Action convergence of operators and graphs

Backhausz, Ágnes; Szegedy, Balázs

doi:10.4153/s0008414x2000070x

Cited by 28 publications

(40 citation statements)

References 26 publications

(54 reference statements)

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“…Indeed, suppose that Φ ∈ A 4 satisfies (33) and (34). Then the measure Φ * * also satisfies these conditions, and the symmetrized measure 1 2 (Φ + Φ * * ) satisfies these equations and, in addition, (32) as well.…”

Section: 21mentioning

confidence: 88%

“…This holds for every U ∈ A, so it follows that for all A ∈ A, ϕ 1 st (A) − ϕ 2 st (A) = 1 J×A (s, t) − 1 A×J (s, t) = δ s (A) − δ t (A), (…”

Section: Proof Of the Multicommodity Flow Theorem I The "Only If " Directionmentioning

confidence: 97%

“…It turns out that double measure spaces play a role in other recent work in graph limit theory, as limit objects for graph sequences that are neither dense nor boundeddegree, but convergent in some well-defined sense: shape convergence [17] or action convergence [1]. We don't describe these limit theories here, but as an example for which a very reasonable limit can be defined in terms of double measure spaces we mention the sequence of hypercubes.…”

Section: Double Measure Spacesmentioning

confidence: 99%

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Flows on measurable spaces

Lovász

2021

Geom. Funct. Anal.

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The theory of graph limits is only understood to a somewhat satisfactory degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that one of the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with a stationary distribution). This motivates our goal to extend some important theorems from finite graphs to Markov spaces or, more generally, to measurable spaces. In this paper, we show that much of flow theory, one of the most important areas in graph theory, can be extended to measurable spaces. Surprisingly, even the Markov space structure is not fully needed to get these results: all we need a standard Borel space with a measure on its square (generalizing the finite node set and the counting measure on the edge set). Our results may be considered as extensions of flow theory for directed graphs to the measurable case.

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Section: 21mentioning

confidence: 88%

“…This holds for every U ∈ A, so it follows that for all A ∈ A, ϕ 1 st (A) − ϕ 2 st (A) = 1 J×A (s, t) − 1 A×J (s, t) = δ s (A) − δ t (A), (…”

Section: Proof Of the Multicommodity Flow Theorem I The "Only If " Directionmentioning

confidence: 97%

Section: Double Measure Spacesmentioning

confidence: 99%

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Flows on measurable spaces

Lovász

2021

Geom. Funct. Anal.

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“…(Again, Lemma 3.1 in [9] is formulated for functions 1] but the same proof clearly works for [0,1] as well. )…”

Section: Preliminariesmentioning

confidence: 91%

“…be nonnegative continuous functions on [0,1] such that their sum is the constant function 1 and such that…”

Section: Comparison Of K-shapes and Shapesmentioning

confidence: 99%

Graph limits: An alternative approach to s‐graphons

Doležal

2021

Journal of Graph Theory

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Multi-population phase oscillator networks with higher-order interactions

Bick

Böhle

Kuehn

2022

Nonlinear Differ. Equ. Appl.

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The classical Kuramoto model consists of finitely many pairwisely coupled oscillators on the circle. In many applications a simple pairwise coupling is not sufficient to describe real-world phenomena as higher-order (or group) interactions take place. Hence, we replace the classical coupling law with a very general coupling function involving higher-order terms. Furthermore, we allow for multiple populations of oscillators interacting with each other through a very general law. In our analysis, we focus on the characteristic system and the mean-field limit of this generalized class of Kuramoto models. While there are several works studying particular aspects of our program, we propose a general framework to work with all three aspects (higher-order, multi-population, and mean-field) simultaneously. In this article, we investigate dynamical properties within the framework of the characteristic system. We identify invariant subspaces of synchrony patterns and study their stability. It turns out that the so called all-synchronized state, which is one special synchrony pattern, is never asymptotically stable. However, under some conditions and with a suitable definition of stability, the all-synchronized state can be proven to be at least locally stable. In summary, our work provides a rigorous mathematical framework upon which the further study of multi-population higher-order coupled particle systems can be based.

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Action convergence of operators and graphs

Cited by 28 publications

References 26 publications

Flows on measurable spaces

Flows on measurable spaces

Graph limits: An alternative approach to s‐graphons

Multi-population phase oscillator networks with higher-order interactions

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