Multi-Population Phase Oscillator Networks with Higher-Order Interactions

Bick, Christian; Böhle, Tobias; Kuehn, Christian

doi:10.48550/arxiv.2012.04943

Cited by 9 publications

(16 citation statements)

References 33 publications

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“…The approach in this paper naturally extends to the setting on a compact Riemannian manifold, e.g., the torus T m or sphere S m , which are typically used in the Kuramoto models of lower or higher order interactions [2], the swarm sphere model [31], as well as models for opinion dynamics [10].…”

Section: Discussion and Outlooksmentioning

confidence: 99%

“…We also point out that the IPS we study allow for distinguishability of particles in terms of a directed generalized graph, whereas indistinguishability seems predominant in the literature where graphs are assumed to be symmetric. It is worth mentioning as well that the approach established in this paper readily adapts to dynamical systems with phase space being Riemannian manifolds such as sphere or torus, and hence apply potentially to e.g., Kuramoto models of high-order interactions [2], or the opinion dynamics model on the sphere [10]. Once we have successfully addressed the two main technical challenges, we carefully demonstrate how to apply our results to a wide variety of models ranging from epidemiology, ecology to social sciences.…”

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confidence: 99%

“…. Let u(t) = t 0 w(s)2 ds + b t 0 w(s)ds + c, for t ∈ T . Then u(t) ≥ c for all t ∈ T since w(t) ≥ 0.…”

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confidence: 99%

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Vlasov equations on digraph measures

Kuehn¹,

Xu²

2021

Preprint

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Many science phenomena are described as interacting particle systems (IPS). The mean field limit (MFL) of large all-to-all coupled deterministic IPS is given by the solution of a PDE, the Vlasov Equation (VE). Yet, many applications demand IPS coupled on networks/graphs. In this paper, we are interested in IPS on directed graphs, or digraphs for short. It is interesting to know, how the limit of a sequence of digraphs associated with the IPS influences the macroscopic MFL of the IPS. This paper studies VEs on a generalized digraph, regarded as limit of a sequence of digraphs, which we refer to as a digraph measure (DGM) to emphasize that we work with its limit via measures. We provide (i) unique existence of solutions of the VE on continuous DGMs, and (ii) discretization of the solution of the VE by empirical distributions supported on solutions of an IPS via ODEs coupled on a sequence of digraphs converging to the given DGM. The result substantially extends results on one-dimensional Kuramoto-type models and we allow the underlying digraphs to be not necessarily dense. The technical contribution of this paper is a generalization of Neunzert's in-cell-particle approach from a measure-theoretic viewpoint, which is different from the known techniques in L p -functions using graphons and their generalization via harmonic analysis of locally compact Abelian groups. Finally, we apply our results to various models in higher-dimensional Euclidean spaces in epidemiology, ecology, and social sciences.

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Section: Discussion and Outlooksmentioning

confidence: 99%

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confidence: 99%

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Vlasov equations on digraph measures

Kuehn¹,

Xu²

2021

Preprint

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“…Due to the fact that higher-order Kuramoto models describe interactions beyond usual graph structures, they have recently raised remarkable interest, see [4,35] and references given therein. However, as their incorporation requires laborious notation and would obstruct a comprehensible presentation of the key idea, we do not include detailed calculations here.…”

Section: The Associated Matrixmentioning

confidence: 99%

On the reliable and efficient numerical integration of the Kuramoto model and related dynamical systems on graphs

Böhle

Kuehn

Thalhammer

2021

International Journal of Computer Mathematics

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In this work, a novel approach for the reliable and efficient numerical integration of the Kuramoto model on graphs is studied. For this purpose, the notion of order parameters is revisited for the classical Kuramoto model describing all-to-all interactions of a set of oscillators. First numerical experiments confirm that the precomputation of certain sums significantly reduces the computational cost for the evaluation of the right-hand side and hence enables the simulation of high-dimensional systems. In order to design numerical integration methods that are favourable in the context of related dynamical systems on network graphs, the concept of localized order parameters is proposed. In addition, the detection of communities for a complex graph and the transformation of the underlying adjacency matrix to block structure is an essential component for further improvement. It is demonstrated that for a submatrix comprising relatively few coefficients equal to zero, the precomputation of sums is advantageous, whereas straightforward summation is appropriate in the complementary case. Concluding theoretical considerations and numerical comparisons show that the strategy of combining effective community detection algorithms with the localization of order parameters potentially reduces the computation time by several orders of magnitude.

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“…While a class of systems can be modeled by networks with pairwise interactions [7,8], where nodes of the network are connected by links, higher-order interactions are prevailing in various systems, including the network of neurons [9], the contagion network [10,11], and social networks [12]. An emerging direction in network science started to uncover the significance of higher-order interactions [13][14][15][16][17][18], which induce diverse phenomena beyond pairwise interactions.…”

Section: Introductionmentioning

confidence: 99%

Optimizing higher-order network topology for synchronization of coupled phase oscillators

Tang¹,

Shi²,

Lü³

2021

Preprint

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Networks in nature have complex interactions among agents. One significant phenomenon induced by interactions is synchronization of coupled agents, and the interactive network topology can be tuned to optimize synchronization. The previous studies showed that the optimized conventional network with pairwise interactions favors a homogeneous degree distribution of nodes when the interaction is undirected, and is always structurally asymmetric when the interaction is directed. However, the optimal control on synchronization for networks with prevailing higher-order interactions is less explored. Here, by considering the higher-order interactions in hypergraph and the Kuramoto model with 2-hyperlink interactions, we find that the network topology with optimized synchronizability may have distinct properties. For the undirected interaction, optimized networks with 2-hyperlink interactions by simulated annealing tend to become homogeneous in the nodes' generalized degree, consistent with 1-hyperlink (pairwise) interactions. We further define the directed hyperlink, and rigorously demonstrate that for the directed interaction, the structural symmetry can be preserved in the optimally synchronizable network with 2-hyperlink interactions, in contrast to the conclusion for 1-hyperlink interactions. The results suggest that controlling the network topology of higher-order interactions leads to synchronization phenomena beyond pairwise interactions.

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Multi-Population Phase Oscillator Networks with Higher-Order Interactions

Cited by 9 publications

References 33 publications

Vlasov equations on digraph measures

Vlasov equations on digraph measures

On the reliable and efficient numerical integration of the Kuramoto model and related dynamical systems on graphs

Optimizing higher-order network topology for synchronization of coupled phase oscillators

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