HYPERNETWORKS: CLUSTER SYNCHRONIZATION IS A HIGHER-ORDER EFFECT

Sören von der Gracht; Eddie Nijholt; Bob Rink

doi:10.1137/23M1561075

HYPERNETWORKS: CLUSTER SYNCHRONIZATION IS A HIGHER-ORDER EFFECT

Sören von der Gracht, Eddie Nijholt, Bob Rink

Mathematics

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Abstract

Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronization patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronization patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order, i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronization patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by demonstrating how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree.

Original language	English
Pages (from-to)	2329-2353
Number of pages	25
Journal	SIAM journal on applied mathematics
Volume	83
Issue number	6
Early online date	28 Nov 2023
DOIs	https://doi.org/10.1137/23M1561075
Publication status	Published - Dec 2023

Bibliographical note

Publisher Copyright:
© Society for Industrial and applied Mathematics.

Funding

\ast Received by the editors March 23, 2023; accepted for publication (in revised form) August 9, 2023; published electronically November 28, 2023. https://doi.org/10.1137/23M1561075 Funding: The work of the first author was partially supported by the Deutsche Forschungsge-meinschaft (DFG, German Research Foundation) grant 453112019. The work of the second author was partially supported by the Serrapilheira Institute grant Serra-1709-16124. \dagger Department of Mathematics, Paderborn University, Paderborn, Germany (soeren.von.der. [email protected]). \ddagger Department of Mathematics, Imperial College London, London, SW7 2RH, United Kingdom ([email protected]). \S Department of Mathematics, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands ([email protected]).

Funders	Funder number
Deutsche Forschungsgemeinschaft	453112019
Instituto Serrapilheira	Serra-1709-16124

Keywords

bifurcation theory
dynamical systems
hypernetworks
synchronization

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10.1137/23M1561075

23m1561075_HYPERNETWORKSFinal published version, 806 KBLicence: Article 25fa Dutch Copyright Act

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title = "HYPERNETWORKS: CLUSTER SYNCHRONIZATION IS A HIGHER-ORDER EFFECT",

abstract = "Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronization patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronization patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order, i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronization patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by demonstrating how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree.",

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AU - Rink, Bob

N1 - Publisher Copyright: © Society for Industrial and applied Mathematics.

PY - 2023/12

Y1 - 2023/12

N2 - Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronization patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronization patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order, i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronization patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by demonstrating how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree.

AB - Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronization patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronization patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order, i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronization patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by demonstrating how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree.

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HYPERNETWORKS: CLUSTER SYNCHRONIZATION IS A HIGHER-ORDER EFFECT

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