Hypernetworks: Cluster Synchronization Is a Higher-Order Effect

IF 1.9 4区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Applied Mathematics Pub Date : 2023-11-28 DOI:10.1137/23m1561075

Sören von der Gracht, Eddie Nijholt, Bob Rink

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引用次数: 3

Abstract

SIAM Journal on Applied Mathematics, Volume 83, Issue 6, Page 2329-2353, December 2023.
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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超级网络:集群同步是一种高阶效应

应用数学学报，第83卷，第6期，2329-2353页，2023年12月。摘要。许多网络系统是由节点之间的非成对交互控制的。由此产生的高阶交互结构可以通过超网络进行编码。本文通过为每一个超网络定义一类可容许映射来考虑超网络上的动态系统。我们解释了如何通过寻找平衡分区来对超网络上的健壮集群同步模式进行分类，并将图纤化的概念推广到超网络上下文中。我们还证明了鲁棒同步模式只能完全由多项式允许的高阶映射确定。这意味着，与二元网络不同，超网络上的集群同步是一种高阶的，即非线性的效应。根据超网络的阶，我们给出了确定鲁棒同步模式的多项式允许映射的程度的一个公式。我们还通过调查一类例子证明了这个度是最优的。最后，我们证明了这种效应如何导致显著的同步中断分岔，发生在高多项式度。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Applied Mathematics 数学-应用数学

CiteScore

3.60

自引率

0.00%

发文量

审稿时长

12 months

期刊介绍： SIAM Journal on Applied Mathematics (SIAP) is an interdisciplinary journal containing research articles that treat scientific problems using methods that are of mathematical interest. Appropriate subject areas include the physical, engineering, financial, and life sciences. Examples are problems in fluid mechanics, including reaction-diffusion problems, sedimentation, combustion, and transport theory; solid mechanics; elasticity; electromagnetic theory and optics; materials science; mathematical biology, including population dynamics, biomechanics, and physiology; linear and nonlinear wave propagation, including scattering theory and wave propagation in random media; inverse problems; nonlinear dynamics; and stochastic processes, including queueing theory. Mathematical techniques of interest include asymptotic methods, bifurcation theory, dynamical systems theory, complex network theory, computational methods, and probabilistic and statistical methods.