Synchronization in Networks with Higher Order Interactions

同步在生物、工程和物理系统中普遍存在。同步的例子包括电网中的交流电、大脑节律、昼夜节律和激光。传统上，同步是通过假设振荡系统成对地相互作用来研究的。然而，在许多情况下，这些系统以三人或更多人为一组进行交互。这个项目将研究群体相互作用系统中的同步。除了促进对同步过程的基本理解外，本项目开发的理论和技术还将促进同步模式的预测和控制，在许多科学和工程领域具有潜在的好处。该研究项目将为研究生提供一个建立复杂和跨学科动力系统模型的机会。研究成果将通过在期刊上发表和在会议上介绍的方式传播。将定期为高级本科生和研究生开设关于网络动力学的课程，为对网络动力学感兴趣的学生提供一种凝聚力。用网络来描述复杂系统是非常成功的。将复杂系统描述为网络假设代理之间的交互只是成对的。然而，在一些重要的情况下，交互作用同时包含多个单元。这样的系统可以被描述为超图，这是一个网络的泛化，其中可以在两个以上的单元之间发生交互。这个项目将研究编码振荡器耦合的超图的结构如何决定振荡器的同步。虽然众所周知，对于一些简单的超图，基团相互作用可以改变同步过程的动力学，但超图结构对同步的全部影响还不是很清楚。这项工作将通过开发一个全面而广泛的框架来研究超图中振子的同步来填补这一空白。一个理论框架，可以解释不同的超图结构，振荡器相互作用类型，以及它们之间的相关性。该方法基于适当的超图生成模型和降维技术的使用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Synchronization is pervasive in biological, engineered, and physical systems. Examples of synchronization include alternating current in power grids, brain rhythms, circadian rhythms, and lasers. Traditionally, synchronization has been studied by assuming that the oscillating systems interact in pairs. However, in many situations these systems interact in groups of three or more. This project will study synchronization in systems with group interactions. In addition to advancing basic understanding of synchronization processes, the theories and techniques developed in this project will facilitate prediction and control of synchronization patterns with potential benefits in many areas of science and engineering. The research project will provide an opportunity to train a graduate student on modeling complex and interdisciplinary dynamical systems. The research results will be disseminated by publication in journals and presentation in conferences. A class on dynamics on networks for advanced undergraduate and graduate students will be taught regularly, providing a coalescing force for the students interested in dynamics on networks. The description of complex systems in terms of networks has been extremely successful. Describing a complex system as a network assumes that interactions between agents are only pairwise. However, there are important cases where interactions encompass multiple units simultaneously. Such systems can be described as hypergraphs, a generalization of a network where interactions can occur between more than two units. This project will study how the structure of the hypergraph that encodes oscillator coupling determines the synchronization of the oscillators. While it is known that group interactions can modify the dynamics of synchronization processes for some simple hypergraphs, the full effect of hypergraph structure on synchronization is not well understood yet. This work will fill this gap by developing a comprehensive and broad framework to study synchronization of oscillators in hypergraphs. A theoretical framework that can account for different hypergraph structures, oscillator interaction types, and correlations between them will be developed. The approach is based on the use of appropriate hypergraph generative models and dimensional reduction techniques.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.