it was shown that right-convergence is equivalent to left-convexity, both for simple graphs and for graphs with nontrivial nodeweights and edgeweights.
References
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Recurrence of Distributional Limits of Finite Planar Graphs
Itai Benjamini,Oded Schramm +1 more
TL;DR: In this article, the authors introduce the notion of a distributional limit of a connected planar graph, and prove that with probability one of the vertices in such graphs is recurrent.
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Quasi-random graphs
TL;DR: A large equivalence class of graph properties is introduced, all of which are shared by so-called random graphs, and it is often relatively easy to verify that a particular family of graphs possesses some property in this class.
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Quick Approximation to Matrices and Applications
Alan Frieze,Ravi Kannan +1 more
TL;DR: The matrix approximation is generalized to multi-dimensional arrays and from that derive approximation algorithms for all dense Max-SNP problems and the Regularity Lemma is derived.
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Asymptotic Enumeration of Spanning Trees
TL;DR: In this paper, the authors give new formulas for the asymptotics of the number of spanning trees of a large graph and derive tree entropy, which is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs.