Large Networks and Graph Limits

Lovász, László

doi:10.1090/coll/060

Cited by 1,072 publications

(1,919 citation statements)

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“…Various notions of sparse graphs and intermediate classes have recently been studied extensively (see, e.g., [NO12] and [NO13]); for a presentation of graph limits for bounded-degree graphs, see [Lov12].…”

Section: Kaleidoscope Theoriesmentioning

confidence: 99%

Invariant measures via inverse limits of finite structures

Freer

Neetil

Patel

2016

European Journal of Combinatorics

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Abstract. Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are invariant under all permutations of the underlying set that fix all constants. These measures are constructed from inverse limits of measures on certain finite structures. We use this construction to obtain invariant probability measures concentrated on the classes of countable models of certain first-order theories, including measures that do not assign positive measure to the isomorphism class of any single model. We also characterize those transitive Borel G-spaces admitting a G-invariant probability measure, when G is an arbitrary countable product of symmetric groups on a countable set.

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Section: Kaleidoscope Theoriesmentioning

confidence: 99%

Invariant measures via inverse limits of finite structures

Freer

Neetil

Patel

2016

European Journal of Combinatorics

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“…Lovász introduced a kind of Hankel matrices for graph parameters [48] which were used to study real-valued graph parameters and their relation to partition functions, [30,49]. In [33], the definability of graph parameters in monadic second order logic was related to the rank of their Hankel matrices.…”

Section: Hankel Matrices and Weighted Word Automatamentioning

confidence: 99%

Language and Automata Theory and Applications

Labai

Makowsky

2016

Lecture Notes in Computer Science

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Hankel matrices (aka connection matrices) of word functions and graph parameters have wide applications in automata theory, graph theory, and machine learning. We give a characterization of real-valued functions on nested words recognized by weighted visibly pushdown automata in terms of Hankel matrices on nested words. This complements C. Mathissen's characterization in terms of weighted monadic second order logic.

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“…Consider a {0, 1}-valued function W Γ defined on the unit square [0, 1] 2 as follows The plot of the support of W Γ is called the pixel picture of Γ [24]. It provides a convenient visualization of the structure of the graph.…”

Section: Preliminariesmentioning

confidence: 99%

“…Below, we review some facts about graph limits that will be needed in the remainder of this paper. For more details, we refer the interested reader to [25,7,24].…”

Section: Graph Limitsmentioning

confidence: 99%

Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

Medvedev

Tang

2015

J Nonlinear Sci

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The Kuramoto model of coupled phase oscillators on complete, Paley, and Erdős-Rényi (ER) graphs is analyzed in this work. As quasirandom graphs, the complete, Paley, and ER graphs share many structural properties. For instance, they exhibit the same asymptotics of the edge distributions, homomorphism densities, graph spectra, and have constant graph limits. Nonetheless, we show that the asymptotic behavior of solutions in the Kuramoto model on these graphs can be qualitatively different. Specifically, we identify twisted states, steady state solutions of the Kuramoto model on complete and Paley graphs, which are stable for one family of graphs but not for the other. On the other hand, we show that the solutions of the initial value problems for the Kuramoto model on complete and random graphs remain close on finite time intervals, provided they start from close initial conditions and the graphs are sufficiently large. Therefore, the results of this paper elucidate the relation between the network structure and dynamics in coupled nonlinear dynamical systems. Furthermore, we present new results on synchronization and stability of twisted states for the Kuramoto model on Cayley and random graphs.

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Large Networks and Graph Limits

Cited by 1,072 publications

References 0 publications

Invariant measures via inverse limits of finite structures

Invariant measures via inverse limits of finite structures

Language and Automata Theory and Applications

Stability of Twisted States in the Kuramoto Model on Cayley and Random Graphs

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